Operator Theory: Advances and Applications Vol. 181
Editor: I. Gohberg (GLWRULDO2I½FH School of Mathematical Science...

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Operator Theory: Advances and Applications Vol. 181

Editor: I. Gohberg (GLWRULDO2I½FH School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel Editorial Board: D. Alpay (Beer-Sheva) J. Arazy (Haifa) A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Böttcher (Chemnitz) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) R. E. Curto (Iowa City) K. R. Davidson (Waterloo, Ontario) R. G. Douglas (College Station) A. Dijksma (Groningen) H. Dym (Rehovot) P. A. Fuhrmann (Beer Sheva) B. Gramsch (Mainz) J. A. Helton (La Jolla) M. A. Kaashoek (Amsterdam)

H. G. Kaper (Argonne) S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) B. Mityagin (Columbus) V. Olshevsky (Storrs) M. Putinar (Santa Barbara) L. Rodman (Williamsburg) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) I. M. Spitkovsky (Williamsburg) S. Treil (Providence) H. Upmeier (Marburg) S. M. Verduyn Lunel (Leiden) D. Voiculescu (Berkeley) D. Xia (Nashville) D. Yafaev (Rennes) Honorary and Advisory Editorial Board: C. Foias (Bloomington) T. Kailath (Stanford) H. Langer (Vienna) P. D. Lax (New York) H. Widom (Santa Cruz)

Operator Algebras, Operator Theory and Applications Maria Amélia Bastos Israel Gohberg Amarino Brites Lebre Frank-Olme Speck Editors

Birkhäuser Basel · Boston · Berlin

Editors: Maria Amélia Bastos Amarino Brites Lebre Frank-Olme Speck Departamento de Matemática Instituto Superior Técnico, U.T.L. $YHQLGD5RYLVFR3DLV /LVERD3RUWXJDO HPDLODEDVWRV#PDWKLVWXWOSW DOHEUH#PDWKLVWXWOSW IVSHFN#PDWKLVWXWOSW

Israel Gohberg School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University 5DPDW$YLY,VUDHO HPDLOJRKEHUJ#PDWKWDXDFLO

0DWKHPDWLFDO6XEMHFW&ODVVL½FDWLRQ''' /LEUDU\RI&RQJUHVV&RQWURO1XPEHU Bibliographic information published by Die Deutsche Bibliothek. 'LH'HXWVFKH%LEOLRWKHNOLVWVWKLVSXEOLFDWLRQLQWKH'HXWVFKH1DWLRQDOELEOLRJUD½H detailed bibliographic data is available in the Internet at http://dnb.ddb.de

,6%1%LUNKlXVHU9HUODJ$*%DVHO- Boston - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of WKHPDWHULDOLVFRQFHUQHGVSHFL½FDOO\WKHULJKWVRIWUDQVODWLRQUHSULQWLQJUHXVHRI LOOXVWUDWLRQVUHFLWDWLRQEURDGFDVWLQJUHSURGXFWLRQRQPLFUR½OPVRULQRWKHUZD\VDQG storage in data banks. For any kind of use permission of the copyright owner must be obtained.

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Contents Editorial Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

List of Participants of WOAT 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Summer School: Lecture Notes S.C. Power Subalgebras of Graph C*-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

B. Silbermann C ∗ -algebras and Asymptotic Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . .

33

H. Upmeier Toeplitz Operator Algebras and Complex Analysis . . . . . . . . . . . . . . . . . . .

67

Workshop: Contributed Articles F.P. Boca Rotation Algebras and Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 121 G. Bogveradze and L.P. Castro On the Fredholm Index of Matrix Wiener-Hopf plus/minus Hankel Operators with Semi-almost Periodic Symbols . . . . . . . . . . . . . . . . 143 L.P. Castro and D. Kapanadze Diﬀraction by a Strip and by a Half-plane with Variable Face Impedances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159

A.C. Concei¸c˜ ao and V.G. Kravchenko Factorization Algorithm for Some Special Matrix Functions . . . . . . . . . . 173 E. Gots and L. Lyakhov On a Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 R. El Harti Extensions of σ-C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201

A.Y. Karlovich Higher-order Asymptotic Formulas for Toeplitz Matrices with Symbols in Generalized H¨older Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 207

vi

Contents

Yu.I. Karlovich Nonlocal Singular Integral Operators with Slowly Oscillating Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Y.I. Karlovich and L.V. Pessoa Poly-Bergman Projections and Orthogonal Decompositions of L2 -spaces Over Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 V. Kokilashvili and S. Samko Vekua’s Generalized Singular Integral on Carleson Curves in Weighted Variable Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 V. Manuilov On Homotopical Non-invertibility of C ∗ -extensions . . . . . . . . . . . . . . . . . . . 295 S. Mendes Galois-ﬁxed Points and K-theory for GL(n) . . . . . . . . . . . . . . . . . . . . . . . . . . 309 K.M. Mikkola and I.M. Spitkovsky Spectral Factorization, Unstable Canonical Factorization, and Open Factorization Problems in Control Theory . . . . . . . . . . . . . . . . . 321 K. Nourouzi Compact Linear Operators Between Probabilistic Normed Spaces . . . . 347 V.S. Rabinovich and S. Roch Essential Spectra of Pseudodiﬀerential Operators and Exponential Decay of Their Solutions. Applications to Schr¨ odinger Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 V.S. Rabinovich, S. Roch and B. Silbermann On Finite Sections of Band-dominated Operators . . . . . . . . . . . . . . . . . . . . 385 H. Rafeiro and S. Samko Characterization of the Range of One-dimensional Fractional Integration in the Space with Variable Exponent . . . . . . . . . . . . . . . . . . . . . 393 C.C. Ramos, N. Martins and P.R. Pinto Orbit Representations and Circle Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 N. Samko and B. Vakulov On Generalized Spherical Fractional Integration Operators in Weighted Generalized H¨older Spaces on the Unit Sphere . . . . . . . . . .

429

Editorial Introduction This volume is devoted to the International Summer School and Workshop on Operator Algebras, Operator Theory and Applications, WOAT 2006, held at Instituto Superior T´ecnico in Lisbon, Portugal on 1–5 September 2006. WOAT 2006 was a satellite conference of the International Congress of Mathematicians 2006 that was held in Madrid, Spain. Operator Algebras and Operator Theory are important areas of Mathematics that play an important role in diﬀerent mathematics areas and its applications, particularly in Mathematical Physics and Numerical Analysis. The main aim of WOAT 2006 was to bring together researchers in the Operator Algebras and Operator Theory areas. This volume contains three lecture notes of the Summer School courses and nineteen articles, contributions to the workshop of the WOAT 2006. The lecture notes, written by leading experts in the ﬁelds, are focused on: • Subalgebras of Graph C ∗ -Algebras (S. Power) A self contained introduction to two novel classes of non self-adjoint operator algebras, namely the generalized analytic Toeplitz algebras associated with the Fock spaces of a directed graph and subalgebras of graph C ∗ algebras, are given. The topics are independent but in both cases the focus is on techniques and problems related to classifying isomorphism types and to the recovery of underlying foundational structures, be they graphs or groupoids. • C ∗ -Algebras and Asymptotic Spectral Theory (B. Silbermann) An introduction to asymptotic spectral theory is presented using the elementary theory of C ∗ algebras. Given a bounded sequence of matrices with increasing size the spectra, ε-pseudospectra and the singular values of theses matrices are characterized. Three fundamental notions are discussed: stability, fractality and Fredholm sequences. The theory is applied to ﬁnite sections of quasidiagonal operators, Toeplitz operators, and operators with almost periodic diagonals. • Toeplitz Operator Algebras and Complex Analysis (H. Upmeier) Recent investigations are presented concerning Hilbert spaces of holomorphic functions on hermitian symmetric domains of arbitrary rank and dimension, in relation to operator theory (Toeplitz C ∗ -algebras and their representations), harmonic analysis (discrete series of semi-simple Lie groups) and quantization (covariant functional calculi and Berezin transformation).

viii

Editorial Introduction

The articles were based contributions to the workshop, the majority of them being centered on the main topics of the workshop: • Crossed product C ∗ -algebras. C ∗ -algebras of operators on Hardy and Bergman spaces. Invertibility theory for non-local C ∗ -algebras. Von Neumann algebras. • Approximate methods in operator algebras. Asymptotic properties of approximation operators. • Toeplitz, Hankel, and convolution type operators and algebras. Symbol calculi. Invertibility and index theory. • Operator theoretical methods in diﬀraction theory. Factorization theory and integrable systems. Applications to Mathematical Physics. The organizers gratefully acknowledge the support of the WOAT 2006 sponsors: the Portuguese Foundation for Science and Technology, Center for Mathematics and its Applications, Center for Mathematical Analysis, Geometry and Dynamical Systems, Research Project FCT/FEDER/POCTI/MAT/59972/2004, as well as Caixa Geral de Dep´ositos, Cˆamara Municipal de Lisboa, Funda¸c˜ao Calouste Gulbenkian, Embassy of Germany in Portugal, Funda¸c˜ao Luso-Americana, and Reitoria da Universidade T´ecnica de Lisboa. Lisbon, September 2007 The Editorial Board

WOAT 2006 – Program Friday, 1 September 2006 Workshop – Room ACI Registration Opening Session Session W1 Nikolai Nikolski Coﬀee break David Evans Lewis Coburn Lunch

09:00–09:30 09:30–09:50 09:55–10:45 10:50–11:10 11:10–12:00 12:05–12:55 13:00–14:00 14:00–15:30 15:30–15:45 15:45–17:15 17:15–17:30 17:30–19:00

Summer School – Room ACI Stephen Power Break Course B/Lecture 1 Konrad Schm¨ udgen Coﬀee break Course C/Lecture 1 Bernd Silbermann Course A/Lecture 1

Saturday, 2 September 2006

09:00–09:50 09:55–10:45 10:50–11:05

11:05–11:30 11:35–12:00 12:05–12:30 12:35–13:00 13:00–14:00 14:00–15:30 15:30–15:45 15:45–17:15 17:15–17:30 17:30–19:00

Course Course Course Course

Session W3 Room QA1.1 D. Mushtari I. Todorov R. Popescu V. Strauss

Workshop Session W2 – Room QA02.3 Ilya Spitkovsky Vladimir Manuilov Coﬀee break Session W4 Session W5 Room QA1.2 Room QA1.3 M. Ptak E. Gots H. Kaptanoglu P. Lopes M.C. Cˆ amara A. Montes-Rodr´ıguez A. Karlovich H. Rafeiro Lunch

Summer School – Room ACI Harald Upmeier Break Course A/Lecture 2 Stephen Power Coﬀee break Course B/Lecture 2 Konrad Schm¨ udgen Course D/Lecture 1

A: Subalgebras of Graph C ∗ -algebras B: C ∗ -algebras – Selected topics C: C ∗ -algebras and Asymptotic Spectral Theory D: Toeplitz Operator Algebras and Multivariable Complex Analysis

WOAT 2006 – Program Monday, 4 September 2006

09:00–09:50 09:55–10:20 10:25–10:50 10:50–11:05

11:05–11:30 11:35–12:00 12:05–12:30 12:35–13:00 13:00–14:00 14:00–15:30 15:30–15:45 15:45–17:15 17:15–17:30 17:30–19:00

Workshop Session W6 – Room QA1.1 Session W7 – Room QA1.2 Florin Boca Yuri Karlovich Session W8 – Room QA1.1 Session W9 – Room QA1.2 A. Katavolos R. Duduchava L. Marcoux L. Castro Coﬀee break Session W10 Session W11 Session W12 Room QA1.1 Room QA1.2 Room QA1.3 M. Dritschel J. Rodriguez A. Concei¸ca ˜o H. Tandra R. Marreiros C. Diogo E. Lopushanskaya K. Nourouzi T. Malheiro H. Mascarenhas M.C. Martins Lunch Summer School – Room ACI Bernd Silbermann Break Course D/Lecture 2 Harald Upmeier Coﬀee break Course A/Lecture 3 Stephen Power Course C/Lecture 2

Tuesday, 5 September 2006

09:00–09:50 09:55–10:45 10:50–11:05 11:05–11:30 11:35–12:00 12:05–12:30 12:35–13:00 13:00–14:00 14:00–15:30 15:30–15:45 15:45–17:15 17:15–17:30 17:30–19:00

Workshop Session W13 – Room QA1.1 Session W14 – Room QA1.2 Mikhail Agranovich Steﬀen Roch Nikolai Rabinovich Stefan Samko Coﬀee break Session W15 – Room QA1.1 Session W16 – Room QA1.2 C. Fernandes N. Samko R. El Harti L. Pessoa C. Ramos G. Bogveradze S. Mendes A. Nolasco Lunch Summer School – Room ACI Konrad Schm¨ udgen Break Course C/Lecture 3 Bernd Silbermann Coﬀee break Course D/Lecture 3 Harald Upmeier Course B/Lecture 3

List of Participants of WOAT 2006 Agranovich, Mikhail Moscow Institute of Electronics and Mathematics, Russia Al-Rashed, Maryam Imperial College London, United Kingdom Bastos, M. Am´elia Universidade T´ecnica de Lisboa/IST, Portugal Becher, Florian University of Freiburg, Germany Boca, Florin University of Illinois-Urbana-Champaign, USA Bogveradze, Giorgi Universidade de Aveiro, Portugal Bravo, Ant´ onio Universidade T´ecnica de Lisboa/IST, Portugal Cˆ amara, M. Cristina Universidade T´ecnica de Lisboa/IST, Portugal Campos, Hugo Universidade do Algarve/FCT, Portugal Campos, Lina Universidade do Algarve/FCT, Portugal Carvalho, Catarina Universidade T´ecnica de Lisboa/IST, Portugal Castro, Lu´ıs Universidade de Aveiro, Portugal Coburn, Lewis The State University of New York at Buﬀalo, USA Concei¸ca ˜o, Ana Universidade do Algarve/FCT, Portugal Diogo, Cristina Instituto Superior de Ciˆencias do Trabalho e da Empresa, Portugal Dritschel, Michael University of Newcastle, United Kingdom Duduchava, Roland A. Razmadze Mathematical Institute, Georgia

xii

List of Participants of WOAT 2006

Efendiev, Messoud GSF/TUM-Munich, Germany El Harti, Rachid University Hassan I, Morocco Erkursun, Nazife Middle East Technical University, Turkey Esteves, Ana Universidade de Aveiro, Portugal Evans, David Cardiﬀ University, United Kingdom Fernandes, Cl´ audio Universidade Nova de Lisboa/FCT, Portugal Ferreira dos Santos, Ant´ onio Universidade T´ecnica de Lisboa/IST, Portugal Gots, Ekaterina Voronezh State Technological Academy, Russia Habgood, Joe Queens University Belfast, United Kingdom Kaptanoglu, H. Turgay Bilkent University, Turkey Karlovich, Yuri Universidad Aut´ onoma del Estado de Morelos, Mexico Karlovych, Oleksiy Universidade do Minho, Portugal Katavolos, Aristides University of Athens, Greece Kravchenko, Viktor Universidade do Algarve/FCT, Portugal Lazar, Aldo Tel Aviv University, Israel Lebre, Amarino Universidade T´ecnica de Lisboa/IST, Portugal Lopes, Paulo Universidade T´ecnica de Lisboa/IST, Portugal Lopushanskaya, Ekaterina Voronezh State University, Russia Malheiro, Teresa Universidade do Minho, Portugal Manuilov, Vladimir Moscow State University, Russia Marcoux, Laurent W. University of Waterloo, Canada Marreiros, Rui Universidade do Algarve/FCT, Portugal

List of Participants of WOAT 2006 Martins, Maria do Carmo Universidade dos A¸cores, Portugal Mascarenhas, Helena Universidade T´ecnica de Lisboa/IST, Portugal Mendes, S´ergio University of Manchester, United Kingdom Montes-Rodr´ıguez, Alfonso Universidad de Sevilla, Spain Moura Santos, Ana Universidade T´ecnica de Lisboa/IST, Portugal Mushtari, Daniar Kazan State University, Russia Nikolski, Nikolai Universit´e Bordeaux, France and Steklov Institute of Mathematics, Russia Nolasco, Ana Universidade de Aveiro, Portugal Nourouzi, Kourosh K.N.Toosi University of Technology, Iran Oliveira, Isabel Universidade T´ecnica de Lisboa/IST, Portugal Oliveira, Lina Universidade T´ecnica de Lisboa/IST, Portugal Pereira, Paulo Universidade de Aveiro, Portugal Pessoa, Lu´ıs Universidade T´ecnica de Lisboa/IST, Portugal Pinto, Paulo Universidade T´ecnica de Lisboa/IST, Portugal Popescu, Radu Universidade T´ecnica de Lisboa/IST, Portugal Power, Stephen Lancaster University, United Kingdom Ptak, Marek University of Agriculture of Krakow, Poland Quint˜ ao Braga, Maria Jo˜ ao Universidade Cat´ olica Portuguesa, Portugal Rabinovich, Vladimir Instituto Politecnico Nacional, ESIME, Mexico Rafeiro, Humberto Universidade do Algarve/FCT, Portugal Ramos, Carlos ´ Universidade de Evora, Portugal Roch, Steﬀen Technical University Darmstadt, Germany

xiii

xiv

List of Participants of WOAT 2006

Rodrigues, C´ atia Universidade de Aveiro, Portugal Rodr´ıguez, Juan Universidade do Algarve/FCT, Portugal Rodr´ıguez Mart´ınez, Alejandro Universidad de Sevilla, Spain Samko, Natasha Universidade do Algarve/FCT, Portugal Samko, Stefan Universidade do Algarve/FCT, Portugal Sangha, Amandip University of Oslo, Norway Santos, Pedro Universidade T´ecnica de Lisboa/IST, Portugal Schm¨ udgen, Konrad University of Leipzig, Germany Silbermann, Bernd Technical University of Chemnitz, Germany Skill, Thomas Philipps-University Marburg, Germany Speck, Frank-Olme Universidade T´ecnica de Lisboa/IST, Portugal Spitkovsky, Ilya College of William & Mary, USA Strauss, Vladimir Simon Bolivar University, Venezuela Tandra, Haryono Bandung Institute of Technology, Indonesia Teixeira, Francisco Universidade T´ecnica de Lisboa/IST, Portugal Todorov, Ivan Queen’s University Belfast, United Kingdom Upmeier, Harald Marburg University, Germany

Operator Theory: Advances and Applications, Vol. 181, 3–32 c 2008 Birkh¨ auser Verlag Basel/Switzerland

Subalgebras of Graph C*-algebras Stephen C. Power Abstract. I give a self-contained introduction to two novel classes of nonselfadjoint operator algebras, namely the generalised analytic Toeplitz algebras LG , associated with the “Fock space” of a graph G, and subalgebras of graph C*algebras. These two topics are somewhat independent but in both cases I shall focus on fundamental techniques and problems related to classifying isomorphism types and to the recovery of underlying foundational structures, be they graphs or groupoids. Mathematics Subject Classiﬁcation (2000). Primary 47L40. Keywords. Operator algebras, directed graphs.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2 The Cuntz algebras, intuitively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3 Toeplitz algebras for countable directed graphs . . . . . . . . . . . . . . . . . . . . . . . . .

17

4 Subalgebras of On . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

5 Appendix: Digraph algebras and limit algebras . . . . . . . . . . . . . . . . . . . . . . . . .

27

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

1. Introduction I give a self-contained introduction to two novel classes of nonselfadjoint operator algebras, namely the generalised analytic Toeplitz algebras LG , associated with the “Fock space” of a graph G, and subalgebras of graph C*-algebras. These two topics are somewhat independent but in both cases I shall focus on fundamental techniques and problems related to classifying isomorphism types and to the recovery of underlying foundational structures, be they graphs or groupoids.

4

S.C. Power

1.1. Generalities on operator algebras Let us set the scene with a bird’s eye view of how operator algebras “come about” and comment on their morphisms. I shall take the term operator algebra to mean a complex algebra of bounded linear operators on a separable complex Hilbert space. For example, the operator algebra A could be the set of all complex single variable polynomials in a given operator T ; that is A = {p(T ) = a0 I + a1 T + · · · + an T n : p(z) = a0 1 + a1 z + · · · + an z n }. Often, the operator algebras of interest are manufactured by specifying a set of generators (such as the set {I, T } in the example) on a Hilbert space both of which (set and space) arise from a “foundational” mathematical structure, such as a group, or a graph, or a dynamical system. We might call this a spatial setting since the Hilbert space is in place at the outset and the operator algebra is taken to be the algebra generated by the given generators. The term “generated” may mean simply the unclosed complex algebra or it may refer to the closure of this algebra in some topology, usually either the operator norm topology or the weak operator topology. We do not assume that the generated algebra is self-adjoint, that is, closed under the conjugate transpose operation X → X ∗ , although of course that will follow if the set of generators is a self-adjoint set. Operator algebras are also constructed in a Hilbert space-free way, for example, as a particular operator algebra, within some huge general class of operator algebras, satisfying a universal property for (perhaps) a set of generators and relations. Alternatively the algebra A might be constructed in the category of normed algebras with the expectation that A is isometrically isomorphic to an operator algebra by virtue of the fact that the ingredients for A are operator algebras. For example A might be a Banach algebra direct limit of operator algebras, or, again, simply a quotient of operator algebras. We shall focus on spatial viewpoints. However, let us recall that the celebrated Gelfand–Naimark theorem can bring us back to a spatial context from a space free one. In truth, there are usually more ready-to-hand ways of providing a Hilbert spaces on which an indirectly constructed operator algebra can sit. Theorem 1.1. Let A be a C*-algebra (an involutive complete normed algebra with ab ≤ ab and a∗ a = a2 for all a, b ∈ A). Then there is a Hilbert space H (a separable one is possible if A is separable) and an isometric star homomorphism A → B(H). A fundamental question for a class of operator algebras is: When are two operator algebras A1 and A2 isomorphic? The strongest sense of isomorphism is undoubtedly unitary equivalence, that is, A1 = U A2 U ∗ for some isometric onto map U from the Hilbert space of A2 to that of A1 . Here the operator algebras really are the same if only we would identify the Hilbert spaces. A weaker form of isomorphism which also takes account of how the operator algebra sits on the underlying Hilbert space is star-extendible isomorphism. This requires that there is a map φ : A1 → A2 which is the restriction

Subalgebras of Graph C*-algebras

5

of an adjoint respecting algebra isomorphism φ : C ∗ (A1 ) → C ∗ (A2 ) between the generated C*-algebras. Weaker still, and now ignoring how the operator algebras are manifested, an isometric algebra isomorphism is simply an algebra isomorphism which is an isometric linear map. For nonselfadjoint operator algebras this form of isomorphism is usually the essential case to elucidate. In truth, while these forms of isomorphisms certainly are diﬀerent, in the case of operator algebras constructed from the same spatial scheme, as alluded to above, the resulting forms of isomorphism type are usually the same. By this I mean that if A1 and A2 are isomorphic in one of the three sense above then they are isomorphic in the other senses. Is there a metatheorem here I wonder? Let us also note a companion question to that above, which is generally deeper, concerning the symmetries of an operator algebra A. What is the isometric automorphism group of an operator algebra? Naturally one expects that when two instances of a foundational structure are isomorphic then this entails an isometric isomorphism between the associated operator algebras and indeed this is generally a routine veriﬁcation. (We might more formally realize the association as a functor.) But how about the converse direction? If A(S1 ) and A(S2 ) are the (norm closed say) operator algebras obtained from the foundation structures S1 and S2 , and if Φ : A1 → A2 is an isometric algebra isomorphism then does this somehow induce an isomorphism between S1 and S2 ? This would provide a satisfyingly deﬁnitive answer to the isomorphism question and it is in this connection that there is, as we say, clear blue water between the non-self-adjoint and the self-adjoint theory. Algebras of the former category seem to remember their foundations while the self-adjoint algebras need not.1 As an indication of the general landscape ahead here is a list of the ingredients of several operator algebra contexts concerning a hierarchy of Toeplitz algebras. A: The classical context : The Hardy–Hilbert space H 2 for the unit circle, the unilateral shift operator S, with dim(I − SS ∗ ) = 1, the disc algebra A(D), the function algebra H ∞ (D), realizations of A(D) and H ∞ (D) as “analytic” Toeplitz algebras, and the Toeplitz C*-algebra TZ+ . Let us outline some important classical facts. The Hilbert space 2 (Z+ ) is unitarily equivalent to the Hardy space H 2 of the Hilbert space L2 (T) of square integrable functions on the circle. Here H 2 denotes the closure in L2 (T) of the space of polynomials in z. The basis matching unitary U : 2 (Z+ ) → H 2 which does this (with U en := z n for n ≥ 0) eﬀects a unitary equivalence between the unilateral shift S (with Sen = en+1 , n ≥ 0) and the multiplication operator Tz : f → zf, f ∈ H 2 . That is U SU ∗ = Tz . C*-algebras it is K-theory and associated invariants that often lead to classifying invariants. In general such invariants are insuﬃcient as they are generally determined by the diagonal part A ∩ A∗ of the operator algebra A.

1 For

6

S.C. Power

More generally, the Toeplitz operator Tφ with symbol function φ in C(T) or L∞ (T) is given by Tφ : f → PH 2 φf . When φ is analytic, that is, is the boundary function of an analytic function on the disc, then Tφ is simply the restriction of the multiplication operator Mφ to H 2 . In this way we have a representation of the so called disc algebra A(D) as an algebra of operators on H 2 . A good exercise to do at least once is to show that the Toeplitz algebra TZ+ = C ∗ (I, Tz ) (which is equal to U C ∗ (I, S)U ∗ ) contains every compact operator K on H 2 . The idea is to start with the rank one operator I − Tz Tz∗ and “move it around” with the shifts Tz , Tz∗ to obtain every rank one operator of the form f → f, z k z l . For example, show that Tzn (I − Tz Tz∗ ) is the matrix unit operator En,0 : f → f, z 0 z n . Then from these operators and their adjoints create all the matrix units Ei,j . Finally, take linear combinations to approximate any ﬁnite rank operator. Once this is done one a little more work leads to the following theorem. Theorem 1.2. (i) For each Toeplitz operator Tφ and compact operator K we have Tφ + K ≥ Tφ . (ii) The Toeplitz algebra TZ+ is equal to the set of operators Tφ +K with φ in C(T) and K compact and the quotient TZ+ /K is naturally isomorphic to C(T). On the other hand the norm closed operator algebra generated by Tz is abelian and isometrically isomorphic to the disc algebra. Indeed, it is the algebra {Tφ : φ ∈ A(D)}. The weak operator topology closed algebra is similarly a copy of H ∞ (D), namely, {Tφ : φ ∈ H ∞ (D)}. On occasion we simply write H ∞ for this operator algebra when the context is clear. Recall that the weak operator topology is the weakest topology for which the spatial linear functionals T → T f, g are continuous. There are a great many ways in which one can move on from the Toeplitz context above and below I discuss some aspects of the following operator algebra directions. B: The (spatial) free semigroup context (Section 3.): The Fock space 2 (F+ n) for the free semigroup on n generators, the freely noncommuting shifts L1 , . . . , Ln with dim(I − (L1 L∗1 + · · · + Ln L∗n )) = 1, the noncommutative disc algebra An and free semigroup algebra Ln , and the Cuntz–Toeplitz C*-algebra on 2 (F+ n ). C: The (spatial) graph context (Section 3.): The Fock space of a directed graph G = (V, E), the freely noncommuting partial isometries Le , e ∈ E, the tensor algebra AG , the free semigroupoid algebras LG , and the Cuntz–Krieger– Toeplitz C*-algebras TG = C ∗ (AG ). D: The (universal) free semigroup context (Sections 2, 4): The freely noncommuting isometries S1 , . . . , Sn with S1 S1∗ + · · · + Sn Sn∗ = I, the Cuntz algebras On = C ∗ (S1 , . . . , Sn ).

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E: The (universal) graph context (Section 4): The (universal) graph C*algebra C*(G) of a countable directed graph G = (V, E) with partial isometry generators Se , for e ∈ E, and relations Se Se∗ = Px , Se∗ Se = Ps(e) , r(e)=x

where {Px : x ∈ V } is a family of orthogonal projections and e = (r(e), s(e)).

2. The Cuntz algebras, intuitively The Cuntz algebra On is a certain C*-algebra generated by n isometries, S1 , . . . , Sn say, satisfying S1 S1∗ + · · · + Sn Sn∗ = 1. That is, their range projections Si Si∗ are orthogonal and sum to the identity operator. In fact I could have dropped the word “certain” because of the following remarkable uniqueness property. Theorem 2.1. If n ≥ 2 and s1 , . . . , sn is any family of n isometries in a unital C*algebra with s1 s∗1 + · · · + sn s∗n = I, then C ∗ ({s1 , . . . , sn }) is naturally isometrically isomorphic to On . Our main aim is to obtain tools and results that will help in understanding norm closed subalgebras of the Cuntz algebras. In this connection we are prepared to consider operator algebras generated by semigroups of words in the generators and to contemplate quite general subalgebras. Perhaps it is fair to say that a C*-algebraist is largely happy with the state of knowledge of the Cuntz algebras. He/she is more interested in looking for generalisations and wider classes to classify and understand (such as Graph C*-algebras, or C*-algebras allied to dynamical systems). Our view here is quite diﬀerent – we are intending to linger with On and look inside it with a view to understanding classes of nonselfadjoint operator algebras. Our orientation and motivation comes partly from the existing theory of limit algebras which are found as nonselfadjoint subalgebras of approximately ﬁnite C*-algebras. We shall focus on the Cuntz algebras for clarity but the methods we discuss do extend to more general graph C*-algebras. 2.1. Cuntz algebra basics One direct way to deﬁne On is to look into the next section, take the freely noncommuting shifts L1 , . . . , Ln on the Fock space for the free semigroup on n generators, take the generated C*-algebra and divide out by the ideal of compact operators. (This should sound familiar if n = 1!) In taking the quotient we lose the Hilbert space and gain equality in place of the one-dimensional defect dim(I − L1 L∗1 + · · · + Ln L∗n ) = 1. The uniqueness allows us to consider two other models for On which will in fact be our viewpoint. I call these models the interval picture and the Cantorised interval picture. The ﬁrst uses isometric operators S1 , . . . , Sn on L2 [0, 1] whose i ranges are the orthogonal subspaces L2 [ i−1 n , n ]. For deﬁniteness we may deﬁne √ i−1+x i Si ( n ) = nf (x), for x ∈ [0, 1], and Si f )(t) = 0 for t not in [ i−1 n , n ].

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Notice that a product Si1 Si2 . . . Sik has range L2 [Ek ] where Ek is an interval with |Ek | = n−k . Write Sµ for this product where µ is the word i1 i2 . . . ik . These k-fold products have distinct ranges and so if the lengths of µ and ν are |µ| = |ν| = k then Sµ∗ Sν is the zero operator when µ and ν diﬀer. (Exercise: Prove this algebraically.) It follows in general that if µ and ν have diﬀering lengths and Sµ∗ Sν is nonzero then either Sµ∗ Sν = Sµ∗ where µ = νµ , or Sµ∗ Sν = Sν where ν = µν . On the other hand products with stars on the right are always non-zero. Indeed, from the interval picture we see that for |µ| = |ν| = k the operator Sµ Sν∗ acts as an isometry from L2 [Eν ] to L2 [Eµ ] for certain intervals Eµ , Eν with lengths |n|−k . Moreover, the set of operators {Sµ Sν∗ : |µ| = |ν| = k} satisﬁes the relations of a matrix unit system. The span of this set, Fkn say, is thus a copy of the matrix algebra Mnk , and we have the matrix algebra tower F1n ⊆ F2n ⊆ F3n · · · . n We see then that the generators of On provide a distinguished subalgebra F∞ = ∞ n ∞ n ∪k=1 Fk which we refer to as a matricial star algebra of type n . Write F for the closure of this subalgebra in On . Notice that the diagonal matrix units are those of the form Sµ Sµ∗ . The closed linear span of these is an abelain algebra, C say, in F n and On which plays a distinguished role.

0

1

1

0 1/4

1/2

Figure 1. Interval picture for the operator S1 S2 in O2 . Theorem 2.1 gives rise immediately to an important family of automorphisms of On , the so called gauge automorphisms γz , for z ∈ T, which satisfy γz (Si ) = zSi , 1 ≤ i ≤ n. (Alternatively, these automorphisms are inherited from easily deﬁned unitarily implemented automorphisms of the Cuntz-Toeplitz C*-algebra.)

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Proposition 2.2. (i) Each operator a in the star algebra generated by S1 , . . . , Sn has a representation N N (S1∗ )i a−i + a0 + ai S1i a= i=1

i=1

n F∞

where ai ∈ for each i. This representation is unique if for each i ≥ 0 we have ai = ai Pi and a−i = Pi a−i where Pi is the ﬁnal projection of S1i . (ii) The linear maps Ei , deﬁned by Ei (a) = ai , extend to continuous, contractive, linear maps from On to F n . (iii) The generalised Cesaro sums Σk (a) =

N

1−

k=1

N |k| ∗ k |k| (S1 ) E−k (a) + Ek (a)sk1 1− N N k=0

converge to a as N → ∞. Proof. Our observations above show that the linear span of the operators Sµ Sν∗ is the algebra generated by the generators. The representation in (i) (with the dilation actions carried by S1 alone) follows from this by means of formulae such as Sµ = (Sµ (S1∗ )k )S1k = aS1k where k = |µ|. The key to uniqueness is to make use of “recovery formulae” such as 2π dθ a0 = γz (a) 2π 0 where the integral is a Riemann integral of a norm continuous function. The representation in (i) can be viewed as a generalised Fourier series representation for the operator a. In fact to any operator a in On one may assign generalised Fourier coeﬃcients ak in F n by means of the maps Ek (.). The operators ak S1k (k > 0) and (S1∗ )k a−k (k < 0) appear as the conventional Fourier series coeﬃcients for the norm continuous operator-valued function fa : z → γz (a). The Cesaro polynomials pn (z) for the continuous function fa converge uniformly in operator norm on |z| = 1 by classical theory. Finally, the specialisation z = 1 gives the desired norm convergence of the generalised Ces´aro polynomials. Exercises. (i) Show that E0 is faithful, that is, if a ≥ 0, a = 0 then E0 (a) = 0. (ii) Show that ∗k γz (aS1 )dz S1k , k > 0. ak = T

The Cantorised interval picture for On is a reﬁnement of the interval presentation. The beauty of this perspective is that it provides a context for deﬁning binary relation (and groupoid) invariants for subalgebras. The essence of the idea is captured in Figure 2, shown with 2-fold branching relevant to the n = 2 case.

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0

1 01

00

11

10 101

100

0

x

1

X

Figure 2. Cantorised interval picture. Let X be the set of inﬁnite paths on the tree, starting at vertex 0 or vertex ∞ 1. This set is identiﬁable with the direct product k=1 {0, 1}, consisting of points x which are zero-one sequence x1 x2 . . . . Each vertex word w in the tree, such as 1001, gives rise to an “interval” Ew of points x whose product expansion starts with w. With the usual product topology the direct product is a Cantor space whose topology has the set of intervals as a base of closed-open sets. For each pair of vertex words w1 , w2 there is a partial homeomorphism αw2 ,w1 from Ew1 to Ew2 deﬁned by matching tails: if x = w1 xp xp+1 . . .

then

αw2 ,w1 (x) = w2 xp xp+1 . . .

Notice that for |w1 | = |w2 | = k the partial homeomorphism has an action on ∗ the set X that bears close analogy with Sw2 Sw and its interval picture, where we 1 have relabeled the generators as S0 and S1 . In fact we can add the natural product probability measure to X and present On on L2 (X) in terms of (newly labeled) generators S0 and S1 induced by the partial homeomorphisms α∅,0 : x → 0x, α∅,1 : x → 1x. That is, for i = 1, 2 we have (Si f )(α∅,0 (x)) =

√ 2f (x),

and (Si f )(y) = 0 if y is not in the range of α∅,i . Exercise. Obtain the partial homeomorphism that is associated with the partial ∗ . (Here we have the 0-1 subscript labeling as opposed to isometry S1 S0∗ + S00 S11 the 1-2 labeling.) 2.2. Normalising partial isometries We now come to an important class of partial isometries associated with the abelian diagonal algebra C.

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Deﬁnition 2.3. A partial isometry in F n (or, more generally, in On ) is C-normalising if vCv ∗ ⊆ C and v ∗ Cv ⊆ C. The obvious examples are the matrix units Sµ Sν∗ for |µ| = |ν| = k, and the sums v of these when the initial projections are orthogonal and the ﬁnal projections are orthogonal. Also we may multiply such a v by a unitary diagonal element d. The resulting C-normalising partial isometry has support which can be indicated pictorially, as shown. The coordinates have been arranged so that the support picture represents v intuitively as a continuous matrix (although the d information is now lost). The picture should be “Cantorised” and viewed as a subset of X × X. 0

1

0

1 ∗ Figure 3. The support the partial isometry S2 S1∗ + S11 S22 .

The next theorem is an extremely useful characterisation. It shows in particular that for subalgebras of F n an isometric isomorphism α : A1 → A2 with α(C) = C preserves the set of normalising partial isometries in the algebras. Theorem 2.4. Let v be an element of F n . Then the following assertions are equivalent: (i) v is a C-normalizing partial isometry. (ii) v is an orthogonal sum of a ﬁnite number of partial isometries of the form dSµ Sν∗ , where |µ| = |ν| and d ∈ C. (iii) For all projections p, q ∈ C, the norm qvp is equal to 0 or 1. First we note some general “recovery facts” about F n which show how operators may be approximated in an explicit manner. For b ∈ F n the “diagonal part” ∆(b) ∈ C may be deﬁned as the limit of the block diagonal operators bk := Σi ekii bekii

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where ekii are the diagonal matrix units of Fkn . The map ∆ : F n → C is a projection and moreover is faithful in the sense that ∆(b∗ b) = 0 entails b = 0. Likewise one can use block diagonal maps (via matrix unit projections taken from the commutants n n n of Fm in Fkn , k = m, m + 1, . . . ) to deﬁne explicit maps ∆m : F n → F˜m where F˜m ∗ n n m m is the C*-algebra C (C, Fm ) (which in fact is identiﬁable with Fm ⊗ (e11 Ce11 ) and we have ∆m (b) → b as m → ∞ for all b ∈ F n . In this way (and analogously to Cesaro convergence) we can approximate a general element b in F n by explicitly n . constructed approximants ∆m (b) in F˜m Proof of Theorem 2.4: The directions (ii) =⇒ (i) =⇒ (iii) are elementary so assume that v is an element which satisﬁes the zero one norm condition. Choose m large so that v = ∆m (v) + v , v < 1. Since v satisﬁes the zero one norm condition this is also true of ∆m (v) and v . Indeed, this holds for operators in the matrix subalgebras and by approximation holds in general. The implication (iii) implies (ii) is straightforward for elements of F˜kn . Thus it remains to show that if v has norm less than one and satisﬁes the zero one norm condition then v = 0. This too follows from approximation. Exercise. Show that F n has no proper closed two-sided ideals. (Hint: E0 is faithful.) Remark. The map a → ∆(E0 (a)) is a positive faithful contraction onto the diagonal algebras but it is not (as above) deﬁned as a limit of block diagonal parts. (eg consider a = S1 ). (However, as we note in the Notes below, it may be deﬁned in a more subtle algebraic manner.) n 2.3. Subalgebras of F∞ With the two models for On above and generalised Fourier series we are almost ready to contemplate the following vague question. What are the natural subalgebras of On ?

Before turning to this we should of course look ﬁrst inside the C*-algebra n Mn and the algebras F∞ , F n. The most natural subalgebras of Mn are perhaps those unital subalgebras A which are spanned by a subset of the standard matrix unit system {eij : 1 ≤ i, j ≤ n}. In fact these subalgebras are precisely those that contain the diagonal algebra C spanned by {eii }. (It is this latter property that we essentially use to deﬁne inﬁnite-dimensional variants.) In this case the set E = {(i, j) : eij ∈ A} can be viewed as the set of edges of a directed graph G with n vertices. It follows that (i, i) ∈ E for all i and that (i, k) ∈ E if (i, j), (j, k) ∈ E. Conversely, if G is the graph (V, E), with E such a reﬂexive and transitive relation, then A(G) := span{eij : (i, j) ∈ E} is a complex unital subalgebra containing C. These algebras are the building block algebras of limit algebras. See the Appendix for a fuller discussion.

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It is an elementary but worthwhile exercise to show that the operator algebra A(G) remembers the graph G: Proposition 2.5. (Recovery theorem.) If A(G1 ) and A(G2 ) are isometrically isomorphic algebras then the graphs G1 , G2 are isomorphic. Sketch proof: (i) Projections must map to projections (since projections are the idempotents with norm one), (ii) minimal projections map to minimal projections, (iii) one can reduce (via unitary equivalence) to the case that diagonal projections map to diagonal projections and this gives the needed vertex map. We can think of E both as a “support set” for the algebra (should we view a matrix (aij ) as a function ij → aij ), and also, more usefully, as a binary relation that comes with the algebra. In these terms the proposition says that a digraph algebra has isometric isomorphism type determined by the isomorphism type of its binary relation. In Section 4 I give a generalisation of this fact for a wide class of subalgebras of On . First however, let us look inside the matricial star algebra n F∞ and its closure F n . In fact we may as well consider a more general class of approximately ﬁnite algebras. Deﬁnition 2.6. (a) A unital matricial star algebra is a complex algebra B for which there exists a spanning set {ekij : 1 ≤ i, j ≤ nk , k = 1, 2, . . . } such that (i) for each k the set {ekij : 1 ≤ i, j ≤ nk } is a matrix unit system for Mnk , (ii) for each k, Mnk ⊆ Mnk+1 and moreover the inclusion map is a C*algebra injection which maps each ekij to a sum of matrix units from {ek+1 : 1 ≤ i, j ≤ nk }. ij (b) A regular matricial algebra is a complex algebra A which is a unital subalgebra of a matricial star algebra containing the diagonal subalgebra C = span{ekij }. It is straightforward to see that the regular matricial algebra A in the matricial star algebra B is the union of the algebras Ak = A ∩ Mnk = span{ekij : ekij ∈ A} and that each Ak is a digraph algebra A(Gk ) relative to the given matrix unit sysn tem. Our subalgebra F∞ is a particular unital matricial star algebra in which each inclusion map has the same multiplicity n. Further examples of such algebras can be obtained with the technology of direct limits: ﬁrst take a tower of appropriate inclusions maps, A(G1 ) → A(G2 ) → . . . . Algebraic direct limits then provide the algebra A = lim→ A(Gk ) as well as B and C.

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2.4. Subalgebras of unital AF C*-algebras Suppose we have the purely algebraic setting C ⊆ A ⊆ B given in Deﬁnition 2.4. The matricial star algebra carries a unique C*-algebra norm and taking operator norm closures gives the triple inclusion C ⊆ A ⊆ B. Here B is a UHF C*-algebra, C is a particularly nice abelian subalgebra in B and A is an instance of a limit algebra A = lim(A(Gk ), φk ) →

where the inclusion maps φk : A(Gk ) → A(Gk+1 ) are particularly nice. (In the terminology of the Appendix, they are star-extendible and regular.) We give three key results for such limit algebras. The ﬁrst, rather surprisingly perhaps, shows that any norm closed algebra A with C ⊆ A ⊆ B is necessarily the closure of a regular matricial algebra. This is justiﬁcation for the opinion that the subalgebras of UHF C*-algebras which contain the distinguished masa are the natural generalisations of ﬁnite-dimensional digraph algebras. The following “local recovery” theorem gives a key step towards understanding the limit algebras A. Theorem 2.7. (Inductivity principle.) Let B be a unital matricial star algebra with subalgebra chain {Mnk } and diagonal C and let B, C be their norm closures. If A ⊆ B is a norm closed subalgebra containing C then A is the closed union of the digraph algebras Ak = A ∩ Mnk , k = 1, 2, . . . . The next two theorems (and that above) have more general forms for subalgebras of AF C*-algebras but we state them here for subalgebras of the UHF C*-algebra F n . We ﬁrst need to deﬁne the appropriate substitute for the graph that underlies a digraph algebra and for this the Cantorised interval picture provides what we need, both for F n and for On . In fact we are going to deﬁne an isometric isomorphism invariant for the algebra A together with its diagonal C which is in the category of topological binary relations. Often (always?!) the binary relation is a complete isometric isomorphism invariant for the algebra alone. ∞ Let X be the Cantor space k=1 {0, 1, . . . , n − 1}. For words µ, ν with the same length k recall that Sµ Sν∗ is a partial isometry on L2 (X) which is induced by the partial homeomorphisms αµ,ν : νxk+1 xk+2 · · · → µxk+1 xk+2 . . . . Deﬁne the topological space R to be the set in X × X which is the union of the graphs of these partial homeomorphism: R = {(α(x), x) : x ∈ dom(α), α = αµ,ν , |µ| = |ν| = k, k = 1, 2, . . . } (k)

k for the graph of the partial homeomorphism for the matrix unit eij Write Ei,j n in Fk and one can conceive of these sets as the “support set” of the matrix (k) units. The diagonal matrix units eij provide closed-open sets Eiik in the diagonal ∆ = {(x, x) : x ∈ X} and these give a base for the natural Cantor space topology. k We topologise R by taking the sets Eij as a base for the topology.

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Exercise. Show that R is an equivalence relation. Show that the topology is not the relative product topology. It follows from Theorem 2.7 that if C ⊆ A ⊆ F n then A, and the given subalgebra chain, determines a subset R(A) of R, namely k R(A) := R(A) := ∪{Eij : ekij ∈ Ak }.

With the relative topology, R(A) is known as the topological binary relation of A. In fact the topological binary relation R(A) is determined by the pair (A, C) and serves as the analogue of the graph of a digraph algebra. The following uniqueness theorem also follows from Theorem 2.7. Theorem 2.8. (Spectral theorem for subalgebras.) Let A1 , A2 be norm closed subalgebras of F n which contain the canonical diagonal algebra C. If R(A1 ) = R(A2 ) then A1 = A2 . That R(A) is intrinsic to the pair A, C is also revealed by the following proposition. We identify X with the Gelfand space of C. Proposition 2.9. As a set, R(A) is the set of points (x, y) in X × X for which there exists a ∈ A and δ > 0 such that paq ≥ δ for all projections p, q in C with x(p) = y(q) = 1. We can now state the following classiﬁcation theorem which, loosely paraphrased, asserts that a triangular subalgebra of F n remembers its topological binary relation. Theorem 2.10. Let A1 and A2 be norm-closed subalgebras of F n with Ai ∩ A∗i = C for i = 1, 2. (Such algebras are said to be triangular.) Then the following statements are equivalent n n and A2 ∩ F∞ are isometrically isomorphic normed algebras. (i) A1 ∩ F∞ (ii) A1 and A2 are isometrically isomorphic operator algebras. (iii) The topological binary relations R(A1 ), R(A2 ) are isomorphic, that is, there is a homeomorphism α : M (C) → M (C) such that α × α : R(A1 ) → R(A2 ) is a homeomorphism.

The key to the proofs of Theorem 2.7, Theorem 2.8, Proposition 2.9 and Theorem 2.10 is the structure of partial isometries given in Theorem 2.4. It is this which which makes the link between operator algebra entities and the underlying topological binary relation. Remarkably, perhaps, there is a close generalisation of Theorem 2.10 to gauge invariant subalgebras of On . (Theorem 4.3.) Sketch proof of Theorem 2.10. Let Φ : A1 → A2 be an isometric isomorphism. By triangularity, the set of projections p in Ai generate Ci . Since the projections are the norm one idempotents it follows that Φ(C1 ) = C2 . By the zero-one characterisation

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in Theorem 2.4 it follows that Φ maps the C1 -normalising partial isometries to C2 normalising partial isometries. Considering the support of a normalising partial isometry as a closed-open set in M (Ci ) × M (Ci ) it follows (from the ﬁniteness of Theorem 2.4 (ii)) that Φ maps a base of closed open sets to a base of closed-open sets. Moreover Φ induces a point bijection α : R(A1 ) → R(A2 ). (Take intersections of neighbourhoods.) The point bijection is a topological homeomorphism since its map on sets extends the original bijection of closed-open sets. Open problems. (See also the Notes below.) 1. For the algebraic context C ⊆ A ⊆ B given at the end of Section 2.3 is C unique in A up to automorphisms of A? If this is the case then R(A) becomes a well-deﬁned invariant for the algebra A. 2. Is R(A) an algebraic isomorphism invariant for a regular matricial algebra A? 2.5. Notes Theorem 1.2 is usually referred to as Coburn’s theorem. For more on this, Cuntz algebras and other C*-algebras, see, for example, Davidson [3] and the references therein. On is usually deﬁned as the universal C*-algebra generated by isometries S1 , . . . , Sn with S1 S1∗ + · · · + Sn Sn∗ = I. The direct sum of irreducible realisations of such relations gives generators for this algebra; the universal property is that to any isometric realisation s1 , . . . , sn of the relations there should exist a canonical homomorphism On → C ∗ ({s1 , . . . , sn }) and this is readily checked for this universal direct sum. The uniqueness theorem, Theorem 2.1, will follow now from the simplicity of On , and this in turn follows readily from an algebraic formulation of the map E0 : On → Fn . (See [3],[1].) For if J is an ideal and a ∈ J is nonzero, then a∗ a is a positive nonzero element in J and so it follows from the algebraic formulation that a0 = E0 (a) is in J. Since Fn is simple the simplicity of On follows. In fact more is true in that one can ﬁnd elements x, y ∈ On such that xab = I. To do this one uses the algebraic deﬁnitions of E0 and ∆ to get, in J, an operator d = pdp = x1 ay2 close to a diagonal matrix unit p. Then one ﬁnds the appropriate isometry Sµ to conjugate p to an operator close to the identity. The normalising partial isometry theorem, from which Theorem 2.7, Theorem 2.8 and Theorem 2.10 are easily obtained, are discussed further in [21]. Here one can also ﬁnd their natural extensions to AF C*-algebras. The open problems are essentially the problems 7.8, 7.9 of [21]. Those problems are stated for closed algebras but because of inductivity the problems really reside in pure algebra and are stated in these terms here. If A is self-adjoint then the masa C is unique up to automorphism. More is true: for any other matrix unit system for A, with subalgebra system {Ak } and masa C (as in the deﬁnition), there is an automorphism A → A which maps C to C . (See [21].) However in this case the automorphism, which is determined by an intertwining diagram A1 → An1 → Am1 → An2 → · · · , can be constructed in such a way so that there is an intertwining diagram with regular maps in the sense that the normaliser of each diagonal algebra maps into

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17

the normaliser of the diagonal of the next algebra. (See the Appendix.) Part of the diﬃculty of the general nonselfadjoint problem is that it is known that there are diagonal masas (of the type above) which although automorphically equivalent are not equivalent through an intertwining diagram of regular maps. See [9] for this subtlety, and see [10], [24] for further discussions. The importance of the problem is that all kinds of putative invariants can be associated with classes of regular direct system and one would like these constructs (partial isometry homology groups for example) to be invariants for the algebra rather than the pair (A, C).

3. Toeplitz algebras for countable directed graphs We now take up a diﬀerent topic and formally deﬁne the analytic Toeplitz algebras AG and LG . Let G be a ﬁnite or countable directed graph, with edge set E(G) and vertex set V (G). The free semigroupoid F+(G) determined by G is a set with partially deﬁned associative multiplication. The set consists of the vertices, which act as multiplicative units, and all ﬁnite directed paths in G. The partially deﬁned product is the natural operation of concatenation of paths, with a vertex considered as a path. Thus a nonunit element of F+(G) is a path (or word) w = e1 e2 . . . en where the initial vertex of each ei , for i < n, is equal to the ﬁnal vertex of ei+1 . Vertices may appear in a word of edges, redundantly, to indicate information. For example given a path w in F+(G) we have w = ywx when the initial and ﬁnal vertices of w are, respectively, x and y. Let HG = 2 (F+(G)) be the Hilbert space with orthonormal basis of vectors ξw indexed by elements w of F+(G). For each edge e ∈ E(G) and vertex x ∈ V (G) deﬁne partial isometries and projections on HG by the following left-sided actions on basis vectors: ξew if ew ∈ F+(G) Le ξw = 0 otherwise and ξxw = ξw if w = xw ∈ F+(G) Lx ξw = 0 otherwise. We also write Px for the projection Lx . If G has a single vertex x then ξx is referred to as the vacuum vector and the operators Lw are isometries. If there are, additionally, only ﬁnitely many edges e1 , . . . , en then each path w is a free word in these edges and the semigroupoid of paths is simply the free (unital) semigroup F+ n with n generators. For n = 2 one can visualise the action of the two isometries Le1 and Le2 as downward left and right shifts of basis vectors placed at the vertices of a downward branching tree. Deﬁnition 3.1. (i) The free semigroupoid algebra LG is the weak operator topology closed algebra generated by the projections Lx and the (partial) shift operators Le ; LG

= wot–Alg {Le , Lx : e ∈ E(G), x ∈ V (G)}.

18

S.C. Power

(ii) The algebra AG is the norm closed algebra generated by {Le , Lx : e ∈ E(G), x ∈ V (G)}. This Toeplitz algebra is also referred to as the tensor algebra for G. The algebra LG can also be thought of as being generated by the left “regular” representation λG : F+ G → B(HG ), λG (e) = Le , which faithfully represents the as partial isometries. In the case of a noncomposable elements semigroupoid F+ G w1 and w2 one can check that the corresponding partial isometries have zero product. 3.1. Examples and matrix function algebras It should be apparent that the algebra LG for the graph with a single vertex x and single loop edge e = xex is unitarily equivalent to the analytic Toeplitz algebra TH ∞ ; the Fock space naturally identiﬁes with the Hardy space H 2 , and Le is then unitarily equivalent to the unilateral shift Tz . More generally, the noncommutative analytic Toeplitz algebras Ln , n ≥ 2, also known as the free semigroup algebras, arise from the graphs with a single vertex and n distinct loop edges, while L∞ comes from the single vertex graph with countably many loops. (i) If G is a ﬁnite graph with no directed cycles, then the Fock space HG is ﬁnite-dimensional and so too is LG . As an example, consider the graph with three vertices and two edges, labelled x1 , x2 , x3 , e, f where e = x2 ex1 , f = x3 f x1 . Then the Fock space is spanned by the vectors {ξx1 , ξx2 , ξx3 , ξe , ξf } and with this basis the general operator X = αLx1 + βLx2 + γLx3 + λLe + µLf in LG is represented by the matrix ⎤ ⎡ α 0 0 0 0 ⎢ 0 β 0 0 0⎥ ⎥ ⎢ ⎥ X⎢ ⎢ 0 0 γ 0 0⎥ . ⎣λ 0 0 β 0 ⎦ µ 0 0 0 γ Here, LG is isometrically isomorphic to (but not unitarily equivalent to) the socalled digraph algebra (see later) in M3 (C) consisting of the matrices ⎡ ⎤ α 0 0 ⎣λ β 0⎦ . µ 0 γ (ii) Consider the graph G with two vertices x, y, a loop edge e = xex and the edge f = yex directed from vertex x to vertex y. The tree graph for Fock space takes the form shown in Figure 4. The semigroupoid algebra LG is generated by {Le , Lf , Px , Py }. If we make the natural identiﬁcations HG = Px HG ⊕ Py HG H 2 ⊕ H 2 , then Tz 0 0 0 I 0 0 0 , Px Le , Lf , Py . 0 0 Tz 0 0 0 0 I Thus, LG is seen to be unitarily equivalent to a matrix Toeplitz algebra, which we can also view as (isometrically and weak star – weak star isomorphic to) the

Subalgebras of Graph C*-algebras

x

19

y

e

f

ee

fe

eee

fee

Figure 4. Fock space graph. matrix function algebra

∞ H H0∞

0 CI

where H0∞ is the subalgebra of H ∞ formed by functions h with h(0) = 0. Exercise. Add to G a “returning” directed edge g = xgy to obtain a graph G . Show that LG contains a “copy” of L2 by virtue of the fact that it contains isometries with mutually orthogonal ranges. (iii) Let n ≥ 1 and consider the cycle graph Cn which has n vertices x1 , . . . , xn and n edges en = x1 en xn and ek = xk+1 ek xk for k = 0, . . . , n − 1. Identify Lxi HG with H 2 for each i in the natural way (respecting word length). Then HG = Lx1 HG ⊕ . . . ⊕ Lxn HG Cn ⊗ H 2 and the operator α1 Le1 + . . . + αn Len is identiﬁed with the operator matrix ⎡ ⎤ 0 αn Tz ⎢α1 Tz ⎥ 0 ⎢ ⎥ ⎢ ⎥ α2 Tz 0 ⎢ ⎥. ⎢ ⎥ .. .. ⎣ ⎦ . . αn−1 Tz Hn∞ ∞

∞

0

Write for the subalgebra of H arising from functions of the form h(z n ) with h in H . It follows that the algebra LCn is isomorphic to the matrix function algebra ⎡ ⎤ Hn∞ z n−1 Hn∞ . . . zHn∞ ⎢ .. ⎥ ⎢ zHn∞ Hn∞ . ⎥ ⎢ ⎥. ⎢ ⎥ .. .. ⎣ ⎦ . . ... Hn∞ z n−1 Hn∞

20

S.C. Power

3.2. Fourier series There is a companion algebra to LG coming from “right shifts”. Consider the right regular representation ρG : F+(G) → B(HG ) determined by a directed graph G. This yields partial isometries ρG (w) ≡ Rw for w ∈ F+(G) acting on HG deﬁned by the equations Rw ξv = ξvw , where w is the word w in reverse order. The corresponding algebra is RG

= wot–Alg {Re , Rx : e ∈ E(G), x ∈ V (G)}.

Given edges e, f ∈ E(G), observe that Le Rf ξw = ξewf = Rf Le ξw , for all w ∈ F+(G), so that Le Rf = Rf Le , and similarly for the vertex projections. In fact, we have Proposition 3.2. The commutant LG of LG is equal to RG . The proof of this proposition makes use of an important tool, namely the Fourier series expansion of elements of LG . Recall that Px is the projection for the subspace spanned by basis vectors ξw with w = xw. Write Qx for the projection onto the subspace spanned by basis vectors ξw with w = wx. (The projections Qx correspond to the “components” of the Fock space when basis elements are linked by the natural tree structure.) The proposition below takes a simple form when there is a single vertex, i.e., LG = Ln . Proposition 3.3. Let A ∈ LG , x ∈ V (G), and let aw ∈ C be the coeﬃcients for which Aξx =Qx Aξx = w=wx aw ξw . Then the Cesaro sums associated with the formal sum w∈F+(G),w=wx aw Lw , given by |w| aw L w 1− Σk (Qx A) = k w=wx;|w|≤k

converge in the strong operator topology to Qx A. 3.3. LG remembers the graph G Let us ﬁrst observe why Ln and Lm are not isomorphic when n = m. This can be seen from the following theorems the ﬁrst of which introduces another important tool, namely the eigenvectors for L∗G . By an eigenvector for L∗n we really mean a vector ν which is a joint eigenvector for the n-tuple of generators L∗e1 , . . . , L∗en , so that L∗ei ν = αi ν, , 1 ≤ i ≤ n, for some complex numbers αi ∈ C. The notation below is that w(λ) is the complex number obtained on substituting λi for ei in the word w. Theorem 3.4. The eigenvectors for L∗n are complex multiples of the unit vectors w(λ)ξw , νλ = (1 − λ2 )1/2 w∈F+ n

for λ = (λ1 , . . . , λn ) in the open unit ball Bn ⊆ Cn . Furthermore L∗ei νi = λi νλ , for each i.

Subalgebras of Graph C*-algebras

21

Note that for λ in the open unit ball n λi Lei 2 = |λi |2 = λ2 < 1 so that I −

1

n

1 λi Lei is invertible, with inverse −1 k I− λi Lei = λi Lei = w(λ)Lw .

e

k≥0

e

w

In particular w w(λ) is convergent, and a similar shows the norm to be (1 − λ2 )−1/2 . Eigenvectors are important since they are allied to characters, i.e., multiplicative linear functionals φ : Ln → C, φ : An → C. Indeed, the map φλ : An → C deﬁned by φλ (A) = Aνλ , νλ satisﬁes φλ (p(Le1 , . . . , Len )) = νλ , p(L∗e1 , . . . , L∗en )νλ = νλ , p(λ1 , . . . , λn )νλ = p(λ1 , . . . , λn ). It follows that the vector functional actually deﬁnes a character. One can go on to show that the character space of An is homeomorphic to closed unit ball. The dimension of the character space serves as a classifying invariant for the algebras An and Ln . More generally one has the following theorem, and an analogous result for the weakly closed free semigroup algebras. Theorem 3.5. Let G, G be directed graphs. Then the following assertions are equivalent. (i) G and G are isomorphic graphs. (ii) AG and AG are unitarily equivalent. (iii) AG and AG are isometrically isomorphic. 3.4. Notes There are a number of approaches to the classiﬁcation theorem above. In Kribs and Power [14], following the free semigroup algebra analysis of Davidson and Pitts [5], [6], wandering vectors are analysed to obtain a Beurling type theorem for invariant subspaces of the algebra. Using this one obtains unitarily implemented automorphisms of LG that act transitively on the set of eigenvectors. Now the eigenvectors are parametrised by the union of unit balls for each vertex with loop edges. In particular isomorphisms can be normalised to the special case where vacuum vectors map to vacuum vectors. Consequently the ideal A0G generated {Le : e ∈ E(G)} is preserved. Theorem 3.5 is straightforward in this case. See also Solel [29] and Katsoulis and Kribs [13]. Other topics that can be found in these papers, and others, are the determination of unitary automorphisms, the structure of partial isometries, the reﬂexivity and hyper-reﬂexivity of the algebras LG , and determination of the Jacobson radical and semisimplicity.

22

S.C. Power The Hilbert space Hn is readily identiﬁable with the Fock space ⊕(Cn )⊗k Hn = C ⊕ k∈Z+

formed by the direct sum of multiple tensor products of Cn . With this formulation the operators Le are conveniently speciﬁed by the shift property Le (ξ1 ⊗ · · · ⊗ ξk ) = ξe ⊗ ξ1 ⊗ · · · ⊗ ξk where ξ1 ⊗ · · · ⊗ ξk is an elementary tensor in the k-fold tensor product summand. In general the generating operators Le are partial isometries acting on a natural generalized Fock space Hilbert space, in which not all tensors are admissible. Although we have not needed the tensor formalism this does provide a fundamental construction allowing for further generalisations, most notably for the tensor algebras of correspondences. See, for example, Muhly and Solel [17]. We remark that there is presently a theory of higher rank versions of such non-selfadjoint operator algebras being developed which is associated with higher rank graphs and with higher rank correspondences (product systems). For more on this see Kribs and Power [16], Power [25], Solel [30] and Power and Solel [26]. The following result from [15] gives a graph theoretic condition corresponding, roughly speaking, to the separation of the algebras LG into two classes, those which are “matrix function like” and those that are “free semigroup like”. The following notion parallels somewhat the requirement that a C∗ -algebra contain O2 , or that a discrete group contain a free group. Deﬁnition 3.6. A wot-closed algebra A is partly free if there is an inclusion map L2 → A which is the restriction of an injection between the generated von Neumann algebras. If the map can be chosen to be unital, then A is said to be unitally partly free. A directed graph G is said to have the double-cycle property if there are distinct minimal cycles w = xwx, w = xw x over some vertex x in G. Theorem 3.7. The following assertions are equivalent for a countable directed graph G with a ﬁnite number of vertices. (i) G has the double-cycle property. (ii) LG is partly free.

4. Subalgebras of On Returning to the themes of Section 2, we are now ready to look inside On . Our context is that of a norm closed subalgebra A ⊆ On which contains the canonical diagonal subalgebra C associated with the given generators of On . Let us ﬁrst note that there are a number of ways such algebras arise. (i) Generator constraints: (a) If S is a semigroup of operators of the form Sµ Sν∗ which contains all the projections Sµ Sµ∗ then the norm closed linear span is

Subalgebras of Graph C*-algebras

23

a subalgebra of On which contains the canonical diagonal subalgebra. Note that this algebra is left invariant by the gauge automorphisms of On . (b) Let A1 be the norm closed algebra generated by the diagonal algebra C and the single operator S1 . Let A2 be the (nonunital) subalgebra which is the ideal in A1 generated by 1 − S1 . The abelain algebra A1 /com(A1 ) can be naturally identiﬁed with the disc algebra A(D), while A2 /com(A2 ) identiﬁes with the ideal of functions h(z) with h(1) = 0. The gauge automorphisms of On rotate the ideals of A(D) and so the algebra A2 is not gauge invariant. (ii) Fourier series constraints: Let A ⊆ F n be a triangular subalgebra with A ∩ A∗ = C. Then A = {a ∈ On : E0 (a) ∈ A, Ek (a) = 0, k < 0} is a triangular subalgebra of On . Once again, A is gauge invariant. (iii) Extrinsic constraints: Let N ⊆ C be a totally ordered family of projections. For example, N could consist of the projections corresponding to the intervals [0, k/2n] in the interval picture of On . To such a nest of projections one can assign the nest subalgebra A = On ∩ AlgN = {a ∈ On : (1 − p)ap = 0, for all p ∈ N }. Once again, masa normalising partial isometries give a key tool for recovering the underlying “coordinates” from the algebra structure. Each partial isometry Sµ Sω∗ is C-normalising, as are ﬁnite sums of these when they have orthogonal ranges and orthogonal domains. Also we may multiply these sums by unitary elements of C to obtain further examples. These turn out to be all the normalising partial isometries and they may be characterised in intrinsic terms as in the following theorem. In terms of the interval picture, or the Cantor interval picture one can, once again, indicate pictorially the support of such a partial isometry, as shown. 0

1

0

1

Figure 5. The support a normalising partial isometry in O2 .

24

S.C. Power

Theorem 4.1. Let v be a contraction in On . Then the following assertions are equivalent: (i) v is a C-normalizing partial isometry. (ii) v is an orthogonal sum of a ﬁnite number of partial isometries of the form dSµ Sν∗ , where d ∈ C. (iii) For all projections p, q ∈ C, the norm qvp is equal to 0 or 1. Proof. The implications from (ii) to (i) to (iii) are routine and left to the reader. Let v be a contraction with the zero one norm condition. We claim ﬁrst that E0 (v) is a C-normalizing partial isometry in F n . The intuitive reason for this follows on contemplating the (Cantor space) support supp(v) of v which is deﬁned as the set of points (y, x) in X × X such that for some δ > 0 qvp = 1 for all projections p, q in C with y(q) = x(p) = 1. Considering continuous matrices we can see that the support supp(w) of a ﬁnite sum w of products of the generators and their adjoints will be the union of the sets supp(E0 (w)) and supp(w − E0 (w)), the former set consisting of the diagonal segments parallel to the main diagonal. Because of the essential disjointness of these sets it follows from (iii) that E0 (w) satisﬁes the zero-one condition and so by Theorem 2.10 is a normalising partial isometry. For a rigorous proof we can argue as follows. If E0 (v) were not a normalising partial isometry then by Theorem 2.10 we would be able to ﬁnd δ > 0 and projections p, q in C so that δ ≤ qE0 (v)p ≤ 1 − δ. Moreover, for each N it is possible to choose the projections in such a way so that qwp = 0 for any standard partial isometry w of the form Sµ Sν∗ with |µ| = |ν| and 0 ≤ |µ|, |ν| < N . Choose v in the star algebra generated by the generators with v − v < δ/3. Since v − E0 (v ) is a linear combination of Sµ Sν∗ with |µ| = |ν| we may choose p, q as above so that pvq − pv q = pvq − pE0 (v )q. It follows that pvq is not zero or one, a contradiction. Now suppose that m > 0. If |ν| = m and |λ| − |µ| = m, then the product Sν∗ Sλ Sµ∗ is either zero or of the form Sλ1 Sµ∗ with |λ1 | = |µ|. It follows that if Φm (v) is the mth term in the series expansion of v (Φm (v) = am S1m for m ≥ 0), then Sν∗ Φm (v) = E0 (Sν∗ v). Since v satisﬁes the zero one condition, so does Sν∗ v and the argument above shows that Sν∗ Φm (v) is C-normalizing and so has the desired form. This, in turn, implies that Sν Sν∗ Φm (v) is C-normalizing and has the required form for any word ν of length m. Consequently, Φm (v) has the desired form. In a similar fashion, we can show that when m < 0, Φm (v) also has the desired form (consider adjoints, for example). Finally, if w is a partial isometry and ww∗ xw∗ w = 0 then w + ww∗ xw∗ w > 1. From this observation and the Ces´ aro convergence of generalised Fourier series, it follows that the operators Φm (v) are non-zero for only ﬁnitely many values of m and that v is the orthogonal sum of these operators. Thus v itself has the desired form.

Subalgebras of Graph C*-algebras

25

Recall the Cantor interval picture and the partial homeomorphisms αµν . The dilation factor of αµν we deﬁne to be k = |ν| − |µ|. Previously we focused on the case k = 0 appropriate to F n . We now deﬁne the counterpart to R(F n ). The Cuntz groupoid R(On ) is, intuitively speaking, the support of the algebra in the Cantorised interval picture, with record taken of the dilation factors. More formally it is the set R(On ) = {(x, k, y) : x = αµν (y) for some αµν }, together with (i) the totally disconnected topology with (as before) the set of graphs Eµν , for the partial homeomorphisms αµν , as a base, (ii) the natural partially deﬁned multiplication coming from composition of appropriate partial homeomorphisms. It is natural now to seek to obtain for subalgebras of On results analogous to those in Section 2. It turns out that there is a complication in that “synthesis”, as expressed in Theorem 2.7, may fail. However, it is precisely the gauge invariant closed subalgebras containing C that are determined by their groupoid support: Theorem 4.2. Let A be a closed subalgebra of On containing the canonical diagonal masa C. Then A is generated by the partial isometries Sµ Sν∗ belonging to A if and only if A is invariant under the gauge automorphisms γz for |z| = 1. For a gauge invariant algebra as above we deﬁne an associated topological semigroupoid R(A). This is the set R(A) = {(x, k, y) : x = αµν (y) for µ, ν such that Sµ Sν∗ ∈ A} with the relative topology and partially deﬁned multiplication. With the characterisation of normalising partial isometries given above it is now possible to prove the following analogue of Theorem 2.10, and in a similar manner to the proof of that theorem. To paraphrase, gauge invariant triangular subalgebras of On remember their semigroupoids and are classiﬁed by them. Theorem 4.3. Let A1 and A2 be norm-closed subalgebras of On with Ai ∩ A∗i = C for i = 1, 2. Then the following statements are equivalent (i) A1 and A2 are isometrically isomorphic operator algebras. (ii) The semigroupoids R(A1 ), R(A2 ) are isomorphic, that is, there is a homeomorphism α : R(A1 ) → R(A2 ) which respects the partially deﬁned multiplication. Proof. The direction (i) =⇒ (ii) is similar to that of the proof of Theorem 2.10. The direction (ii) =⇒ (ii) is more straightforward. One lifts the map α to a map on the semigroup of normalising partial isometries generated by the Sµ Sν ∗ and this can be extended to an isometric algebra isomorphism. Suppose now that G is a countable directed graph (V, E) with range and source maps r, s : E → V . One can generalise the Cuntz relations as we indicated above in context E of the introduction. To each edge e there is a partial isometry

26

S.C. Power

Se and to each vertex x a projection Px . The initial projection of Se is Ps(e) while the range projection is dominated by Pr(e) . Moreover, under a given Py the range projections sum to that projection: Se Se∗ = Py e:r(e)=y

with weak operator topology convergence if the edge incidence is inﬁnite. We see then that it is simply the graph that encodes partial isometry generators and relations. The graph C*-algebra C ∗ (G) is deﬁned to be the associated universal C*algebra. Much is known about the structure of this diverse class. See for example [27]. Once again, words in the generators and their adjoints have a reduced form Sα Sβ∗ where, in the graph case, α = α1 . . . αn is a directed path in G (directed from left to right in our convention) and there are natural counterparts to methods and results in Section 2.1. Moreover, if G has no source vertices, in the sense that r is onto, then the abelian C*-algebra generated by the projections Sα Sα∗ is a masa. In this setting, with the simplifying assumption of ﬁnite incidence at every vertex (so-called row ﬁniteness) one has exact counterparts to all the results of this section. 4.1. Notes Theorems 4.1, 4.2 and 4.3 are taken from Hopenwasser, Peters and Power [11] where one can also ﬁnd the more general variants for graph C*-algebras. The methods related to the AF classiﬁcation, Theorem 2.10, has assisted in the analysis of many particular families of subalgebras of AF C*-algebras and much is known of the structure of ideals and representations, for example. On the other hand subalgebras of graph C*-algebras have not received such attention but it may be timely to do so. In these lectures we have been led from operator algebra considerations to speciﬁc topological groupoids and semigroupoids, for F n and On . It is an important and natural consideration to complete the circle and construct operator algebras associated with general abstract topological groupoids. For this see Renault [28], Paterson [18] and Raeburn [29]. For further perspectives on non-self-adjoint operator algebras see the recent article of Donsig and Pitts [8] who also comment on variants of the open problems in Section 2.

Subalgebras of Graph C*-algebras

27

5. Appendix: Digraph algebras and limit algebras We give a brief self-contained account of digraph algebras A(G) and some examples of direct limit algebras. 5.1. Digraph Algebras A digraph is a directed graph G = (V, E) with no multiple directed edges, so that E ⊆ V × V and each edge e in E can be written as (x, y) with initial vertex y and ﬁnal vertex x. If V = {1, . . . , n} then the subspace A(G) ⊆ Mn (C) is deﬁned by A(G) := span{ei,j : (i, j) ∈ E}, where {eij } is the standard matrix unit system for Mn (C). Suppose further that E, when regarded as a binary relation, is reﬂexive. Thus (v, v) ∈ E for all v ∈ V . Then Dn ⊆ A(G), where Dn = span {eii }. Note that A(G) is a complex algebra if and only if G is transitive, that is, if (i, j) and (j, k) are edges then so is (i, k). We say that A(G) is a standard digraph algebra in this case. Examples. (i) If Km is the complete digraph on {1, . . . , m} then A(Km ) = Mm (C). (ii) Let D2m be a 2m-sided polygon with alternating directions on the edges and loops at each vertex. Then A(D2m ) is identiﬁable as a rather sparse algebra of matrices. (iii) Let Tn be the subalgebra of upper triangular matrices in Mn (C). Then Tn is a digraph algebra. (iv) Given the digraph G, we can construct G × Km , the relative product by replacing each vertex of G by Km , and replacing each proper edge of G by all the n2 edges between the new vertices. The digraph algebra A(G × Km ) is identiﬁable with A(G) ⊗ Mm (C) . Deﬁnition 5.1. A ⊆ Mn (C) is a digraph algebra if A is a complex algebra and A contains a maximal abelian self-adjoint algebra (masa) D. Proposition 5.2. If D ⊆ Mn (C) is a masa then there is a unitary matrix u ∈ Mn (C) such that uDu∗ = Dn , the standard masa. Proof. D being a masa means that if D properly contains D and D is also a self-adjoint algebra, then D is not abelian. Let {p1 , . . . , pt } ⊆ D be a maximal set of pairwise orthogonal projections. Then rank pi = 1, for all i, by maximality of D. For if not split the projection into a sum of two projections to obtain a larger abelian algebra. It follows, again by maximality, that t = n and so there is a unitary u such that for all i, upi u∗ = eii as required. Proposition 5.3. If D ⊆ A(G) is a masa then there exist a unitary u ∈ A(G) such that uDu∗ = Dn . Thus, maximal abelian self-adjoint subalgebras of digraph algebras are unique up to inner conjugacy (inner unitary equivalence). Proof. Use the previous proposition, combining unitaries in each block of the block diagonal self-adjoint subalgebra A(G) ∩ A(G)∗ . Since an algebra in Mn (C) which contain the standard masa is a standard digraph algebra we have the following corollary.

28

S.C. Power

Corollary 5.4. Every digraph algebra A ⊆ Mn (C) is inner conjugate to a standard digraph algebra. That is uAu∗ = A(G) for some digraph G and some unitary u in A. If standard digraph algebras are unitarily equivalent then by Proposition 5.2 we can assume that the unitary equivalence maps the standard masa to the standard masa and it follows readily that the graphs are isomorphic. 5.2. Maps between digraph algebras To begin to understand algebras of the form A = k A(Gk ), where the building block algebras are nested, i.e., A(Gk ) ⊆ A(Gk+1 ), we must consider the nature and m variety of the possible inclusion maps A(Gk ) → A(Gk+1 ). Let (fij )i,j=1 ⊆ Mn (C) be operators with the relations of an m × m matrix unit system, that is, ⎤ fij fjk = fik ∀ ijk ⎦ fij∗ = fji (∗) fij fkl = 0 if j = k Then the map φ : Mm → Mn which is deﬁned to be the linear extension of the correspondences eij → fij , is an injective C ∗ -algebra homomorphism. Conversely, it is straightforward to show that if φ is such a map, then {φ(eij )} satisfy the relations (∗). Deﬁnition 5.5. Let A(G1 ) ⊆ Mm , A(G2 ) ⊆ Mn be unital standard digraph algebras with connected graphs. Then an algebra homomorphism φ : A(G1 ) → A(G2 ) is said to be star-extendible if φ is the restriction of a C ∗ -algebra map between the ﬁnite-dimensional C*-algebras C ∗ (A(G1 )) and C ∗ (A(G2 )). Example (i) The map φ : T2 → T4 given by ⎛ a 0 ⎜ 0 a a b φ =⎜ ⎝ 0 0 0 c 0 0 is a star-extendible algebra injection. Example (ii) The map φ : T2 → T4 given by ⎛ a ⎜ a b φ: →⎜ ⎝ c

√ b/√2 b/ 2 c 0

b c

0 0 a

√ −b/√ 2 b/ 2 0 c

⎞ 0 0 ⎟ ⎟ 0 ⎠ 0

is not star-extendible (and is an isometric algebra injection). Example (iii) φ : T2 → T4 given by ⎡ ⎤ a b ⎢ ⎥ a b a b ⎥ φ: →⎢ ⎣ ⎦ c c c is a star-extendible algebra injection.

⎞ ⎟ ⎟ ⎠

Subalgebras of Graph C*-algebras

29

Remark 1 Star-extendible injective maps between digraph algebras are necessarily isometric: since they are restrictions of C*-algebra maps. This is an elementary fact from spectral theory. Indeed, note ﬁrst that if p ∈ Mm is a nonzero selfadjoint projection then ||p|| = 1, by the deﬁnition of the operator norm. If φ is a star homomorphism then φ(p) is a projection, so φ(p) = 1 = p. We claim that φ(a) = a if a is a self-adjoint operator in Mm . By the spectral theorem a = λ1 p1 + · · · + λm pm , and we can suppose a = |λ1 | ≥ |λk | for k = 2, . . . , m, where p1 , . . . , pm are pairwise orthogonal projections. Let qi = φ(pi ) 1 ≤ i ≤ m. Then φ(a) = λ1 q1 + · · · + λm qm , with q1 . . . qm pairwise orthogonal projections. Thus (exercise) φ(a) = |λ1 | = a. Finally if b ∈ Mm is a general element, then φ(b)2 = φ(b)∗ φ(b) = φ(b∗ )φ(b) = φ(b∗ b) = b∗ b = b2 . Deﬁnition 5.6. Let A(G1 ), A(G2 ) be digraph algebras with standard matrix unit systems {ekij : (ij) ∈ E(Gk )}, k = 1, 2, as usual. (i) An algebra injection φ : A(G1 ) → A(G2 ) is a standard regular injection, with respect to {e1ij }, and {e2ij }, if φ is star-extendible and maps each e1ij to a sum of matrix units in {e2kl } (ii) An algebra injection ψ : A(G1 ) → A(G2 ) is a regular (star-extendible) injection if there exists a unitary operator u in A(G2 ) such that ψ(a) = uφ(a)u∗ ∀a ∈ A(G1 ), where φ is a standard regular injection. We say that ψ is inner equivalent (or, less precisely, unitarily equivalent) to φ when such a relationship holds. Remark. The unitary u belongs to A2 ∩ A∗2 . Indeed, by the spectral theorem, u = λ1 p1 + · · · + λr pr where pi is the spectral projection for the eigenspace for λi and each pi lies in the self-adjoint subalgebra. Exercises. (i) Prove that there are uncountably many inner conjugacy classes of embedding φ : T2 → T4 which are star-extendible and unital. (ii) Prove that there are only ﬁnitely many inner conjugacy classes of regular embeddings between two digraph algebras. 5.3. Direct limits We leave it to the reader to recall the deﬁnition of the direct limit algebra of a direct system and we now give some standard direct systems of triangular matrix algebras. The examples are all determined by regular star extendible inclusion maps. Limits of ﬁnite-dimensional operator algebras with respect to isometric irregular embeddings are less well understood and have not received much attention. The interested reader can ﬁnd some classiﬁcations in this direction in [24]. Standard limits. Let n2 = rn1 and let σ : Mn1 → Mn2 be the inclusion map such that (1)

(2)

(2)

(2)

σ(ei,j ) = ei,j + ei+n1 ,j+n1 + · · · + ei+(r−1)n1 ,j+(r−1)n1

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or equivalently, identifying Mn2 with Mr ⊗ Mn1 , σ(a) = I ⊗ a, so that, in block matrix terms, ⎡ ⎤ a 0 ... 0 ⎢ 0 a ⎥ ⎥. σ(a) = ⎢ ⎣ ⎦ 0 a Then σ(Tn1 ) ⊆ Tn2 , and so, repeating, we may construct the regular matricial algebra Aσ = lim(Tnk , σ), when nk |nk+1 for all k. Reﬁnement limits. Let n2 = rn1 and let ρ : Mn1 → Mn2 be the inclusion map such that (1)

(2)

(2)

(2)

ρ(ei,j ) = e(i−1)n1 +1,(j−1)n1 +1 + e(i−1)n1 +2,(j−1)n1 +2 + · · · + e(i−1)n1 +r,(j−1)n1 +r or equivalently, identifying Mnk with Mn1 ⊗Mr , ρ(a) = a⊗I, so that, ρ((aij )) is the inﬂated matrix (aij Ir ). Again, ρ(Tn1 ) ⊆ Tn2 , and for a sequence with nk |nk+1 for all k we can deﬁne the regular matricial algebra Aρ = lim(Tnk , ρ). This algebra is not isometrically isomorphic to the Aσ algebra for the same sequence (nk ) despite the fact that their generated C*-algebras are isomorphic. Countable total order limits. The standard embeddings σ and the reﬁnement embeddings ρ can also be alternated in which case one obtains a more general class of algebras, the so called alternation algebras. The three classes described correspond to a Cantor space product coordinate indexing by Z− , Z+ , and Z respectively. One can generalise this further to obtain strange triangular algebras whose background Cantor product is indexed over an arbitrary countable order. Moreover this countable order is an algebra isomorphism invariant. For further detail see [23].

Subalgebras of Graph C*-algebras

31

References [1] J. Cuntz, Simple C*-algebras generated by isometries, Comm. Math. Phys. 57 (1977), 173–185. [2] K.R. Davidson, Free Semigroup Algebras: a survey. Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), 209–240, Oper. Theory Adv. Appl. 129, Birkh¨ auser, Basel, 2001. [3] K.R. Davidson, C*-algebras by Example, Fields Institute Monograph Series, vol. 6, American Mathematical Society, 1996. [4] K.R. Davidson, E. Katsoulis, Nest representations of directed graph algebras, Proc. London Math. Soc., to appear. [5] K.R. Davidson, D.R. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras, Math. Ann. 311 (1998), 275–303. [6] K.R. Davidson, D.R. Pitts, Invariant subspaces and hyper-reﬂexivity for free semigroup algebras, Proc. London Math. Soc., 78 (1999), 401–430. [7] K.R. Davidson, E. Katsoulis and J. Peters, Meet-irreducible ideals and representations of limit algebras. J. Funct. Anal. 200 (2003), no. 1, 23–30. [8] A.P. Donsig and D.R. Pitts Coordinate Systems and Bounded Isomorphisms for Triangular Algebras. math.OA/0506627 66 pages. [9] A.P. Donsig and S.C. Power, The failure of approximate inner conjugacy for standard diagonals in regular limit algebras. Canad. Math. Bull. 39 (1996), no. 4, 420–428. [10] P.A. Haworth and S.C. Power, The uniqueness of AF diagonals in regular limit algebras. J. Funct. Anal. 195 (2002), no. 2, 207–229. [11] A. Hopenwasser, J. Peters and S.C. Power, Subalgebras of Graph C*-Algebras, New York J. Math. 11 (2005), 1–36. [12] A. Hopenwasser and S.C. Power,Limits of ﬁnite dimensional nest algebras, Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 1, 77–108. [13] E. Katsoulis, D.W. Kribs, Isomorphisms of algebras associated with directed graphs, Math. Ann., 330 (2004), 709–728. [14] D.W. Kribs, S.C. Power, Free semigroupoid algebras, J. Ramanujan Math. Soc. 19 (2004), 75–117. [15] D.W. Kribs, S.C. Power, Partly free algebras, Operator Theory: Advances and Applications, Birkh¨ auser-Verlag Basel/Switzerland, 149 (2004), 381–393. [16] D.W. Kribs and S.C. Power, The H ∞ algebras of higher rank graphs, Math. Proc. of the Royal Irish Acad., 106 (2006), 199–218. [17] P. Muhly, B. Solel, Tensor algebras, induced representations, and the Wold decomposition, Can. J. Math. 51 (4), 1999, 850–880. [18] A. L. Paterson, Groupoids, inverse semigroups and their operator algebras, Progress in Mathematics, Vol. 170, Birkh¨ auser, Boston, 1999. [19] G. Popescu, Multi-analytic operators on Fock spaces, Math. Ann. 303 (1995), 31–46. [20] G. Popescu, Noncommuting disc algebras and their representations, Proc. Amer. Math. Soc. 124 (1996), 2137–2148.

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[21] S.C. Power, Limit algebras: an introduction to subalgebras of C ∗ -algebras, Pitman Research Notes in Mathematics Series, vol. 278, (Longman Scientiﬁc & Technical, Harlow, 1992) CRC Press ISBN: 0582087813. [22] S.C. Power, Lexicographic semigroupoids, Ergodic Theory Dynam. Systems 16 (1996), no. 2, 365–377. [23] S.C. Power, Inﬁnite lexicographic products of triangular algebras, Bull. London Math. Soc. 27 (1995), no. 3, 273–277. [24] S.C. Power, Approximately ﬁnitely acting operator algebras, J. Funct. Anal. 189 (2002), no. 2, 409–468. [25] S.C. Power, Classifying higher rank analytic Toeplitz algebras, preprint 2006, preprint Archive no., math.OA/0604630. [26] Operator algebras associated with unitary commutation relations, preprint March 2007. [27] I. Raeburn, Graph algebras C.B.M.S lecture notes, vol. 103, Amer. Math. Soc., 2006. [28] J. Renault, A groupoid approach to C ∗ -algebras, Springer, Berlin, 1980. [29] B. Solel, You can see the arrows in a quiver algebra, J. Australian Math. Soc., 77 (2004), 111–122. [30] B. Solel, Representations of product systems over semigroups and dilations of commuting CP maps, J. Funct. Anal.235 (2006), 593–618. Stephen C. Power Department of Mathematics and Statistics Lancaster University England LA1 4YF e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 33–66 c 2008 Birkh¨ auser Verlag Basel/Switzerland

C ∗ -algebras and Asymptotic Spectral Theory Bernd Silbermann Abstract. The presented material is a slightly polished and extended version of lectures given at Lisbon, WOAT 2006. Three basic topics of numerical functional analysis are discussed: stability, fractality, and Fredholmness. It is further shown that these notions are corner stones in order to understand a few topics in asymptotic spectral theory: asymptotic behavior of singular values, ε-pseudospectra, norms. Four important examples are discussed: Finite sections of quasidiagonal operators, Toeplitz operators, band-dominated operators with almost periodic coeﬃcients, and general band-dominated operators. The elementary theory of C ∗ -algebras serves as the natural background of these topics. Mathematics Subject Classiﬁcation (2000). 47B35. Keywords. C∗-algebras, operator sequences, asymptotics, ﬁnite sections.

1. Introduction One goal of functional analysis is to solve equations with “inﬁnitely” many variables, and that of linear algebra to solve equations in ﬁnitely many variables. Numerical analysis builds a bridge between these ﬁelds. Functional numerical analysis is concerned with the theoretical foundation of numerical analysis. Given a bounded linear operator A acting on some Hilbert space H, that is A ∈ B(H), consider the equation Ag = h ,

(1.1)

where h ∈ H is given and g is to ﬁnd if this equation is supposed to be uniquely solvable. Even if the operator A is continuously invertible (and this will be assumed in what follows), it is as a rule impossible to compute the solution A−1 h. Then one tries to solve (1) approximately. For, one chooses a sequence (hn ) ⊂ H of elements which approximates the right-hand side h, and a sequence (An ) of operators

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which approximates the operator A, and one replaces (1.1) by the approximation equations An gn = hn , n = 1, 2, . . . (1.2) the solutions gn of which are sought in H (or in certain subspaces Hn of H) again. Approximation of h by hn means that h − hn H → 0 as n → ∞. It is tempting to suppose that the operators An also approximate A in the norm, but this assumption does not work in practice. The point is that usually A acts on an inﬁnite-dimensional space, whereas one will, of course, try to choose the An as acting on spaces of ﬁnite dimension, i.e., as ﬁnite matrices. But the only operators which can be approximated in norm by ﬁnite rank operators, are the compact ones. The kind of approximation which ﬁts much better to the purpose of numerical analysis is that of pointwise or strong convergence: the sequence (An ) converges strongly to the operator A if Ah − An hH → 0 for every h ∈ H (notation: s-lim An = A). We write s∗ -lim An = A, if s-lim An = A and s-lim A∗n = A∗ . Basic question: suppose that An is invertible for all n ≥ n0 . Does the sequence (gn ) of solutions of (1.2) converge to the solution g of (1.1)? The answer is NO! Let l2 := {(x0 , . . . , xn , . . . ) :

|xk |2 < ∞},

k∈Z+

˜l2 := {(. . . , x−n , . . . , x0 , . . . , xn , . . . ) :

|xk |2 < ∞}.

k∈Z

Example 1. Let ε = (εn ) be a sequence with εn > 0 and lim εn = 0 and Aε an inﬁnite matrix given by ⎞ ⎛ ε0 1 ⎟ ⎜ 1 ε0 0 ⎟ ⎜ ⎟ ⎜ 0 ε 1 1 ⎟ ⎜ ⎟ ⎜ 1 ε 0 1 ⎟ ⎜ (with respect to the standard basis of l2 ). ⎟ ⎜ 1 0 ε 2 ⎟ ⎜ ⎜ .. ⎟ ⎜ . ⎟ 1 ε 2 ⎠ ⎝ .. .. . . Consider Pn Aε Pn , Pn : l2 → l2 , (x0 , . . . , xn , xn+1 , . . . ) → (x0 , . . . , xn , 0, 0, . . . ). Then: ≤2 if n is odd (Pn Aε Pn )−1 Pn = ≥ ε−1 if n is even. n Thus, if n is even, then (Pn Aε Pn )−1 Pn → ∞, that is there is an h ∈ H such that gn = (Pn Aε Pn )−1 Pn h A−1 h = g by the Steinhaus-Banach Theorem.

C ∗ -algebras and Asymptotic Spectral Theory

35

Let us turn back to the general situation: Suppose that (Pn ) is a sequence of orthogonal projections such that s-lim Pn = I, and (An ) a bounded sequence of operators An : imPn → imPn with the property that there is an n0 such that An is invertible for n ≥ n0 and sup A−1 n Pn < ∞. If s-lim An Pn = A, then n≥n0

−1 A−1 x → 0 for every x ∈ H : n Pn x − A −1 −1 A−1 x ≤ A−1 x + Pn A−1 x − A−1 x n Pn x − A n Pn x − Pn A −1 x + Pn A−1 x − A−1 x → 0 . ≤ A−1 n Pn x − An Pn A ∗ Remark. Suppose again that sup A−1 n Pn < ∞ and s -lim An Pn = A. Then A n≥n0

is invertible. Indeed, we have An Pn x ≥ CPn x

(n ≥ n0 , C > 0) .

Passing to limits gives Ax ≥ Cx . Thus im A = im A and ker A = {0}. Using A∗n Pn x ≥ CPn x we get in the same manner A∗ x ≥ Cx . Thus A∗ = im A∗ and ker A∗ = {0}. Deﬁnition 1. A sequence of operators An ∈ B(imPn ) is called stable if there exists a number n0 such that the operators An are invertible for every n ≥ n0 and if the norms of their inverses are uniformly bounded: sup A−1 n Pn < ∞ .

n≥n0

The above discussion shows the crucial role of stability in analysis. How to prove stability? There is no general idea. In most cases it is very complicated. Easy cases: • A = B + iS, B positive and S selfadjoint, • A = I + T , T compact, and An = Pn APn , where (Pn ) is a sequence of orthogonal projections with slim Pn = I. Exercise: prove that in both cases the sequence (An ) is stable.

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We will show that the stability problem can frequently be tackled by the help of C ∗ -algebra techniques. Recall that a complex Banach algebra is called C ∗ algebra if there is an involution a → a∗ such that aa∗ = a2 . Given two C ∗ algebras A and B, a ∗-homomorphism ϕ : A → B is a continuous homomorphism such that ϕ(a∗ ) = ϕ(a)∗ for all a ∈ A.

2. Algebraization of stability Let H be a (separable) Hilbert space and (Ln ) be a sequence of orthoprojections on H with s-lim Ln = I. Deﬁnition 2. Let F be the set of all sequences (An )∞ n=0 of operators An ∈ B (im Ln ) which are uniformly bounded: sup An Ln < ∞ . n≥0

The natural operations (An ) + (Bn ) := (An + Bn ), (An )(Bn ) := (An Bn ), λ(An ) := (λAn ) , (An )∗ := (A∗n ) make F to an algebra with involution. Proposition 1. F is a C ∗ -algebra (prove it). We are mainly interested in the asymptotic behavior of the sequences belonging to F . This means that sequences which diﬀer in a ﬁnite number of entries only will have the same asymptotic behavior, and therefore can be identiﬁed. For this goal we introduce the set G of all sequences (Gn ) in F with lim Gn Ln = 0. n→∞

Proposition 2. G is a closed ideal in F (prove it). The following theorem reveals a perfect frame to study stability problems in an algebraic way. Theorem 1. (A. Kozak) A sequence (An ) ∈ F is stable ⇔ the coset (An ) + G is invertible in the quotient algebra F /G. Proof. ⇒: If (An ) is stable, then (A−1 n )n≥n0 is bounded for some suﬃciently large n0 by deﬁnition. We make (A−1 ) n n≥n0 to a bounded sequence −1 , A (B0 , B1 , . . . , Bn0 −1 , A−1 n0 n0 +1 , . . . ) in F by freely choosing operators Bi ∈ B(imLi ). It is evident that this sequence is an inverse of (An ) modulo G. ⇐ Let conversely, (An )+ G be invertible in F /G. Then there are sequences (Bn ) ∈ F as well as (Gn ) and (Hn ) in G such that An Bn = Ln +Gn , Bn An = Ln +Hn . If n is large enough, then Gn < 12 , Hn < 12 , and a Neumann series argument yields the invertibility of Ln + Gn and Ln + Hn as well as the uniform boundedness of their inverses by 2. Hence, Bn (Ln +Gn )−1 , (Ln +Hn )−1 Bn are uniformly bounded. Thus, the operators An are invertible for all suﬃciently large n, and their inverses are uniformly bounded.

C ∗ -algebras and Asymptotic Spectral Theory

37

Proposition 3. For all (An ) ∈ F, (An + G)F /G = lim sup An Ln n→∞

(2.1)

(where lim sup stands for lim superior).

Proof. Exercise.

Formula (2.1) gives raise to ask if there are interesting sequences in F for which lim sup in (2.1) can be replaced by lim. This question is important in order to prove that the condition numbers of a stable sequence converge. Recall, that the condition number (cond A) for an invertible matrix (operator) A is deﬁned by cond A := A A−1 (for computational purposes: cond A should be small). The right tool to study this and related questions is another fundamental notion of numerical analysis – that of a fractal sequence, which we are now going to discuss. It is not important in this place that the elements of the sequences under consideration are operators. So we will use slightly generalized deﬁnitions of the C ∗ -algebras F and G, namely, given unital C ∗ -algebras Cn , n = 0, 1, 2, . . . , with identity elements en , let F stand for the set of all bounded sequences (c0 , c1 , . . . ) with cn ∈ Cn , and let G refer to the set of all sequences (c0 , c1 , . . . ) in F with cn → 0 as n → ∞. Deﬁning elementwise algebraic operations and an elementwise involution, and taking the supremum norm, we make F to a C ∗ -algebra and G to a closed ideal of F . Thus, F is the product of the C ∗ -algebras Cn , and G-their restricted product. Given a strongly monotonically increasing sequence η : Z+ → Z+ , let Fη and Gη denote the product and the restricted product of the C ∗ -algebras Cη(0) , Cη(1) , . . . , respectively, and let Rη stand for the restriction mapping Rη : F → Fη , (an ) → (aη(n) ). The mapping Rη is a ∗-homomorphism from F onto Fη . Further, given a C ∗ -subalgebra A of F , let Aη refer to the image of A under Rη . By the ﬁrst isomorphy theorem for C ∗ -algebras ([7], Theorem 1.45), Aη actually is a C ∗ -algebra. Deﬁnition 3. Let A be a C ∗ -subalgebra of F . (a) A ∗-homomorphism W : A → B of A into a C ∗ -algebra B is fractal if for every strongly monotonically increasing sequence η, there is a ∗-homomorphism Wη : Aη → B such that W = Wη Rη . (b) The algebra A is fractal, if the canonical homomorphism π : A → A/A ∩ G is fractal. (c) A sequence (an ) ∈ F is fractal, if the smallest C ∗ -subalgebra of F containing (an ), is fractal. Roughly spoken: given a subsequence (aη(n) ) of a sequence (an ) which belongs to a fractal algebra A, it is possible to reconstruct the original sequence (an ) from its subsequence modulo sequences in A ∩ G.

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Consequences: • (aη(n) ) ∈ Gη ⇒ (an ) ∈ G ([7], Theorem 1.66),

(2.2)

• (aη(n) ) stable ⇒ (an ) stable

(2.3)

(see Theorem 4 below). ∗

Theorem 2. ([7], Theorem 1.7.1) Let A be a fractal C -subalgebra of F . If (an ) ∈ A, then the limit lim an exists and is equal to (an ) + G. |an |2 < ∞} and the bounded Example 2. Consider again l2 := {(an )n∈Z+ : n∈Z+

linear operators Pn , Rn : l2 → l2 given by (ak ) → (a0 , a1 , . . . , an , 0, 0, 0, . . . ) , (ak ) → (an , an−1 , . . . a0 , 0, 0, 0, · · · ) , respectively. Let Cn = B(imPn ), and let F W refer to the set of all sequences (An ) ∈ F ˜ (An ) := s-lim Rn An Rn for which the strong limits W (An ) := s-lim An Pn and W ∗ ∗ ˜ as well as the strong limits W (An ) and W (An ) exist. The set F W actually forms a C ∗ -subalgebra of F (prove it or compare the proof of Theorem 1.18 (a) in [7]). ˜ : F W → B(l2 ) turn out to be fractal: given a strongly The ∗-homomorphism W, W ˜ η : Fη → B(l2 ) via monotonically increasing sequence η, we can deﬁne Wη , W Wη (Aη(n) ) := s- lim Aη(n) Pη(n) and ˜ η (Aη(n) ) := s- lim Rη(n) Aη(n) Rη(n) . W ˜ =W ˜ η Rη . The algebra F W is not fractal: consider Then, obviously, W = Wη Rη , W the sequence (An ) ∈ F, where A2n+1 = 0 and A2n = diag (0, . . . , 0, 1, 0, . . . , 0), where the 1 stands in the center of this diagonal matrix. It is easily seen, that ˜ (An ) = 0, but (An ) ∈ (An ) ∈ F W , W (An ) = 0, W / G(⊂ F W ). For the special choice η(n) = 2n + 1 one obtains Rη (An ) = (A2n+1 ) ∈ Gη . By (2.2) F W cannot be fractal. One the other hand, F W contains interesting fractal subalgebras as we will see later on.

3. Asymptotic behavior Given a sequence (An ) ∈ F one can ask how the spectra (ε-pseudospectra) of the entries develop. We need some deﬁnitions. Let (Mn )∞ n=1 be a set sequence with values in the set of all subsets of the complex plane. For instance, if (An ) ∈ F, then the mapping n → sp An is a set sequence in this sense. Deﬁnition 4. Let (Mn )∞ n=1 be a set sequence. The partial limiting set or limes superior lim sup Mn (resp. the uniform limiting set or limes inferior lim inf Mn ) of the sequence (Mn ) consists of all points m ∈ C which are a partial limit (resp. limit) of a sequence (mn ) of points mn ∈ Mn (partial limit of a sequence (mn ) is by deﬁnition a limit of some subsequence of (mn )).

C ∗ -algebras and Asymptotic Spectral Theory

39

Observe that the partial limiting set lim sup Mn is non-empty if inﬁnitely many of the Mn are non-empty and if Mn is bounded, whereas the uniform n

limiting set can be empty even under these restrictions as the trivial example Mn = {(−1)n } shows. Let CC denote the set of all non-empty and compact subsets of C. The Hausdorﬀ distance of two elements A and B of CC is deﬁned by h(A, B) := max max dist (a, B), max dist (b, A) , a∈A

b∈B

where dist (a, B) = min |a − b|. The function h is actually a metric on CC . We b∈B

denote limits with respect to this metric by h-lim. Proposition 4. ([7], Proposition 3.6) Let (Mn ) be a set sequence taking values in CC . Then lim sup Mn and lim inf Mn coincide if and only if the sequence (Mn ) is h-convergent. In that case lim sup Mn = lim inf Mn = h- lim Mn . Example 3. Let V : l2 → l2 be the shift operator acting by (a0 , a1 , a2 , . . . ) → (0, a0 , a1 , a2 , . . . ) and consider (Pn V Pn ). It is easy to see that the matrix representation of Pn V Pn with respect to the standard basis of im Pn equals ⎞ ⎛ 0 ⎟ ⎜ 1 0 ⎟ ⎜ ⎟ ⎜ 1 0 0 ⎟. ⎜ ⎟ ⎜ 0 1 0 ⎟ ⎜ ⎠ ⎝ 1 0 1 0 Hence, sp Pn V Pn = {0} for all n and lim inf sp Pn V Pn = lim sup sp Pn V Pn = {0}, but sp V = {z ∈ C : z ≤ 1} ⊂ spF /G ((Pn V Pn ) + G). Therefore h-lim Pn V Pn exists but this limits has almost nothing to do with sp V . What is the reason for this unpleasant fact? One can prove that for (an ) ∈ F a point s ∈ C belongs to the partial limiting set lim sup sp an if and only if the sequence (an −sen ) is not spectrally stable (Theorem 3.17 in [7]). (A sequence (an ) is spectrally stable if its entries an are invertible for suﬃciently large n and if the spectral radii ρ(a−1 n ) of their inverses are uniformly bounded.) Spectral stability is a very involved notion and not much is known. We accomplish this discussion with Theorem 3. ([7], Theorem 3.19) Let Cn = Cn×n and (An ) ∈ F. Then lim sup sp (An + Cn ) = spF /G ((An ) + G). (Cn )∈G

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One conclusion: Spectral stability is very sensitive with respect to perturbations from G (contrary to stability). These diﬃculties disappear if we restrict our attention to sequences for which stability and spectral stability coincide. Corollary 1. ([7], Corollary 3.18) If (an ) ∈ F is a sequence of normal elements, then lim sup sp an = spF /G ((an ) + G). For fractal algebras we get reﬁnements. Theorem 4. ([7], Theorem 3.20) Let A be a fractal C ∗ -subalgebra of F which contains the identity. (a) A sequence (an ) ∈ A is stable if and only if it possesses a stable (inﬁnite) subsequence. (b) If (an ) ∈ A is normal, then lim sup sp an = lim inf sp an = h-lim sp an . (c) If (an ) ∈ A is normal, then the limit lim ρ(an ) exists and is equal to ρ((an ) + G) (ρ-spectral radius). Let us shortly discuss limiting sets of singular values (because of their importance in numerical analysis). Let B be a unital C ∗ -algebra and a ∈ B. The set (a) of the singular values of a is deﬁned to be {λ ∈ R+ : λ2 ∈ sp (a∗ a)}. Since the determination of the singular values is equivalent to the determination of the spectrum of a self-adjoint element, the previous results have the following evident analogues for singular value sets. Theorem 5. If (an ) ∈ F, then lim sup (an ) = ((an ) + G). Theorem 6. If A is a fractal C ∗ -subalgebra of F containing the identity and if (an ) ∈ A, then lim sup (an ) = lim inf (an ) = h- lim (an ) . The last topic in this section is ε-pseudospectra. A computer working with ﬁnite accuracy cannot distinguish between a noninvertible matrix and an invertible matrix the inverse of which has a very large norm. This suggests the following deﬁnition reﬂecting ﬁnite accuracy. Deﬁnition 5. Let B be a C ∗ -algebra with identity e and let ε be a positive constant. An element a ∈ B is ε-invertible if it is invertible and a−1 < 1ε . The ε-pseudospectrum spε (a) of a consists of all λ ∈ C for which a − λe is not εinvertible. It is easily seen that ε-invertible elements of a C ∗ -algebra form an open set, and that ε-pseudospectra are compact and non-empty subsets of C. The following theorem provides an equivalent description of the ε-pseudospectrum which oﬀers a way for numerical computations at least for (ﬁnite) matrices.

C ∗ -algebras and Asymptotic Spectral Theory

41

Theorem 7. ([7], Theorem 3.27) Let B be a unital C ∗ -algebra and ε > 0. Then, for every a ∈ B, the ε-pseudospectrum is equal to spε (a) = sp (a + p). p∈B p≤ε

Let us still remark that (unital) C ∗ -algebras are also inverse closed with respect to ε-invertibility. What about limiting sets of ε-pseudospectra? Theorem 8. ([7], Theorem 3.31) Let (an ) ∈ F and ε > 0. Then /G ((an ) + G). lim sup spCε n (an ) = spF ε

The proof of Theorem 8 is based on the following result. Proposition 5. (Daniluk) ([3], Theorem 3.14) Let B be a C ∗ -algebra with identity e, let a ∈ B, and suppose a − λe is invertible for all λ in some open subset U of the complex plane. If (a − λe)−1 ≤ C for all λ ∈ U, then (a − λe)−1 < C for all λ ∈ U. In other words: the analytic function U → B, λ → (a − λe)−1 satisﬁes the maximum principle. This is a surprising fact since – in contrast to complex-valued analytic functions – the maximum principlefails in general for operator-valued λ 0 analytic functions (consider C → C2×2 , λ → ). 0 1 It is an open question for which Banach algebras Daniluk’s result is true (one particular answer is in [3], Theorem 7.15). In case A is a fractal C ∗ -subalgebra of F we have the following reﬁnement of Theorem 8: /G ((an ) + G) . h- lim spCε n (an ) = spF ε

4. First applications I. Quasidiagonal operators and their ﬁnite sections Recall that a bounded linear operator T on a separable (complex) Hilbert space is said to be quasidiagonal if there exists a sequence (Pn )n∈N of ﬁnite rank orthogonal projections such that s-lim Pn = I and which asymptotically commute with T , that is [T, Pn ] := T Pn − Pn T → 0 as n → ∞ . In particular, every selfadjoint or even normal operator is quasidiagonal as well as their perturbations by compact operators. However it is by no means trivial to single out a related sequence (Pn ). For instance, for multiplication operators in periodic Sobolev spaces H λ related sequences can explicitly be given: these are orthogonal projections on some spline spaces. Let T be quasidiagonal with respect to (Pn ) = (Pn )n∈N . Consider the C ∗-subalgebra F l of F , the last one deﬁned by help of (Pn ), consisting of all sequences of

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F for which s∗ - lim An Pn exist. It is not hard to prove that J := {(Pn KPn )+(Cn ): K-compact, (Cn ) ∈ G} forms a two-sided closed ideal in F l (but not in F !) Let C(Pn ) (T ) denote the smallest C ∗ -subalgebra of F l containing the sequences (Pn T Pn ), (Pn ), and the ideal J. Proposition 6. The quotient algebra C(Pn ) (T )/G is isometrically isomorphic to the smallest C ∗ -subalgebra C(T ) of B(H) containing T, I, and all compact operators. This isomorphism is given by the quotient map induced via s-lim An Pn ((An ) ∈ C(Pn ) (T )). Sketch of the proof. Suppose s-lim An Pn =: A is invertible. Then A−1 ∈ C(T ), and since every element in C(T ) is quasidiagonal, A−1 also owns this property, and Pn − Pn APn A−1 Pn = Pn AA−1 Pn − Pn APn A−1 Pn = Pn (Pn A − APn )A−1 Pn → 0 . Hence, (Pn APn ) is stable. This means that (Pn APn ) + G is invertible if and only if s-lim Pn APn is invertible. Now it is suﬃcient to prove that Pn APn − An ∈ G. For, it is suﬃcient to show this for the special case An = Pn B1 Pn B2 Pn . We have Pn B1 B2 Pn − Pn B1 Pn B2 Pn = Pn (Pn B1 − B1 Pn )B2 Pn → 0 as n → ∞. Corollary 2. A sequence (An ) ∈ C(Pn ) (T ) is stable if and only if s-lim An is invertible. Moreover, C(Pn ) (T ) is fractal. Now it is evident that the theory of Section 3 applies. Proposition 7. Let (An ) ∈ C(Pn ) (T ) and A = s-lim An . Then (a) lim An = s-lim An Pn . (b) lim inf spε An = lim sup spε An = spε (s-lim An ) (ε > 0). (c) If (An ) is normal, then (d) lim inf

lim inf sp An = lim sup sp An = sp (s- lim An ) . (An ) = lim sup (An ) = (s-lim An ).

In the papers [5], [6] Nathaniel Brown proposed further reﬁnements into two directions: speed of convergence and how to choose the sequence (Pn ) of orthoprojections in some special cases such as quasidiagonal unilateral band operators, bilateral band operators or operators in irrational rotation algebras. Remark. If (an ) ∈ F l is stable and s∗ -lim an = A , A+K invertible and K compact, then (an + Pn KPn ) is stable (this sequence equals (an )(Pn + Pn a−1 n Pn KPn ) for n large enough). II. Toeplitz operators and their ﬁnite sections Let a ∈ L∞ (T) and denote by ak the kth Fourier coeﬃcient of a: ak =

1 2π

2π 0

a(eiθ )e−ikθ dθ , k ∈ Z ,

where

T := {z ∈ C : |z| = 1}.

C ∗ -algebras and Asymptotic Spectral Theory

43

Then the Laurent operator L(a) on ˜ l2 , the Toeplitz operator T (a) on l2 , and 2 the Hankel operator H(a) on l are given via their matrix representation with respect to the standard bases of ˜l2 and l2 by L(a) = T (a) =

(ak−j )∞ k,j=−∞ , ∞ (ak−j )∞ k,j=0 , H(a) = (aj+k+1 )k,j=0 .

Here is a list of elementary properties of these operators (see any textbook on Toeplitz operators). • If a ∈ L∞ (T), then the Laurent operator L(a) is bounded on ˜l2 . • (Brown/Halmos) If a ∈ L∞ (T), then the Toeplitz operator T (a) is bounded on l2 , and T (a) = a∞ . • (Nehari) If a ∈ L∞ (T), then H(a) is bounded on l2 , and ∞

H(a) = distL∞ (T) (a, H ). • T (ab) = T (a)T (b) + H(a)H(˜b), where ˜b(t) := b( 1t ). • T (a)∗ = T (a). • (Coburn) Let a ∈ L∞ (T) \ {0}. Then at least one of the spaces ker T (a) and l2 /imT (a) consists of the zero element only. Proposition 8. ([3], Chapter 1) (i) Let a ∈ C(T). The Toeplitz operator T (a) is Fredholm on l2 if and only if 0 ∈ / a(T). In this case, ind T (a) = −wind a, where wind a refers to the winding number of the curve a(T), provided with the orientation inherited by the usual counter-clockwise orientation of the unit circle, around the origin. / a(T) (ii) Let a ∈ C(T). The Toeplitz operator is invertible on l2 if and only if 0 ∈ and wind a = 0. (iii) Let a ∈ C(T). Then H(a) is compact on l2 . (iv) The smallest C ∗ -subalgebra T (C) of B(l2 ) containing all Toeplitz operators with continuous generating functions, decomposes as 2 ˙ T (C) = {T (a) : a ∈ C(T)}+K(l ),

where K(l2 ) stands for the (closed) ideal of all compact operators. Now let us turn to the ﬁnite section method for Toeplitz operators (with continuous generating function). The ﬁrst question is about the stability of the sequence (Pn T (a)Pn ), where Pn : l2 → l2 is the projection deﬁned by (a0 , a1 , . . . , an , an+1 , . . . ) → (a0 , . . . , an , 0, 0, . . . ) . This problem was investigated by many people. G. Baxter, 63 : (Pn T (a)Pn ) stable in l1 if and only if T (a) is invertible (a ∈ W, 0 ∈ / a(T), wind a = 0, where W stands for the Wiener algebra). I. Gohberg, I. Feldmann, 65 : (Pn T (a)Pn ) stable in l2 if and only if T (a) is invertible.

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Later on related results for classes of discontinuous generating functions were achieved: QC, C +H ∞ , P C, P QC. For instance QC stands for the class of all quasi continuous functions, P QC for the smallest closed subalgebra of L∞ (T) containing the algebra P C of all piecewise continuous functions and QC. Treil, 87 : There are generating functions a with only one point of discontinuity such that T (a) is invertible but (Pn T (a)Pn ) is not stable. Recall the deﬁnition of the algebra F W (Example 2): A sequence (An ) ∈ F belongs to F W , if and only if the strong limits W (An ) := s˜ (An ) = s-lim Rn An Rn as well as W (A∗n ) , W ˜ (A∗n ) exist. Because of lim An Pn , W ∞ Rn T (a)Rn = Pn T (˜ a)Pn (a ∈ L (T)) it is easy to see that (Pn T (a)Pn ) ∈ F W . Moreover, Rn KRn tends strongly to zero for every compact operator due to the weak convergence of (Rn ) to zero. Hence, the smallest C ∗ -subalgebra S(C) in F containing all sequences (Pn T (a)Pn ), a ∈ C(T), is actually contained in F W . Our next goal is to describe the structure of the algebra S(C). For, we need Widom’s identity Pn T (ab)Pn = Pn T (a)Pn T (b)Pn + Pn H(a)H(˜b)Pn + Rn H(˜ a)H(b)Rn (prove it). The collection JW := {(Pn KPn + Rn LRn + Cn ) : K, L compact, (Cn ) ∈ G} forms a closed two-sided ideal in F W . Theorem 9. ([7], Theorem 1.5.3) JW ⊂ S(C). Moreover, each element (An ) ∈ S(C) can uniquely be written as An = Pn T (a)Pn + Pn KPn + Rn LRn + Cn , where K, L are compact operators, (Cn ) ∈ G. Using this representation, it is evident that ˜ (An ) = T (˜ a) + L , W (An ) = T (a) + K , W ˜ = G. and ker W ∩ ker W ˜ : S(C) → B(l2 ) and glue them toNow take the ∗-homomorphisms W, W 0 gether to obtain a ∗-homomorphism smb : S(C) → B(l2 ) × B(l2 ), ˜ (An )) . (An ) → (W (An ), W Furthermore, it is clear that ker smb0 = G. Thus, the quotient homomorphism smb : S(C)/G → B(l2 ) × B(l2 ) is correctly deﬁned and is injective. Notice that an injective ∗-homomorphism is isometric.

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45

Theorem 10. (i) The map smb is a ∗-isomorphism from S(C)/G onto the C ∗ -subalgebra of ˜ (An )) with (An ) running B(l2 ) × B(l2 ) which consists of all pairs (W (An ), W through S(C). ˜ (An ) are invertible oper(ii) (An ) ∈ S(C) is stable if and only if W (An ) and W ators. (iii) S(C) is fractal. Now it is clear that the theory of Section 3 applies. Theorem 11. ˜ (An )}. (a) lim An = max{W (An ) , W ˜ (An ). (b) lim inf spε An = lim sup spε An = spε W (An ) ∪ spε W ˜ (An ). a) = spε W If (An ) = (Pn T (a)Pn ), then spε W (An ) = spε T (a) = spε T (˜ ˜ (An ). (c) If (An ) is normal, then lim inf spAn = lim sup sp An = sp W (An )∪sp W ˜ (d) lim inf (An ) = lim sup (An ) = (W (An )) ∪ (W (An )).

5. Fredholm sequences Now we are going to introduce a third fundamental notion, namely that one of Fredholm sequences. First we introduce Fredholm sequences in some restricted form and ﬁnally in full generality. We introduce C ∗ -subalgebras of F which are generalizations of the algebras l F and F W and which give raise to consider Fredholm sequences. Let H be an inﬁnite-dimensional Hilbert space and (Ln ) be a sequence of orthogonal projections such that Ln → I strongly as n → ∞. The related C ∗ -algebra of all bounded sequences is again denoted by F . We shall assume that all projections Ln are ﬁnite rank operators. Let T be a (possibly inﬁnite) index set and suppose that, for every t ∈ T , we are given an inﬁnite-dimensional Hilbert space H t with identity operator I t as well as a sequence (Ent ) of partial isometries Ent : H t → H such that • the initial projections Ltn of Ent converge strongly to I t as n → ∞, • the range projection of Ent is Ln , • the separation condition ∗

(Ens ) Ent → 0 weakly as n → ∞

(5.1)

holds for every s, t ∈ T with s = t. (Recall that an operator E : H → H is a partial isometry if EE ∗ E = E and that E ∗ E and EE ∗ are orthogonal projections which are called the initial t and the range projections of E, respectively). For brevity, write E−n instead t ∗ t t of (En ) , and set Hn := im Ln and Hn := im Ln . Let F T stand for the set of all sequences (An ) ∈ F for which the strong limits t t s-limn→∞ E−n An Ent and s- lim(E−n An Ent )∗

46

B. Silbermann

exist for every t ∈ T , and deﬁne mappings W t : F T → B(H t ) by W t (An ) := t s- limn→∞ E−n An Ent . It is easily seen that F T is a C ∗ -subalgebra of F which contains the identity, and that the W t are ∗-homomorphisms. The separation condition (5.1) ensures that, for every t ∈ T and every comt ) belongs to the algebra F T , pact operator K t ∈ K(H t ), the sequence (Ent K t E−n and that for all s ∈ T t if s = t K t W s (Ent K t E−n )= . (5.2) 0 if s = t Conversely, (5.2) implies (5.1). Moreover, the ideal G belongs to F T . So we t can introduce the smallest closed ideal J T which contains all sequences (Ent K t E−n ) t t with t ∈ T and K ∈ K(H ) as well as all sequences (Gn ) ∈ G. Remark. The algebra F W provides an example of this type. Indeed T consists only ˜ , and of two points, say 1 and 2. Then W 1 = W, W 2 = W J1 J2

= {(Pn KPn + Cn ) : K compact, (Cn ) ∈ G} , = {(Rn LRn + Cn ) : L compact, (Cn ) ∈ G} .

The ideal JT is exactly the earlier introduced ideal JW . The separation condition (5.1) is obviously fulﬁlled (recall that Rn tends weakly to zero). There are examples which show that indeed inﬁnite index sets T are needed ([7], 4.5.1–4.5.2, for instance). Theorem 12. ([7], Theorem 6.1) (a) A sequence (An ) ∈ F T is stable if and only if the operators W t (An ) are invertible in B(H t ) for every t ∈ T and if the coset (An ) + J T is invertible in the quotient algebra F T /J T . (b) If (An ) ∈ F T is a sequence with invertible coset (An ) + J T , then all operators W T (An ) are Fredholm on H t , and the number of the non-invertible operators among the W t (An ) is ﬁnite. Notice that this theorem can be used to give a diﬀerent proof of Theorem 10, (ii). Deﬁnition 6. (a) A sequence (An ) ∈ F T is called Fredholm if the coset (An ) + J T is invertible. (b) If the sequence (An ) ∈ F T is Fredholm, then its nullity α(An ), deﬁciency β(An ) and index ind (An ) are deﬁned by α(An ) := dim ker W t (An ), β(An ) := dim coker W t (An ), t∈T

t∈T

and ind (An ) := α(An ) − β(An ). It is a triviality to carry over well-known properties of Fredholm operators to Fredholm sequences.

C ∗ -algebras and Asymptotic Spectral Theory

47

Remark. As we will see later on, this notion of Fredholm sequence depends on the underlying algebra F T . We shall also see that Fredholmness of a sequence in the sense of Deﬁnition 6 implies its Fredholmness in a general sense which has still to be deﬁned. Let (An ) ∈ F be arbitrary. We order the singular values of An as follows (ln = rank Ln ): 0 ≤ s1 (An ) ≤ · · · ≤ sln (An )(= An ) . For the sake of convenience let us also put s0 (An ) = 0. Recall that usually the singular values are ordered in the reverse manner. Deﬁnition 7. We say that (An ) ∈ F has the k-splitting property if there is a k ∈ Z+ such that lim sk (An ) = 0 , n→∞

while the remaining ln − k singular values stay away from zero, that is sk+1 (An ) ≥ δ > 0 for n large enough. The number k is also called the splitting number. Notice if (An ) has the 0-splitting property then (An ) is stable (hint: if s1 (An ) = 0 then An is invertible and A−1 = s1 (An )−1 ). Theorem 13. (a) Let (An ) ∈ F T be Fredholm. Then (An ) is subject to the k-splitting property t Ht t En Pker W t (An ) E−n ). with k = α(An ) and sα(An ) (An ) ≤ An ( t∈T

(b) If for (An ) ∈ F T there is at least one t1 ∈ T such that W t1 (An ) is not Fredholm, then lim sl (An ) = 0 for all l ∈ Z+ . n→∞

Assertions (a) and (b) can be proved slightly modifying the idea of the proof of Theorem 6.11 and using Theorem 6.67 in [7]. A complete proof of Theorem 13 is contained in [17]. We present here the proof of Theorem 13, (a), for the special case F l (we use here (Pn ) instead of (Ln ) in accordance with the notations of 4.I.). Proof. We shall make use of the following alternative description of the singular values (as approximation numbers): sj (An ) :=

min

B∈Flln

An − B ,

n−j

ln where Fm denotes the collection of all ln × ln -matrices of rank at most m. Let Rn be the orthoprojection onto im (Pn Pker A Pn ), where A = s-lim An Pn . It is easy to check that im Rn = im Pn Pker A Pn ,

rank Rn = rank Pn Pker A Pn = rank Pker A = dim ker A =: k for n large enough, and Rn − Pn Pker A Pn → 0 as n → ∞ .

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Consequently, ||An Rn || → 0 as n → ∞, and (An Rn ) ∈ G. Consider the sequence (Bn ) ∈ F l , Bn := A∗n An (Pn −Rn )+Pn Pker A Pn . Obviously, this sequence is also Fredholm and s-lim Bn Pn = A∗ A + Pker A is invertible. Then (Bn ) is stable by Theorem 12, (a). Since rank (Pn − Rn ) = ln − k for n large enough we get for those n sk (An ) ≤ || An − An A∗n An (Pn − Rn )Bn−1 Pn || ≤ || (An Bn − An A∗n An (Pn − Rn )) Pn || ||Bn−1 Pn || ≤ ||Bn−1 Pn || ||An Pn Pker A Pn || . Since (Bn ) is stable, there exists for n large enough a constant C with ||Bn−1 Pn || ≤ C. Thus we have sk (An ) ≤ C||An Pn Pker A Pn || → 0 as n → ∞ . Now consider sk+1 (An ). By using the well-known inequality sk+1 (A∗n An ) ≤ ≤ sk+1 (An )||A∗n || and that ||A∗n || is bounded (recall that A∗n Pn converges strongly to A∗ = 0) it has to be shown that sk+1 (A∗ An ) is bounded away from zero (n large enough). We have sk+1 (A∗n An ) = = ≥ =

min

|| (A∗n An − B) Pn ||

min

|| ((A∗n An + Pn Pker A Pn ) − B − Pn Pker A Pn ) Pn ||

n B∈Flln −k−1

n B∈Flln −k−1

min

n B∈Flln −1

|| ((A∗n An + Pn Pker A Pn ) − B) Pn ||

s1 (A∗n An + Pn Pker A Pn ) ≥ δ > 0

for n large enough since (A∗n An + Pn Pker A Pn ) is stable, and we are done.

Corollary 3. If (An ) ∈ F T is Fredholm, then ind (An ) = 0 . Proof. One has only to use that the matrices A∗n An and An A∗n are unitarily equivalent. This shows that the splitting numbers of (An ) and (A∗n ) coincide. Example 4. The sequence (Pn V Pn ) belongs to both algebras F l and F W . This sequence is Fredholm in F W but not in F l . If it would be Fredholm in F l then ind V = 0; but ind V = −1. Theorem 13 has remarkable applications. Let us mention some simple results: • If T (a)(a ∈ C(T)) is Fredholm, then the Moore-Penrose inverses (Pn T (a)Pn )+ converge strongly to T (a)+ if and only if dim ker Pn T (a)Pn = α(Pn T (a)Pn ) for n large enough.

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A deeper study of this problem is presented in [3], Chapter 4. • Let T (a) (a ∈ C(T)) be Fredholm and K be compact. Since (Pn (T (a) + K)Pn ) is subject to the splitting property with splitting number α(Pn (T (a)+ a) and dim ker T (˜ a) is known by K)Pn ) = dim ker(T (a) + K) + dim ker T (˜ Coburns Theorem, dim ker(T (a)+K) can be found numerically (in principle). • If T (a) (a ∈ C(T)) is Fredholm and a is smooth then sα (Pn T (a)P n )βtends fast |k| |ak | < ∞ to zero. For instance, if the Fourier coeﬃcients ak of a fulﬁll k∈Z

for some β > 0 then sα (Pn T (a)Pn ) = O(1/nβ ). • It was mentioned before that multiplication operators Ma with continuous functions a are quasidiagonal in L2 (T). The corresponding sequence of ﬁnitedimensional projections can be taken as orthogonal projections on some spline spaces. To be more precise let T := {|z| = 1} be parametrized be ϕ : [0, 1] → T , ϕ(t) = e2πit . A sequence of partitions (∆k )k∈N , ∆k := {σ0k , . . . , σnk k } , 0 = k −σj ) → σ0k < σ1k < · · · < σnk k = 1, is said to be admissible if h∆k := max(σj+1 δ 0 as k tends to inﬁnity. We denote by S˜ (∆k ) the space of all ψ ∈ C(T) such that ψ ◦ ϕ is (δ − 1) times continuously diﬀerentiable and the restriction of k ) is a polynomial of degree ≤ δ (smoothest ψ ◦ ϕ to each interval (σjk , σj+1 splines). Let P∆k denote the orthogonal projections of L2 (T) onto S˜δ (∆k ). Then (see [8], Section 2.14) (I − P∆k )f P∆k → 0 ,

P∆k f (I − P∆k ) → 0

as k → ∞, where – · stands for the operator norm in L2 (T), – f is continuous, – (∆k ) is admissible. Consider the singular integral operator A with continuous coeﬃcients: 1 g(τ ) dτ Ag = ag + bSg , where (Sg)(t) := πi τ −t T

(Recall that this integral exists for g ∈ L (T) almost everywhere in the sense of Cauchy’s principal value). A singular integral operator A is called strongly (locally) elliptic, if there is a continuous function c on T, a linear operator T with T < 1 and a compact operator K such that 2

A = c(I + T ) + K , c(t) = 0 for all t ∈ T . It follows that A is Fredholm with index 0 (even invertible). It is well known that A is strongly elliptic if and only if a(t) + λb(t) = 0 ∀t ∈ T und ∀λ ∈ [−1, 1] . If A is strongly elliptic and (∆k ) admissible then (P∆k AP∆k ) is stable

50

B. Silbermann (use P∆k AP∆k = P∆k cIP∆k (I + T )P∆k + P∆k KP∆k + C∆k , C∆k → 0). If a and b are merely continuous N × N -matrix functions, then A is strongly elliptic if and only if det(a(t) + λb(t)) = 0 for ∀t ∈ T and ∀λ ∈ [−1, 1]

(see [8], Section 13.31). In this case A is Fredholm of index 0, but might be not invertible. In any case, (P∆k AP∆k ) is Fredholm and α(P∆k AP∆k ) = dim ker A where (∆k ) is admissible. Now we turn to general Fredholm sequences. Deﬁnition 8. Let B be a unital C ∗ -algebra. An element k ∈ B is of central rank one if, for every b ∈ B, there is an element µ(b) belonging to the center of B such that kbk = µ(b)k. An element of B is of ﬁnite central rank if it is the sum of a ﬁnite number of elements of central rank one, and it is centrally compact if it lies in the closure of the set of all elements of ﬁnite central rank. We denote the set of all centrally compact elements in B by J (B). It is easy to check that J (B) forms a closed two-sided ideal in B. Proposition 9. ([7], Proposition 6.33) A sequence (An ) ∈ F is centrally compact if and only if, for every ε > 0, there is a sequence (Kn ) ∈ F such that sup An − Kn < ε and sup dim im Kn < ∞ . n

Deﬁnition 9. A sequence (An ) ∈ F is a Fredholm sequence if it is invertible modulo the ideal J(F ) of the centrally compact sequences. Theorem 14. ([7], Theorem 6.35) A sequence (An ) ∈ F is Fredholm if and only if there is an l ∈ Z+ such that lim inf sl+1 (An ) > 0 .

n→∞

Conclusion. If (An ) ∈ F T is Fredholm then it is also Fredholm in the sense of Deﬁnition 9.

6. Applications continued: Around ﬁnite sections of operators with almost periodic diagonals This material is based on [10]. 6.1. Example. Let ˜l2 := {(xn )n∈Z : |xn |2 < ∞}. n∈Z

The Almost Mathieu Operator is the operator Hα,λ,θ : ˜l2 → ˜l2 , which acts on a sequence x = (xn )n∈Z ∈ ˜l2 by (Hα,λ,θ x)n := xn+1 + xn−1 + λxn cos 2π(nα + θ).

C ∗ -algebras and Asymptotic Spectral Theory

51

Only recently the long-standing Ten Martini problem was solved, see [1], [9], and for a introduction to the topic [2]. The problem consists in describing the spectrum of Hα,λ,θ . The result says (in a somewhat incomplete form) that • If α is rational, α = pq and p, q relatively prime with q > 0, then the spectrum of Hα,λ,θ is the union of exactly q closed and pairwise disjoint intervals, θ πp ∈ / Z. • If α ∈ [0, 1) is irrational, then the spectrum is a Cantor type set (it means: nowhere dense, closed, and does not contain isolated points). This result is a qualitative one! It does not allow to say that a given number µ belongs to the spectrum (or not). There is (at least in present time) only one way to tackle this problem, namely the use of approximation methods. For, introduce the projection operators Pn and P˜n : P˜n x = {. . . , 0, x−n , . . . , xn , 0, . . . } Pn x = {. . . , 0, x0 , . . . , xn , 0, . . . } . First idea: consider the operators (matrices) (P˜n Hα,λ,θ P˜n ) (restricted to im P˜n and with respect to the standard basis) and compute the eigenvalues using Matlab or something else. Then the question arises, is this spectrum some how related to the spectrum of the Almost Mathieu Operator Hα,λ,θ ? The following is devoted to some theory around this problem. However, we will merely make use of the projections Pn . The operator Hα,λ,θ is an example of a band operator with almost periodic diagonals. 6.2. Band-dominated operators with almost periodic diagonals and related Toeplitz-like operators Recall that a general band-dominated operator A : ˜l2 → ˜l2 is the norm limit of band operators, that is of operators of the kind k

al U l ,

−k

where Um : ˜l2 → ˜l2 is the shift operator given by (Um x)(n) = x(n − m) and al I : l → l are multiplication operators with al ∈ l∞ (Z). If we replace the elements al ∈ l∞ (Z) by almost periodic ones, that is the set {Um a}m∈Z is relatively compact in the norm of l∞ (Z), then we obtain the class of band and band-dominated operators with almost periodic diagonals. It is easy to see that the collection AP (Z) ⊂ l∞ (Z), of all almost periodic sequences as well as the collection AAP of all band-dominated operators with almost periodic diagonals form C ∗ -algebras. Simple example: The Laurent operator L(a) with continuous generating function a ∈ C(T) is obviously an element of AAP (Z). ˜2

˜2

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Let us introduce the following class of Toeplitz-like operators. Clearly, l2 can be thought of as a subspace of ˜l2 . Let P denote the orthogonal projection onto l2 and Q := I − P . Consider T (A) : l2 → l2 , A ∈ AAP (Z), T (A) := P AP |imP . If A = L(a), a ∈ C(T), then T (L(a)) is denoted simply by T (a), and this is a familiar Toeplitz operator. Introduce J : ˜ l2 → ˜l2 (ﬂip operator) by (xn ) → ˜ (x−n−1 )) and H(A) := P AQJ , A := JAJ. Then one has ˜ (A, B ∈ AAP (Z)) , T (AB) = T (A)T (B) + H(A)H(B) which reminds the basic identity relating Toeplitz and Hankel operators and it is this identity for A = L(a), B = L(b), a, b ∈ C(T). ˜ are compact operators! Notice: H(A), H(B) Let AAP (Z+ ) denote the smallest C ∗ -subalgebra of B(l2 ) containing all operators T (A), A ∈ AAP (Z). Then • T (A) = A (6.1) 2 ˙ ) • AAP (Z+ ) = {T (A) : A ∈ AAP (Z)}+K(l The ﬁrst identity is based on a remarkable fact which plays an important role in what follows. Let us have a closer look. Let H refer to the set of all sequences h : Z+ → Z which tend to +∞ or −∞. Deﬁnition 10. An operator Ah ∈ B(˜l2 ) is called a norm limit operator of the operator A ∈ B(˜l2 ) with respect to the sequence h ∈ H if U−h(k) AUh(k) → Ah as k → ∞ in norm. The set of all norm limit operators is called the norm operator spectrum σop (A). Theorem 15. A ∈ AAP (Z+ ) is Fredholm if and only if each Ah ∈ σop (A) is invertible. Proposition 10. If A ∈ AAP (Z) then A ∈ σop (A). This result is in force since so-called distinguished sequences exist for A ∈ AAP (Z). Deﬁnition 11. A monotonically increasing sequence h : Z+ → Z+ is called distinguished if Ah exists and equals A. The ﬁrst assertion in (6.1) is now easy to prove: Consider U−h(k) P AP Uh(k) = U−h(k) P Uh(k) U−h(k) AUh(k) U−h(k) P Uh(k) ! " ! " ! " ↓ strongly ↓ in norm ↓ strongly I A I and apply the Banach-Steinhaus Theorem.

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Further conclusion: ess sp A = spA for A ∈ AAP (Z) and T (A) is Fredholm if and only if A is invertible (ess sp A = sp(A + K(˜l2 )). Example 5. Almost Mathieu operators. We have ak (n) = =

U−k Hα,λ,θ Uk = U−1 + U1 + ak I , a(n + k) = λ cos 2π((n + k)α + θ) λ(cos 2π(nα + θ) cos 2π(kα) − sin 2π(nα + θ) sin 2π(kα)) .

Let α ∈ (0, 1) be irrational. We write α as a continued fraction with nth approximant pqnn such that 1

α = lim

n→∞

1

b1 + b2 +

1 ..

. bn−1 +

1 bn

with uniquely determined positive integers. Write this continued fraction as pn /qn with positive and relatively prime integers pn , qn . These integers satisfy the recursions pn = bn pn−1 + pn−2 , qn = bn qn−1 + qn−2 with p0 = 0, p1 = 1, q0 = 1 and q1 = b1 , and one has for all n ≥ 1 # # # # #α − pqnn # < q12 , n ⇒ |αqn − pn | ≤ q1n → 0 . Now it is not hard to see that (qn ) is a distinguished sequence for Hα,λ,θ (note: (qn ) is independent on λ and θ). 6.3. Distinguished sequences and ﬁnite sections In what follows we ﬁx a strongly monotonically increasing sequence h : Z+ → Z+ and deﬁne AAP,h (Z) := {A ∈ AAP (Z) : Ah exists and Ah = A} . It is easy to check that AAP,h (Z) is a C ∗ -subalgebra of B(˜l2) which is moreover shift invariant, i.e., U−k AUk again lies in AAP,h (Z+ ) whenever A does. Let AAP,h (Z+ ) refer to the smallest closed subalgebra of B(l2 ) which contains all operators T (A) with A ∈ AAP,h (Z). For instance, all Toeplitz operators with continuous generating functions lie in this algebra. Thus • K(l2 ) ⊂ AAP,h (Z+ ) and 2 ˙ • AAP,h (Z+ ) = {T (A) : A ∈ AAP,h (Z)}+K(l ).

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Let us turn over to ﬁnite sections. For, let Fh denote the set of all bounded sequences (An ) of matrices An ∈ Ch(n)×h(n) . Provided with pointwise deﬁned operations and the supremum norm, Fh becomes a C ∗ -algebra (An – norm of the operator deﬁned by An on im Ph(n) ). Finally, we let SAP,h (Z+ ) denote the smallest closed subalgebra of Fh which contains all sequences (Ph(n) T (A)Ph(n) ) with operators A ∈ AAP,h (Z). Deﬁne Rn : l2 → l2 , (xn )n≥0 → (xn , xn−1 , . . . , x0 , 0, 0, . . . ). Theorem 16. The C ∗ -algebra SAP,h (Z+ ) consists exactly of all sequences of the form Ph(n) T (A)Ph(n) + Ph(n) KPh(n) + Rh(n) LRh(n) + Ch(n) (6.2) with A ∈ AAP,h (Z) , K, L ∈ K(l2 ) , Ch(n) → 0 as n → ∞, and each sequence in SAP,h (Z+ ) can be written in the form (6.2) in a unique way. ˜ : SAP,h (Z+ ) → AAP,h (Z+ ) by Deﬁne mappings W, W W (An ) = ˜ (An ) = W

s- lim Ph(n) An Ph(n) , s- lim Rh(n) An Rh(n) .

These mappings are ∗-homomorphisms. Their importance is given by the following stability theorem. Theorem 17. A sequence (An ) ∈ SAP,h (Z+ ) is stable if and only if the operators ˜ (An ) are invertible, that is, if W (An ), W An = Ph(n) T (A)Ph(n) + Ph(n) KPh(n) + Rh(n) LRh(n) + Ch(n) ˜ +L with K, L, Ch(n) as above, then (An ) is stable if and only if T (A) + K , T (A) are invertible. Moreover, SAP,h (Z+ ) is fractal. The proof is basically the same as in the Toeplitz case. 6.4. Spectral approximations The last theorem in Section 6.3 is one of the keys to study spectral approximations. Theorem 18. Let A := (An ) ∈ SAP,h (Z+ ) be a self-adjoint sequence. Then the ˜ (A). spectra sp An converges in the Hausdorﬀ metric to sp W (A) ∪ sp W Theorem 19. Let A := (An ) ∈ SAP,h (Z+ ). Then the set of the singular values ˜ (A)). (An ) converges in the Hausdorﬀ metric to (W (A)) ∪ (W Theorem 20. A sequence A = (An ) ∈ SAP,h (Z+ ) is Fredholm if and only if its ˜ (A) is a Fredholm operstrong limit W (A) is a Fredholm operator. In this case W ator too, and ˜ α(A) = dim ker W (A) + dim ker W(A) ; moreover, lim sα (An ) = 0. n→∞

These theorems can be completed by results concerning ε-pseudospectra and the so-called Arveson’s dichotomy (the last for self-adjoint sequences).

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Arveson’s dichotomy: Given a self-adjoint sequence A := (An ) ∈ SAP,h (Z+ ) and an open interval U ⊂ R, let Nn (U ) refer to the number of eigenvalues of An in U , counted with respect to their multiplicity. A point λ ∈ R is called essential for A, if for every open interval U containing λ, lim Nn (U ) = ∞ ,

n→∞

and λ ∈ R is called a transient point for A is there is an open interval U containing λ such that sup Nn (U ) < ∞ . n

Theorem 21. Let A := (An ) ∈ SAP,h (Z+ ) be self adjoint, s-lim An = T (A) + K. (s-lim An is necessarily of this form because of Theorem 16). Then every point λ ∈ sp A is essential, and every point λ ∈ R\sp A is transient for A. Moreover, for every point λ ∈ R\sp A, the sequence A−λP where (P := (Ph(n) ) ∈ SAP,h (Z+ )) is Fredholm and there is an open interval U ⊂ R containing λ such that sup Nn (U ) = α(A − λP).

n

Theorem 21 is a consequence of Theorem 7.12 in [7] and of the fact that $ = ess sp A. $ ess sp A = sp A = sp A The ﬁrst assertion of Theorem 21 implies in particular that each real number is either essential or transient for A. This property is usually referred to as the Arveson’s dichotomy of that sequence. Remark. If A ∈ AAP (Z) is selfadjoint and h is a distinguished sequence, then Theorem 21 oﬀers the possibility to determine the spectrum of A numerically. Test calculations will be provided in the next section. Let us mention that we could also use the projection P$n for this aim. The related theory (see Section 7) is not simpler as that for the projections Pn , but gives essentially the same results. A deep study of the ﬁnite sections for general band-dominated operators in ˜l2 is carried out in [15]. Let us mention also the recent book [4], where spectral properties of banded Toeplitz matrices are studied. 6.5. Test calculations Here we shall demonstrate how the theory can be used to determine numerically the spectrum of the Almost Mathieu operator for some choices of the parameters α, λ and θ (using Matlab). For each of the triples √ √ 2 1 1 1 2 5−1 2 2 , 2, 0 , , 2, , , 2, 0 , , 2, , 2, , , 5 5 2 7 2 2 2 2

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in place of (α, λ, θ), we choose a distinguished sequence of the corresponding Almost Mathieu operator which depends only on & % √ √ 2 5−1 2 2 αj ∈ , , , , 5 7 2 2 namely α1 α2 α3

= = =

α4

=

2 5 : h1 (k) = 5k , 2 7 : h2 (k) = 7k , √ √ k √ k 2 1 2) , 5 : h3 (k) = 2 (1 + 2) +(1 − √ √ √ √ k √ k 5−1 5+ 5 1+ 5 5− 5 1− 5 : h (k) = + 4 2 10 2 10 2

.

For irrational αk , this choice has been done via continued fractions. Notice that the sequences h3 and h4 are rapidly growing. For instance, h3 (13) = 47321 , h4 (23) = 46368. The results are plotted in Pictures 1–7. The results for α4 , λ = 2 , θ = 0, 5 and h5 (k) := 2k (non-distinguished!) are plotted in Picture 8. The computations clearly indicate the advantage of distinguished sequences over non-distinguished (compare Pictures 6 and 8). For irrational α the Cantorlike structure of the spectrum is also somehow reﬂected in the computations (see Pictures 4 and 5). The computations also show that the speed of converges is very high. There is only a guess why it should be, but not a proof.

Picture 1: Eigenvalues of Ph1 (k)Hα,λ,θ Ph1 (k) with α = 2/5, λ = 2, θ = 1/2.

C ∗ -algebras and Asymptotic Spectral Theory

Picture 2: Eigenvalues of Ph2 (k)Hα,λ,θ Ph2 (k) with α = 2/7, λ = 2, θ = 0.

Picture 3: Eigenvalues of Ph2 (k)Hα,λ,θ Ph2 (k) with α = 2/7, λ = 2, θ = 1/2.

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Picture 4: Eigenvalues of Ph3 (k)Hα,λ,θ Ph3 (k) with α =

√ 2/2, λ = 2, θ = 0.

Picture 5: The eigenvalues of Ph3 (k)Hα,λ,θ Ph3 (k) with α = which lie in the interval (−2.4, −2.8).

√ 2/2, λ = 2, θ = 0

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59

√ Picture 6: Eigenvalues of Ph4 (k)Hα,λ,θ Ph4 (k) with α = ( 5 − 1)/2, λ = 2, θ = 0.5.

√ Picture 7: The eigenvalues of Ph4 (k)Hα,λ,θ Ph4 (k) with α = ( 5 − 1)/2, λ = 2, θ = 0.5, which lie in the interval (1.8, 2.6).

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√ Picture 8: Eigenvalues of Ph5 (k)Hα,λ,θ Ph5 (k) with α = ( 5 − 1)/2, λ = 2, θ = 0.5.

7. Applications continued: Band-dominated operators and their ﬁnite sections We reproduce in this section some recent results obtained in [12], [13], however with diﬀerent proofs. We give a complete study of the stability problem for ﬁnite sections of band-dominated operators and use these results to get an index formula for Fredholm operators belonging to that class. Recall that a general band-dominated operator A : ˜l2 → ˜l2 is the norm limit of band operators, that is of operators of the kind k

al U l ,

−k

where Ul are the earlier deﬁned shift operators, and al I : ˜l2 → ˜l2 are the multiplication operators (al x)m := al (m)xm with al = (al (m)) ∈ l∞ (Z). Let P, Q be the orthoprojections in ˜l2 , introduced in Section 6.2. Denote by H the set of all sequences h : N → Z which tend to −∞ or +∞. An operator Ah ∈ B(˜l2 ) is called the limit operator of A ∈ B(˜l2 ) with respect to the sequence h ∈ H if U−h(n) AUh(n) tends ∗-strongly to Ah as n → ∞. The set σ op (A) of all limit operators of a given operator A ∈ B(˜l2 ) is called the operator spectrum of A. It is not hard to see that every limit operator of a Fredholm operator is invertible. It is a remarkable fact that for band-dominated operators the reverse

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implication is true. More precisely, the following theorem is in force (see [11], Chapter 2): Theorem 23. Let A ∈ B(˜l2 ) be a band-dominated operator. Then (a) every sequence h ∈ H possesses a subsequence g such that the limit operator Ag exists, (b) the operator A is Fredholm if and only if each of its limit operators is invertible and if the norms of their inverses are uniformly bounded, (c) for each compact operator K one has σop (A + K) = σ op (A). Moreover, (A + K)g = Ag if one of the operators A (A + K)g , Ag with respect to the sequence g ∈ H exists. Notice that the collection of all band-dominated operators actually forms a C ∗ subalgebra S of B(˜l2 ) which contains all compact operators. Since P AQ, QAP are compact operators for all band-dominated operators, the study of the Fredholm properties of A ∈ S can be reduced to those of P AP + QBQ. ˜ n : ˜l2 → ˜l2 by Let P˜n be deﬁned as in Section 6.1 and deﬁne R (xn )n∈Z → (. . . , 0, 0, x−1 , x−2 , . . . , x−n , xn , xn−1 , . . . , x0 , 0, 0, . . . ) . ˜ n play an important role in the study of the stability of (P˜n AP˜n ), The operators R A ∈ S. It is not hard to see that the following version of Widom’s formula holds true (A, B ∈ S): ˜ n LR ˜ n + Cn (7.1) P˜n AB P˜n = P˜n AP˜n B P˜n + R with some compact operator L, and (Cn ) ∈ G (exercise; hint: consider expressions of the type P˜n P ABP P˜n − P˜n P AP P˜n P BP P˜n and those where P is replaced by Q). The ideas of 4.II are again not directly applicable. The point is that the strong ˜ n AR ˜n does not exist for A ∈ S in general. The situation changes when limit s-lim R we consider suitable subsequences. Theorem 24. (a) Let A ∈ S and h ∈ H be a sequence tending to +∞ for which the limit operator Ah of A with respect to the sequence h exists. Then there is a subsequence g of h such that the limit operator A−g of A exists. Moreover, ˜ g(n) AR ˜ g(n) = P JAh JP + QJA−g JQ . s∗ - lim R (b) An analogous result is true for sequences tending to −∞. ˜ g(n) ) tends weakly Proof. (a ) Since A− P AP − QAQ is a compact operator and (R to zero, it needs only to consider P AP + QBQ. Obviously, ˜ g(n) = P JU−g(n) P AP Ug(n) JP + QJUg(n) QAQU−g(n) JQ , ˜ g(n) (P AP + QAQ)R R where g is a subsequence of h for which the limit operator A−g of A exists (Theorem 23, (a)). A straightforward computation shows the claim.

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Let h ∈ H be a sequence tending to +∞ and introduce the algebra Fh with ˜ respect to the sequence (Ph(n) ) and let FhW stand for the C ∗ -subalgebra of Fh constituted by all sequences of Fh for which the limits ˜ h(n) Ah(n) R ˜ h(n) s∗ - lim P˜h(n) Ah(n) P˜h(n) and s∗ - lim R exist. Again, the set ( ' ˜ ˜ h(n) LR ˜ h(n) + Ch(n) : K, L − compact, (Ch(n) ) ∈ Gh JhW := P˜h(n) K P˜h(n) + R ˜

actually forms a closed two-sided ideal in FhW . (Gh stands here of the closed twosided ideal of Fh consisting of all sequences tending to zero in norm). The algebra ˜ FhW is of the type described in Section 5. Thus, the general theory mentioned in Section 5 applies. ˜

Corollary 4. A sequence (Ah(n) ) ∈ FhW is stable if and only if ˜ h(n) Ah(n) R ˜ h(n) are invertible, (a) the operators s-lim P˜h(n) Ah(n) Ph(n) and s-lim R ˜ ˜ ˜ W W W (b) the coset (Ah(n) ) + Jh is invertible in Fh /Jh . Corollary 5. Let h ∈ H be a sequence tending to +∞ and let A ∈ S be an operator for which ˜ h(n) AR ˜ h(n) s∗ - lim R ˜ exists. Then (P˜h(n) AP˜h(n) ) belongs to FhW .

Proposition 11. Let h ∈ H be a sequence tending +∞ and let A ∈ S be Fredholm. Then there exists a subsequence g of h such that (Pg(n) APg(n) ) is a Fredholm ˜ sequence in FgW . Proof. It is easy to see that A has a regularizer B in S, that is AB = I +M1 , BA = I + M2 , where M1 , M2 are compact operators. Using (7.1) we get ˜ n LR ˜ n + Cn , K, L-compact and (Cn ) ∈ G . P˜n AP˜n B P˜n = P˜n + P˜n K P˜n + R By Theorem 24 and Corollary 5 there is a subsequence h1 of h such that ˜ h (n) AR ˜h (n) exists. By the same reasoning there is a subsequence g1 of h1 s∗ -lim R 1 1 ˜ g (n) B R ˜g (n) exists. Hence, (Pg (n) APg (n) ), (Pg (n) BPg (n) ) ∈ such that s∗ -lim R 1 1 1 1 1 1 ˜

)

˜

˜

FgW1 , and (pg1 (n) APg1 (n) ) + JgW1 is invertible from the right in FgW1 /JgW1 . Again choosing a suitable subsequence g of g1 one obtains in the same fashion as above ˜ that (Pg(n) APg(n) ), (Pg(n) BPg(n) ) belong to FgW and ˜

˜

˜

(Pg(n) APg(n) ) + JgW is already invertible in FgW /JgW .

op op (A), σ− (A) denote the sets of all limit operators of A ∈ S Theorem 25. Let σ+ with respect to sequences tending to +∞ and −∞, respectively. Then the sequence (P˜n AP$n ) is stable if and only if the operator A, and all operators op (A) , QAh Q + P with Ah ∈ σ+

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63

and all operators op P Ah P + Q with Ah ∈ σ− (A)

are invertible on ˜l2 . Proof. Suﬃciency: Suppose that (P˜n AP˜n ) is not stable. Then there is a sequence h tending to +∞ such that lim (P˜h(n) AP˜h(n) )−1 P˜h(n) equals +∞ (if P˜h(n) AP˜h(n) is not invertible on im P˜h(n) , then we set (P˜h(n) AP˜h(n) )−1 Ph(n) = ∞). Then there is by Proposition 10 a subsequence g of h such that (Pg(n) APg(n) ) is a Fredholm ˜ sequence in FgW . By the computations in the proof of Theorem 24 we may assume that ˜ g(n) AR ˜ g(n) s∗ - lim R

= P JAg JP + QJA−g JQ = (P JAg JP + Q)(P + QJA−g JQ) .

Since JP J = Q and JQJ = P , we have J(P JAg JP + Q)J = QAg Q + P . Hence, P JAg JP + Q is invertible. By the same reasoning it follows that also P + QJA−g JQ is invertible. Using Corollary 4 we get that (Pg(n) APg(n) ) is stable. But this is a contradiction. The proof of the necessity is much more easier and is left to the reader. Our next topic is the index of a Fredholm band-dominated operator A ∈ S. For A ∈ S we put A+ = P AP + Q and A− = P + QAQ . It is easy to see that op op (A) ∪ {I} and σ op (A− ) = σ− (A) ∪ {I} . σ op (A+ ) = σ+

Further, the operator A ∈ S is Fredholm if and only if A+ and A− are Fredholm operators. Moreover, the equality P AP + QAQ = A+ A− = A− A+ shows that, for a Fredholm operator A ∈ S, ind A = ind A+ + ind A− . In addition, we shall use the notation ind+ A := ind A+ and ind− A := ind A− and call ind+ A and ind− A the plus- and the minus-index, respectively. Theorem 26. op (A) have the same plus1. Let A ∈ S be Fredholm. Then all operators in σ+ index, and this number coincides with the plus-index of A. Analogously, all op (A) have the same minus-index, and this number coincides operators in σ− with the minus-index of A.

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2. If A ∈ S is Fredholm, then ind A = ind+ B + ind− C , op op where B ∈ σ+ (A) and C ∈ σ− (A) are arbitrary taken. op (A) choose a sequence h tending Proof. It needs only to prove 1: Given B ∈ σ+ to + inﬁnity and for which B is the limit operator for A. Then B ∈ σop (A+ ). Notice that the limit operator of A+ with respect to −h exists and equals I. Then ˜ (Ph(n) A+ Ph(n) ) ∈ FhW . The Fredholmness of A+ implies that (Ph(n) A+ Ph(n) ) is a Fredholm sequence in Fh(n) and, by Corollary 3, ind A+ + ind (QBQ + P ) = 0. Since B is invertible it follows that ind (QBQ + P ) = −ind (P BP + Q) (use that ind (P BP + QBQ) = 0). Hence, ind A+ = ind+ B. Analogously, ind A− = ind− C, op where C ∈ σ− (A) is arbitrarily chosen.

Finally, let us mention an application of Theorems 25 and 26 to ﬁnite sections of band-dominated operators with slowly oscillating coeﬃcients. A function a ∈ l∞ (Z) is slowly oscillating if lim (a(x + k) − a(x)) = 0 for all k ∈ Z .

x→±∞

It is one of the remarkable features of band-dominated operators with slowly oscillating coeﬃcients that their limit operators are Laurent operators with continuous generating functions deﬁned on the unit circle in the complex plane (see [11], Chapter 2). Theorem 27. Let A ∈ S be a band-dominated operator with slowly oscillating coeﬃcients. Then the sequence (P˜n AP˜n ) is stable if and only if A is invertible and ind A+ = 0. Proof. Let A be invertible and ind A+ = 0. Then we have also ind A− = 0. By op op (A) (σ− (A)) have the same plus-index (minusTheorem 26 all operators in σ+ index) and this number coincides with the plus-index (minus-index) of A. Thus, all these indices are equal to zero. By a theorem of Coburg (usually formulated op for Toeplitz operators) the operators P BP + Q(P + QCQ) with B ∈ B+ (A) op (C ∈ σ− (A)) are invertible (see also part 4.II). Now it remains to apply Theorem 26. The reverse statement is obvious. Theorem 25 can be found in [11] under the additional assumption that all op op operators QAh Q + P, Ah ∈ σ+ (A) and all operators P Ah P + Q, Ah ∈ σ− (A), are uniformly invertible. Without this assumption it occurs in [12]. Theorem 26 was originally proved in [14] by help of K-theory and reproved recently in [13] using asymptotic spectral theory. Theorem 27 is Theorem 6.2.4 in [11], however for band operators with slowly oscillating coeﬃcients. In the general setting Theorem 27 was proved by S. Roch in [16] and is now simply a corollary to Theorems 25 and 26.

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The approach pointed out in Section 7 can be also used to get relatively simple proofs of some further results obtained by S. Roch in [15]. Let me mention only two: • computation of the α-number of a Fredholm sequence (P˜n AP˜n ), A ∈ S. • Any sequence (P˜n AP˜n ), A ∈ S and A = A∗ , has the Arveson dichotomy.

References [1] A. Avila and S. Jitomirskaja: The Ten Martini problem. -arXiv:math. DS/ 0503363. [2] F.-P. Boca: Rotation C ∗ -algebras and Almost Mathieu Operators. Theta Series in Advanced Mathematics 1, The Theta Foundation, Bucharest 2001. ¨ ttcher and B. Silbermann: Introduction to Large Truncated Toeplitz Ma[3] A. Bo trices. Springer-Verlag, New York 1999. ¨ ttcher and S.M. Grudsky: Spectral Properties of Bounded Toeplitz Ma[4] A. Bo trices. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA 2005. [5] N.P. Brown: Quasidiagonality and the ﬁnite section method. Math. Comp. 76 (2007), no. 257, 339–360 (electronic). [6] N.P. Brown: AF Embeddings and the Numerical Computation of Spectra in Irrational Rotation Algebras. Num. Funct. Anal. Optim. 27 (2006), no. 5-6, 517–528. [7] R. Hagen, S. Roch, and B. Silbermann: C ∗ -Algebras and Numerical Analysis. Marcel Dekker, Inc., New York, Basel 2001. ¨ ssdorf and B. Silbermann: Numerical Analysis for Integral and related [8] S. Pro Operator Equations. Akademie Verlag, Berlin 1991. [9] J. Puig: Cantor spectrum for the almost Mathieu Operator. Comm. Math. Phys. 244 (2004), 2, 297–309. [10] V. Rabinovich, S. Roch, and B. Silbermann: Finite sections of band-dominated operators with almost periodic coeﬃcients. Operator Theory: Advances and Applications 170, Birkh¨ auser Verlag, Basel, Boston, Berlin 2006. [11] V. Rabinovich, S. Roch, and B. Silbermann: Limit Operators and Their Applications. Operator Theory: Advances and Applications 150, Birkh¨ auser Verlag, Basel, Boston, Berlin 2004. [12] V. Rabinovich, S. Roch, and B. Silbermann: On ﬁnite sections of banddominated operators. Preprint 2486 (2006), Fachbereich Mathematik, TU Darmstadt. [13] V. Rabinovich, S. Roch, and B. Silbermann: The ﬁnite section approach to the index formula for band-dominated operators. Preprint 2488 (2006), Fachbereich Mathematik, TU Darmstadt. [14] V. Rabinovich, S. Roch, and J. Roe: Fredholm indices of band-dominated operators. Integral Equations and Operator Theory 49 (2004), 2, 221–238. [15] S. Roch: Finite sections of band-dominated operators. Preprint Nr. 2355 (2004), Fachbereich Mathematik, TU Darmstadt.

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[16] R. Roch: Band-dominated operators on lp -spaces: Fredholm indices and ﬁnite sections. Acta Sci. Math (Szeged) 70 (2004), 783–797. [17] A. Rogozhin and B. Silbermann: Banach algebras of operator sequences: Approximation Numbers. J. Operator Theory, in print. Bernd Silbermann Technische Universit¨ at Chemnitz Fakult¨ at f¨ ur Mathematik D-09107 Chemnitz, Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 67–118 c 2008 Birkh¨ auser Verlag Basel/Switzerland

Toeplitz Operator Algebras and Complex Analysis Harald Upmeier

1. Introduction The aim of this survey article is to present the recent work concerning Hilbert spaces of holomorphic functions on hermitian symmetric domains of arbitrary rank and dimension, in relation to operator theory (Toeplitz C ∗ -algebras and their representations), harmonic analysis (discrete series of semi-simple Lie groups) and quantization (covariant functional calculi and Berezin transformation). Acknowledgment A substantial part of the results presented here is joint work with J. Arazy (University of Haifa). The ﬁnancial support of the German-Israeli foundation (GIF No. 696-17.6/2001) is gratefully acknowledged. The author would like to thank the referee for valuable comments.

2. Complex geometry of bounded symmetric domains In the following consider a complex vector space Z = Cd , endowed with a norm · , and let D = {z ∈ Z : z < 1} be its unit ball. In general, the boundary ∂D = {z ∈ Z : z = 1} will not be smooth. Consider the set S = ∂ex D ⊂ ∂D of all extreme points and let K = U (Z) = {g ∈ GL(Z) : gz = z}

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be the isometry group. Then K ⊂ GL(Z) is a closed subgroup, therefore K is a compact Lie group. Clearly, the natural K-action satisﬁes K · D = D, K · ∂D = ∂D, K · S = S. Example. For Z = Cd , deﬁne the scalar product zi w i (z|w) = zw∗ = i

and let z = (z|z)

1/2

be the Hilbert norm. Then D = {z ∈ Cd : (z|z) < 1}

is the Hilbert unit ball. In this case K = U (d) is the unitary group and S = ∂D = S 2d−1 is the odd sphere. Example. More generally, consider now the matrix space Z = Cr×(r+b) , endowed with the operator norm z = sup σ(zz ∗ )1/2 , and let D = {z ∈ Cr×(r+b) : z < 1} = {z ∈ Cr×(r+b) : I − zz ∗ > 0} be the corresponding matrix unit ball, with closure D = {z ∈ Cr×(r+b) : I − zz ∗ ≥ 0}. In this case we have K = U (r) × U (r + b) (u, v), via the action z → uzv ∗ , and S = {z ∈ Cr×(r+b) : zz ∗ = I} consists of all isometries. In the special case b = 0 we obtain S = U (r). In this case the domain D is said to be of tube type. We now introduce holomorphic functions. By deﬁnition, a (possibly vectorvalued) function f : D → Cm is holomorphic, if it has a power series expansion f (z) = cα z1α1 · · · znαn α∈Nn

(compact convergence) with coeﬃcients cα ∈ Cm . By collecting monomials of equal (total) degree, we obtain the expansion cα z1α1 · · · znαn = pk (z) f (z) = k≥0 |α|=k

k≥0

into a series of k-homogeneous polynomials. Let Pk (Z, Cm ) be the space of all k-homogeneous polynomials pk : Z → Cm satisfying pk (λz) = λk pk (z) ∀ λ ∈ C. The space Pk (Z, Cm ) P(Z, Cm ) = k≥0

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of all polynomials is dense in the space of all holomorphic functions O(D, Cm ) under compact convergence. Deﬁnition 2.1. Let G = Aut (D) = {g : D → D biholomorphic} denote the holomorphic automorphism group. It is a real Lie group (by a deep theorem of H. Cartan). Moreover, K = {g ∈ G : g(0) = 0} G is a compact subgroup. Deﬁnition 2.2. The domain D is called symmetric iﬀ G acts transitively on D, i.e., ∀ z ∈ D ∃ g ∈ G, z = g(0)). Clearly, this is equivalent to D = G/K. For arbitrary domains, the condition of symmetry requires that at each point there exists a globally deﬁned geodesic reﬂection. In general, this is a stronger condition than homogeneity, but in our case the unit ball D is circular and has the obvious reﬂection s0 (z) = −z at the origin. Example. If Z = Cr×(r+b) and D is the matrix unit ball, we obtain the pseudounitary group ∗ ∗ ab ab 1 0 a c 1 0 G = SU (r, r+b) = ∈ GL (2r + b) : = cd b d 0 − 1 b∗ d∗ 0 −1 acting on D via Moebius transformations ab (z) = (az + b)(cz + d)−1 . cd Its maximal compact subgroup is a0 K= : a ∈ U (r), d ∈ U (r + b) 0d Example. If Z = Cd = C1×d and D is the Hilbert unit ball, we obtain as a special case G = SU (1, d). Example. For dimension d = 1, we have Z = C and D is the unit disk. In this case, we may identify G =

SU (1, 1) ≈ SL(2, R),

K

U (1)

=

It is a fundamental fact [K], [K1], [LO2], [U6] that hermitian symmetric domains have an algebraic description in terms of the so-called Jordan algebras and Jordan triples. In order to explain this connection, consider the Lie algebra exp g −→ G of the real Lie group G = Aut (D), which can be realized as follows:

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Consider a 1-parameter group t → gt ∈ G, and deﬁne, for f ∈ O(D, Cm ), the inﬁnitesimal generator ∂ ∂f (gt (z)) ## f (z). = (Xf )(z) = h(z) f (z) = h(z) t=0 ∂t ∂z Here h : D → Z = Cd is a holomorphic vector ﬁeld, with commutator bracket ∂ ∂k ∂h ∂ ∂ , k = h −k . h ∂z ∂z ∂z ∂z ∂z Let s0 (z) = −z be the symmetry at the origin 0 ∈ D. For the adjoint action of G on g, deﬁned via ∂ ∂ = h(g(z)) g (z) , Ad (g) h ∂z ∂z we obtain the Cartan decomposition g = k ⊕ ℘, where k = {X ∈ g : Ad (s0 ) X = X} is the 1-eigenspace and ℘ = {X ∈ g : Ad (s0 ) X = −X} is the (−1)-eigenspace. By deﬁnition, we obtain a Lie triple system [k, k] ⊂ k ⊃ [℘, ℘], [k, ℘] ⊂ ℘ ⊃ [℘, k]. The crucial step towards Jordan triples is the following Theorem 2.3. Let D ⊂ Z be a bounded symmetric domain. Then there exists a Jordan triple product Z × Z × Z → Z, denoted by u, v, w → {uv ∗ w} = {wv ∗ u}, such that ∂ ℘ = (v − {zv ∗ z}) : v∈Z ∂z ∂ k = {h(z) : h ∈ gl(Z) linear , h{zv ∗ z } = {h(z ) h(v )∗ h(z )}}. ∂z In order to deﬁne the Jordan triple property, let Z × Z → End (Z) be the mapping u, v → u v ∗ , given by (u v ∗ ) z = {uv ∗ z}. Since [[℘, ℘] ℘] ⊂ [k, ℘] ⊂ ℘, we obtain the Jordan triple identity [u v ∗ , x y ∗ ] = {uv ∗ x} y ∗ − x {yu∗v}∗ .

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Example. In the matrix case Z = Cr×(r+b) , let at b t (z) = (at z + bt )(ct z + dt )−1 ∈ SU (r, r + b) ct dt be a 1-parameter group of Moebius transformations. Its diﬀerential is computed as d ## ˙ + b˙ − z cz ˙ − z d˙ # (at z + bt )(ct z + dt )−1 = az dt t=0 ˙ with ac˙˙ db˙ ∈ su (r, r + b). The eigenspace decomposition is given by ∂ ˙ ˙ : a˙ ∈ u(r), d ∈ u(r + b) k = (az ˙ − z d) ∂z and

∂ ∗ r×(r+b) ˙ ˙ ˙ : b∈C ℘ = (b − z b z) . ∂z Therefore, according to Theorem 2.3 we obtain {zv ∗ z} {uv ∗ w} and u v∗ =

= =

zv ∗ z, 1 ∗ ∗ 2 (uv w + wv u)

1 (Luv∗ + Rv∗ u ). 2

Any bounded symmetric domain D ⊂ Z has an important K¨ ahler geometric structure called the Bergman metric. To deﬁne this metric, consider the Bergman operators Z × Z → End (Z), denoted by u, v → B(u, v), and deﬁned as B(u, v)z = z − 2{uv ∗ z} + {u{vz ∗ v}∗ u}. Example. For the matrix case Z = Cr×(r+b) , we have B(u, v)z

=

z − uv ∗ z − zv ∗ u + u(vz ∗ v)∗ u

= =

z − uv ∗ z − zv ∗ u + uv ∗ zv ∗ u (1 − uv ∗ ) z(1 − v ∗ u).

Therefore B(u, v) = L1−uv∗ R1−v∗ u . Deﬁnition 2.4. Deﬁne a G-invariant Hermitian metric on D as follows: ∀ z ∈ D ∀ u, v ∈ Tz D ≈ Z we put hz (u, v) = (u|v)z = (B(z, z)−1 u|v), using the normalized K-invariant inner product (u|v) on T0 D = Z given by (u|v) = where p is the “genus” of D.

2 trZ u v ∗ , p

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Example. For matrices Z = Cr×(r+b) we have 2r + b Luv∗ + Rv∗ u = tr uv ∗ 2 2 and hence the Bergman metric is given by trZ u v ∗ = trZ

hz (u, v) = tr (1 − zz ∗ )−1 u(1 − z ∗ z)−1 v ∗ . In terms of the Bergman operator, the Bergman kernel function K : D × D → C of D can be expressed as K(z, w) = det B(z, w)−1 . It is a fundamental fact that there exists a sesqui-polynomial called the Jordan triple determinant N : Z × Z → C such that the Bergman kernel function has the form K(z, w) = det B(z, w)−1 = N (z, w)−p for all z, w ∈ D. Example. In the matrix case Z = Cr×(r+b) we have det B(z, w) = detZ [L1−zw∗ R1−w∗ z ] = detZ L1−zw∗ detZ R1−w∗ z = detCr (1r − zw∗ )r detCr+b (1r+b − w∗ z)r+b = det (1r − zw∗ )2r+b . It follows that N (z, w) = det (1 − zw∗ ). Jordan triples are closely related to Jordan algebras which were originally introduced in quantum mechanics. Deﬁnition 2.5. A real vector space X ≈ Rd is called a real Jordan algebra iﬀ X has ◦ a commutative, non-associative product X × X −→ X, denoted by x, y → x ◦ y = y ◦ x, which satisﬁes the Jordan algebra identity x2 ◦ (x ◦ y) = x ◦ (x2 ◦ y). If, in addition, we require x2 + y 2 = 0 =⇒ x = y = 0, the Jordan algebra X is called formally-real or Euclidean since, in this case it has a strictly positive inner product which agree with (u|v) on the complexiﬁcation Z = X C . We have the following classiﬁcation [JNW]: Every (irreducible) Euclidean Jordan algebra has a realization X ≈ Hr (K) = {self-adjoint r × r-matrices x = (xij ) ∈ Kr×r , x∗ij = xji } for the anti-commutator product 1 (xy + yx). 2 More precisely, we have the following possibilities, depending on the “rank” r x◦y =

if r ≥ 4, then K = R, C, H = quaternions if r = 3, then K = R, C, H, O = octonions.

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If r = 2, then K = (0) is a real Hilbert space, i.e., we obtain “formal” 2 × 2 matrices α b H2 (K) ∗ b δ with α, δ ∈ R, b ∈ K. It follows from the classiﬁcation that X is uniquely characterized by only two invariants: r = rank X and the characteristic multiplicity a = dimR K. Every Euclidean Jordan algebra X has a positive cone Ω = {x2 : x ∈ X \ {0}}, corresponding to the positive deﬁnite matrices Hr+ (K) and a Jordan algebra determinant N : X → R, which is a degree r polynomial deﬁned via Cramer’s rule ∇x N . x−1 = N (x) Example. For the rank 2 case, N

α b = αδ − (b|b) b∗ δ

is the Lorentz metric on R1, 1+a . If (X, ◦) is a real Jordan algebra, the complexiﬁcation Z := X C becomes a complex Jordan ∗-algebra, and the associated Jordan triple (of tube type) on Z is deﬁned as follows: {uv ∗ w} = (u ◦ v ∗ ) ◦ w + (w ◦ v ∗ ) ◦ u − v ∗ ◦ (u ◦ w). For the ﬁne structure of Jordan triples and the corresponding symmetric domains, we need the notion of tripotent (= triple idempotent) and associated Peirce decomposition. This is analogous to the well-known root decomposition of (semi-simple) Lie algebras which underlies the Lie theoretic approach to symmetric spaces. In any Jordan triple Z, an element c ∈ Z is called a tripotent iﬀ {cc∗ c} = c. In this case we have the Peirce decomposition Z = Z1 (c) ⊕ Z1/2 (c) ⊕ Z0 (c), where Zα (c) = {z ∈ Z : {cc∗ z} = αz} is the α-eigenspace for α = 0, 1/2, 1. Example. For matrices Z = Cr×(r+b) a tripotent c = cc∗ c is a partial isometry. In case k r−k b 0 0 r 1k , c= r−k 0 0 0

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the Peirce decomposition is given by Z1/2 (c) Z1 (c) Cr×(r+b) = Z1/2 (c) Z0 (c)

Z1/2 (c) Z0 (c)

.

The Peirce decomposition can be constructed more generally for any maximal frame e1 , . . . , er of orthogonal tripotents, where r = rank Z. Here orthogonality means ei e∗j = 0 for i = j, a stronger condition than orthogonality for the scalar product of Z. In this case ⊕ Zij , Z= 0≤i≤j≤r

where

1 i (δ + δkj ) z}. 2 k Proposition 2.6. If D is an irreducible symmetric domain, the joint Peirce decomposition yields the characteristic multiplicities ⎫ Z00 = (0) ⎪ ⎪ ⎬ Zii = C ei 1 ≤ i ≤ r a 1≤i<j≤r ⎪ dimC Zij = ⎪ ⎭ dimC Z0j = b 1≤j≤r Zij = {z ∈ Z : {ek e∗k z} =

In particular, b = 0 iﬀ D is of tube type. Example. In the matrix case Z = Cr×(r+b) let e1 , . . . , er be the diagonal matrix units. The associated Peirce decomposition looks as follows ⎛ ⎞ Z11 Z12 Z1r Z01 ⎜ Z12 Z22 Z02 ⎟ ⎜ ⎟ ⎜ ⎟ .. ⎝ ⎠ . Z1r

Zrr

Z0r

and we have a = 2 (“complex case”).

3. Weighted Bergman spaces of holomorphic functions We will now introduce an important class of Hilbert spaces belonging to the holomorphic discrete series. Let Z be an irreducible Jordan triple of rank r, with characteristic multiplicities a, b. The dimension is then given by d = r + a r(r−1) + rb 2 or, equivalently, a d = 1 + (r − 1) + b. r 2 The precise meaning of the “second” multiplicity b will be given in Proposition 2.6 below. Deﬁne the genus p = 2 + a(r − 1) + b. The tube type case corresponds to b = 0 or, equivalently, to p2 = dr . Fix a parameter λ > p − 1 = 1 + a(r − 1) + b

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75

and consider the probability measure on D given by dµλ (z) =

ΓΩ (λ) N (z, z)λ−p dz, π d ΓΩ (λ − d/r)

where ΓΩ denotes the Koecher-Gindikin Γ-function of the symmetric cone Ω discussed in more detail below. The weighted Bergman space of holomorphic functions on D is deﬁned as 2 Hλ (D) = {ψ ∈ O(D, C) : dµλ (z) |ψ(z)|2 < ∞}. D

For λ = p, dµp (z) is a multiple of the Lebesgue measure dz and Hp2 (D) = {ψ ∈ O(D, C) : dz |ψ(z)|2 < ∞} D

is the standard (unweighted) Bergman space. The inner product (conjugate-linear in the ﬁrst variable) is given by ΓΩ (λ) dz N (z, z)λ−p ψ1 (z) ψ2 (z). (ψ1 |ψ2 )λ = dµλ (z) ψ1 (z) ψ2 (z) = d π ΓΩ (λ − dr ) D

D

Here we follow the convention of mathematical physics, where Hilbert state spaces, such as Hλ2 (D), have a scalar-product which is conjugate-linear in the ﬁrst variable, whereas the underlying phase space Cn (in its K¨ahler polarization) has the K¨ahler metric conjugate-linear in the second variable. Proposition 3.1. Hλ2 (D) has the reproducing kernel Kλ (z, w) = N (z, w)−λ = det B(z, w)−λ/p . It follows that the orthogonal projection Pλ : L2 (D, dµλ ) → Hλ2 (D) has the form (Pλ ψ)(z) = dµλ (z) N (z, w)−λ ψ(w). D

Deﬁne an irreducible projective unitary representation Uλ : G → U (Hλ2 (D)) of G by putting (Uλ (g −1 ) ψ)(z) = det ψ (z)λ/p ψ(g(z)). Viewed as a quantum deformation, λp = h1 corresponds to the inverse Planck constant. The representations Uλ constitute the scalar holomorphic discrete series of G. Using the Jordan triple determinant, we may describe the analytic continuation of the scale of weighted Bergman spaces [L3], [FK1], [VR], [WA]: Deﬁne theWallach set W (D) = {λ ∈ C : (N (zi , zj )−λ )1≤i, j≤n

0 ∀ z1 , . . . , zn ∈ D}.

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Theorem 3.2. The Wallach set is a disjoint union ' a ( ' ( a W (D) = : 0 ≤ < r ∪ λ > (r − 1) 2 2 consisting of r discrete points and a continuous part. More explicitly, the Wallach parameters are given as follows:

Figure 1. The scale of Wallach parameters z

a (r − 1) 2 }|

{z a (r − 1) 2

0 a 2

1+b }|

{z d r Hardy

a (r − 1) 2 }|

{z p−1

d a + r 2

weighted Bergman }|

{

p Bergman

Example. In the matrix case Z = Cr×(r+b) we have a = 2, and hence dr = r+b, p = 2r + b. The Shilov boundary S is a homogeneous space under the group K, therefore S = K/L where L := { ∈ K : e = e} for some e ∈ S. In case D ⊂ X C is of tube type and e is the unit element of the associated Euclidean Jordan algebra X, L = Aut (X) agrees with the automorphism group of X. The Shilov boundary S = K/L, endowed with the K-invariant probability measure, corresponds to the Wallach parameter d a λ = = 1 + (r − 1) + b r 2 giving rise to the Hardy space H 2 (S) = {ψ ∈ L2 (S) : ψ holomorphic on B}. The associated Szeg˝ o projection P : L2 (S) → H 2 (S) has the form (P ψ)(z) = ds N (z, s)−d/r ψ(s). S

The proof of the fundamental Theorem 3.2 depends on harmonic analysis (PeterWeyl decomposition) for the compact Lie group K. Consider the algebra P(Z) of all polynomials p : Z → C and the natural algebra action K × P(Z) → P(Z) deﬁned by (Uk−1 p)(z) := p(kz). A deep result [S] asserts that Pm1 ≥...≥mr (Z) P(Z) = m1 ≥...≥mr ≥0

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is a direct sum of irreducible K-modules, labelled by all partitions m = (m1 , . . . , mr ) ∈ Nr+ . Moreover, the highest weight vector has the form Nm (z) = N1 (z)m1 −m2 N2 (z)m2 −m3 · · · Nr (z)mr , where Nk is the Jordan determinant of

Zij

1≤i≤j≤k

(generalized minor). This is the main result of [U3]. Example. For the simplest partition m = (1, 0, . . . , 0), P10···0 (Z) = Z is the dual space of linear forms on Z. In this case N10···0 (z) = (e1 |z), where we ﬁx a frame (e1 , . . . , er ). As a Hilbert space completion, we obtain the Segal-Bargmann-Fock space 1 dz e−(z|z) |ψ(z)|2 < ∞} H 2 (Z) = {ψ ∈ O(Z) : d π Z

of entire functions, with reproducing kernel expanded in a series e(z|w) = Km (z, w), m1 ≥···≥mr ≥0

where Km (z, w) denotes the reproducing kernel of Pm (Z) ⊂ H 2 (Z). The FarautKoranyi binomial formula [FK1] is N (z, w)−λ = (λ)m Km (z, w), m

where (λ)m =

r . Γ(λ + mj − (j − 1) a2 ) ΓΩ (λ + m) = ΓΩ (λ) Γ(λ − (j − 1) a2 ) j=1

denotes the Pochhammer symbol. By checking positivity of the coeﬃcients, one obtains the desired conclusion a a W (D) = {λ ∈ C : (λ)m ≥ 0 ∀ m ∈ Nr+ } = { : 0 ≤ < r} ∪ {λ > (r − 1)}. 2 2

4. Toeplitz operators on symmetric domains A deep relationship between analysis on symmetric domains and operator theory concerns C ∗ -algebras generated by Toeplitz operators in the multi-variable setting. In general, for any complex Hilbert space H, let L(H) be the C ∗ -algebra of all bounded linear operators on H. For f ∈ L∞ (D), deﬁne the Bergman-Toeplitz operators Tfλ ∈ L(Hλ2 (D)) for parameter λ > p − 1 by Tfλ ψ = Pλ (f ψ) ∀ ψ ∈ Hλ2 (D).

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More precisely, (Tfλ ψ)(z)

=

dµλ (z) Kλ (z, w) f (w) ψ(w) D

=

ΓΩ (λ) d π ΓΩ (λ − d/r)

dz N (z, z)λ−p N (z, w)−λ f (w) ψ(w).

D

∞

Similarly, for f ∈ L (S), deﬁne Hardy-Toeplitz operators Tf ∈ L(H 2 (S)) for parameter λ = dr by Tf ψ = P (f ψ) ∀ ψ ∈ H 2 (S). More precisely,

(Tfλ ψ)(z) =

ds N (z, s)−d/r f (s) ψ(s).

S

In both cases we consider the Toeplitz C ∗ -algebras Tλ (D) =

C ∗ (Tfλ : f ∈ C(D)) ⊂ L(Hλ2 (D))

T (S) =

C ∗ (Tf : f ∈ C(S)) ⊂ L(H 2 (S))

In principle this could be deﬁned for any Wallach parameter ·

λ ∈ W (D) = Wdisc (D) ∪ Wcont (D), since Hλ is a Hilbert space with reproducing kernel N (z, w)−λ (z, w ∈ D). However, there is a diﬀerence between the discrete and continuous part: Proposition 4.1. Hλ is a Hilbert-module iﬀ λ > a2 (r − 1). It follows that only in this case the Toeplitz operators Tz1 , . . . , Tzd form a d-tuple of bounded operators. For a deeper analysis of Toeplitz C ∗ -algebras, we need some geometric preparations concerning the boundary of symmetric domains. Proposition 4.2. The Shilov boundary has the Jordan theoretic characterization S = ∂ex D = {e ∈ Z : {ee∗ e} = e tripotent, e e∗ invertible, Z0 (e) = (0)}. In the “tube type” case, we obtain the simpler characterization S = {e ∈ Z tripotent : e e∗ = idZ , Z = Z1 (e)}. The boundary structure of a symmetric domain D can now be described in algebraic terms [LO2]. Let D ⊂ Z be the unit ball of a Jordan triple and consider a tripotent c = {cc∗ c}, with Peirce decomposition Z = Z1 (c) ⊕ Z1/2 (c) ⊕ Z0 (c). Then Z1 (c) is a Jordan ∗-algebra with unit element c such that k = rank Z1 (c). On the other hand, Z0 (c) is a Jordan subtriple and the norm satisﬁes z1 + z0 = max (z1 , z0 ) for all z1 ∈ Z1 (c), z0 ∈ Z0 (c). The unit ball D0 (c) = {z0 ∈ Z0 (c) : z0 < 1}

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is a symmetric domain of lower rank r − k and c + D0 (c) ⊂ ∂D, since c + z0 = max ( c , z0 ) = 1. !" !" =1

(r − 1) 2 in the continuous Wallach set (Hilbert modules), or the subclass λ > 1 + a(r − 1) + b corresponding to the weighted Bergman spaces (discrete series). Let c = {cc∗ c} be a tripotent of rank k. Then D0 (c) ⊂ Z0 (c) is an irreducible symmetric domain of rank r − k, with characteristic multiplicities a, b. Moreover, the shifted parameter a a λ − k > (r − k − 1) 2 2

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81

belongs to the “little” continuous Wallach set. Similarly, in the Bergman case a λ − k > 1 + a(r − k − 1) + b 2 belongs to the “little” discrete series. Theorem 4.6. Let Tλ (D) be the Toeplitz C ∗ -algebra acting on Hλ2 (D) (weighted Bergman space). Then (i) For any tripotent c = {cc∗ c}, there exists a unique irreducible ∗-representation σc : Tλ (D) → Tλ−k a/2 (D0 (c)) such that D (c)

0 σc (TλD (f )) = Tλ−k a/2 (fc )

for all f ∈ C(D), with restriction fc ∈ C(D0 (c)) deﬁned by fc (w) := f (c + w) for all w ∈ D0 (c). (ii) The representations σc are mutually inequivalent (iii) each irreducible ∗-representation of Tλ (D) has this form, i.e., similar to the Hardy case, we may identify Spec Tλ (D) with the set of all tripotents c = {cc∗ c} of Z. Note that the structure and representations of these Toeplitz C ∗ -algebras remain “rigid” along the continuous part of the Wallach set, since we are dealing with a “non-commutative” topological structure. The discrete points, however, show a diﬀerent behavior.

5. Convolution C ∗ -algebras on non-commutative Hardy spaces It turns out that the structure theorems for Toeplitz C ∗ -algebras belong to a much more general theory concerning convolution operators on compact symmetric spaces. Thus we may consider situations where only the Shilov boundary S, but not the domain D itself, is a symmetric space. In this section we use “left” quotient spaces which are more convenient for convolution algebras. (Accordingly the domain D, which in this section plays only a minor role, should be realized as D = K \ G.) Let S =L\K be a homogeneous compact manifold, with stabilizer group L = {k ∈ K : e · k = e}. Here e ∈ S is the “unit element”. L2 (K) carries the left-regular K-action (t f )(s) := f (t−1 s) ∀ s, t ∈ K, f ∈ L2 (K). Consider the group C ∗ -algebra C∗ (K) = C ∗ (u : u ∈ L1 (K))

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generated by all left-convolution operators (u f )(s) := du(t)(t f )(s). K

Its weak closure W ∗ (K) = W ∗ (t : t ∈ K) is the group W ∗ -algebra. Identify L2 (S) ≈ {f ∈ L2 (K) : f (k) = f (k) ∀ ∈ L} L2 (K) closed

with all left L-invariant functions, and consider the orthogonal projection π : L2 (K) → L2 (S) deﬁned via averaging over L:

dt f (k).

(πf )(ek) = L

Note that in this setting we write S as a left quotient space, endowed with a right K-action. Proposition 5.1. For symmetric domains, the Szeg˝ o distribution dE(s) = N (s, e)−d/r on S deﬁnes via convolution the orthogonal projector E : L2 (K) → H 2 (S). π

P

Proof. Consider the orthogonal projections L2 (K) −→ L2 (S) −→ H 2 (S). The substitutions t = e · τ and k = στ −1 yield (P ◦ π) f (e · σ) = dt N (e · σ, t)−d/r (πf )(t) = dτ N (e · σ, e · τ )−d/r d f (τ ) S

K

−d/r

dτ N (e · σ, e · τ )

= =

dτ N (e · στ

f (τ ) =

K

L

−1

−d/r

, e)

f (τ )

K

dk N (e · k, e)−d/r f (k −1 σ) = (N (−,e)−d/r f )(σ).

K

In this special case, the binomial formula yields the expansion d Km (s, e) N (s, e)−d/r = r m m1 ≥···≥mr ≥0

into K-spherical functions.

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We will now consider “non-commutative” Hardy spaces more generally. If K is a compact Lie group with involution σ : K → K and K σ = {k ∈ K : σ(k) = k} denotes the ﬁxed point subgroup, then S = K σ \ K = Kσ is a compact symmetric space, and we have the Peter-Weyl decompositions Kα ⊗ Kα , L2 (K) = ˆ α∈K

ˆ is the unitary dual of K (irreducible representations), and where K L2 (S) = Kα ⊗ φα , ˆσ α∈K

where ˆ σ = {α ∈ K ˆ : Kα contains a non-zero K σ -invariant vector} Sˆ = K is the subset of all spherical representations. Here Kα denotes the (ﬁnite-dimenˆ In terms of the Lie algebra decomposition sional) representation space of α ∈ K. k = kσ ⊕ kσ , consider a maximal torus t = tσ ⊕ tσ , where kσ ⊃ tσ = i a is a maximal abelian “Cartan subspace”. Deﬁne r := dim tσ to be the rank of S. Then there exist natural embeddings as discrete subsets ˆ → i t = L(t, i R) (highest weight) K and Sˆ → a (restricted highest weight). Now let V ⊂ a be an arbitrary open polyhedral cone. The basic result for Fourier analysis on H 2 (S) is the following [L1], [L2]. Theorem 5.2. Let V ⊂ a denote the (closed) dual cone. Then the Hardy space corresponding to the (open) cone V has a Peter-Weyl decomposition HV2 (S) = Kα ⊗ φα . ˆ α∈S∩V

Moreover, the Cauchy-Szeg˝ o distribution can be expanded into a series EV (s) = φα (s) ˆ α∈S∩V

of spherical functions, and HV2 (S) can be realized via boundary values of holomorphic functions on the K-invariant domain DV = K exp (V )(e) ⊂ S C .

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Here the complexiﬁcation S C = (K σ )C \ K C exp

is a complex manifold and, choosing a “Weyl chamber” t+ σ ⊂ tσ ⊂ k −→ K, we obtain a polar decomposition S C = K exp (i tσ ) and may deﬁne the complex phase space (“non-commutative tube domain”) by DV := K exp (V ) where V ⊂ t+ σ is an open polyhedral convex cone. Letting Λ = V ⊂ i tσ be the polar cone, the “non-commutative” Hardy space HV2 (S) := Kα ⊗ φα ˆ σ ∩Λ α∈K

plays the role of “Hilbert state space” in this setting. Theorem 5.3. If DV is pseudoconvex (i.e., a domain of holomorphy), then one may identify HV2 (S) = {h ∈ L2 (S) : h holomorphic on DV }. Using Proposition 5.1, we show next that the Hardy space of hermitian symmetric domains (at least of tube type) ﬁts into the more general framework. In fact, let K be the compact Lie group of Jordan (triple) automorphisms, with involution subgroup K σ consisting of all Jordan algebra automorphisms. In this case, on the Lie algebra level, kσ corresponds to the Jordan multiplication operators and the Cartan subspace i tσ is obtained by diagonal multiplication operators, isomorphic to Rr via spectral theory. The associated cone V = Rr> is precisely the positive octant, a very special case of a polyhedral cone. The Hardy space decomposition ⊕ Pm1 ,...,mr H 2 (S) = m1 ≥···≥mr ≥0

into irreducible K-modules labelled by integer partitions, with Pm1 ,...,mr , generated by the highest weight vector N1 (z)m1 −m2 N2 (z)m2 −m3 · · · Nr (z)mr , where N1 , . . . , Nr are the Jordan theoretic minors ⎞ ⎛ N1 ⎟ N2 ⎜ ⎟ ⎜ ⎟, ⎜ N3 ⎝ ⎠ .. .. . .

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is now understood by realizing the partitions within the dual cone of V . Moreover, the Cauchy-Szeg˝ o kernel N (z, w)−d/r = det (I − zw∗ )−d/r will now be interpreted as a convolution kernel on the Shilov boundary. In principle, the same could be done for the Bergman kernel N (z, w)λ = det (I − zw∗ )−λ , for parameter λ > p − 1. Thus also the Berezin quantization could be realized on the Shilov boundary. Returning to the general case of a compact symmetric space S = L \ K, let P : L2 (S) → HV2 (S) be the orthogonal projection, realized as a convolution operator (PV h)(s) := EV (st−1 ) h(t) dt, K

where EV (s) =

φα (s)

ˆ σ ∩V α∈K

is the Cauchy-Szeg˝o distribution. We may regard KV (s, t) := EV (st−1 ) as a generalized “reproducing kernel”. If f ∈ C(S) is a continuous symbol function, let TS (f ) = PV f PV be the associated Toeplitz operator (depending on V ), given by TS (f ) h := PV (f h). These operators are in general non-commuting, but satisfy TS (f )∗ = TS (f ). The Toeplitz C ∗ -algebra TV (S) := C ∗ (TS (f ) : f ∈ C(S)) is the uniform closure of the algebra generated by all operators TS (f ) with continuous symbol f . In order to describe the generalization of Theorem 4.6 to the general case of compact Lie groups K, let again V be a polyhedral cone, with polar cone V realized in the (dual) Cartan subspace. The basic geometric idea, generalizing the boundary stratiﬁcation of symmetric domains, is as follows: For any face A of V there exists a subgroup KA (not necessarily closed) with Cartan subspace A spanned by the face. Consider the associated foliation K/KA of Kronecker type, which is Hausdorﬀ iﬀ KA is closed. + Theorem 5.4. There exist C ∗ -ideals IKA IK TV (S) in the Toeplitz C ∗ -algebra, A + with subquotient IKA /IKA ≈ C ∗ (K/KA ) isomorphic to the foliation C ∗ -algebra.

The proof is based on C ∗ -duality theory [LPRS], [NT]. As a ﬁrst step, TV (S) is identiﬁed with a corner of the co-crossed product Cˆ ∗ (K) ⊗ C 0 (K) δ

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for a suitable co-action δ on a “completed” C ∗ -algebra Cˆ ∗ (K). On the other hand, the foliation C ∗ -algebra C ∗ (K/KA ) = C0 (K) ⊗ C ∗ (KA ) can be realized as a crossed product [CO]. In view of the well-known Katayama duality K(L2 (K)) = C ∗ (K) ⊗ C 0 (K) δ

one obtains C ∗ -representations of TV (S) on L2 (KA ). This is of course the crucial step, and we give a sketch of the construction. Let T (A) be the tangent space of ˜ such that T (A) ⊂ m ˜ is maximal A. Then there exists a Lie subalgebra ˜k = ˜l ⊕ i m, abelian. Now consider the (non-closed) analytic subgroup KA := exp ˜k = Kc × Ke , which can be factored into a compact times Euclidean part. o distribution relative to the face A. Then there Theorem 5.5. Let EA be the Szeg˝ exists a C ∗ -representation σ

A TV (S) −→ L(L2 (KA )),

which maps the Toeplitz operator T (f ) to TA (fA ), where fA := f |KA is the restriction and TA is the Toeplitz operator deﬁned relative to EA . Proof. Let Cˆ ∗ (K) be the C ∗ -algebra generated by all left convolution operators λgE with g ∈ A(K), the Fourier algebra of K. Then there exist representations A(K) f

−→ L(L2 (KA )) → fA

and

Cˆ ∗ (K) −→ L(L2 (KA )) → λgA EA . λgE The latter representation is best described via the Fourier decomposition W (K) ∼ = ∗

(∞)

L∞ (Kα ),

ˆ α∈K

with λu → u ˆα =

u(s)∗ sα ds

K

denoting the Fourier coeﬃcient. Here sα denotes the action of s ∈ K for the representation α. Now let g(s) = (φ|sγ ψ) be a matrix coeﬃcient function, with φ, ψ ∈ Hγ belonging to a not necessarily irreducible K-representation. Then an / ˆ ˆ (γ ⊗ α) ψ. Now let β ∈ K ˆ A , and let α ∈ K easy computation shows gE(α) = φ∗ E ˜ β ⊂ Kα as a KA -invariant be a sequence such that α|KA β. Realizing K / subspace, one shows that gE(α| ˜ β ) g A EA (β). In this way the representation K is constructed via a limiting argument. For more details, cf. [U5], [W].

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Once the representations σA are constructed, Theorem 5.4 can be expressed as follows: Deﬁning 0 + IA = Ker σA IA = Ker σB B A

it follows that

+ IA /IA ∼ = C ∗ (FA ),

where FA = K/KA is the Kronecker foliation associated with the non-closed subgroup KA . This C ∗ algebra is of type I iﬀ KA is closed. The “local version” can be reformulated in the following global version: There exists a C ∗ -ﬁltration K = I1 I2 · · · Ir TV (S), such that for every k ⊕ ∼ Ik+1 /Ik = C ∗ (FA ). codim A=k

In the special case of symmetric domains, the composition factors Ik+1 /Ik ∼ = C(Sk ) ⊗ K of the ﬁltration are even realized as ﬁbre bundles over the compact manifold Sk of all tripotents (partial isometries) of rank k. Hence in the case of symmetric domains we obtain a ﬁbration of boundary faces instead of a foliation in the more general setting. A few remarks about the associated index theory. In full generality the analytic index is given by a mapping ⊕ Ind K1 C ∗ (FB ) −→A K0 (C ∗ (FA )), BA

and should be related to a longitudinally elliptic operator DA along the foliation FA . The details of this construction have only been worked out for a basic class of non-symmetric domains, the so-called Reinhardt domains. In this case K = S = Tn is the n-torus, whereas L = {1}. The complexiﬁcation S C = (C \ {0})n ⊂ Cn is open and dense. A domain D ⊂ Cn is called Reinhardt iﬀ (z1 , . . . , zn ) ∈ D =⇒ (ei t1 z1 , . . . , ei tn zn ) ∈ D. Let |D| = {(|z1 |, . . . , |zn |) : z ∈ D} ⊂ Rn+ be the associated “absolute” domain and deﬁne V = log |D| ⊂ Rn− ,

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H. Upmeier

assuming that D is contained in the unit polydisk. Important examples, for n = 2, are the L-shaped domains and the so-called Hartogs’ wedge, shown in Figure 2. Figure 2. L-shaped domains and the Hartogs’ wedge |z2 |

|z2 |

T2

2 (1, 1) T

T2 log |z2 |

|D| log |z1 |

V = log |D|

|D| log |z2 |

|z1 |

|z1 |

slope − θ1

V = log |D|

log |z1 |

Via groupoid methods, Toeplitz operators on Reinhardt domains have been studied in [CM], [SSU]. In the special case of Hartogs’ wedge, one obtains the following Theorem 5.6. [SSU]: Let V = VΘ1 , Θ2 be the convex cone with slopes Θ1 and Θ2 , coming from Hartogs’ wedge. Then the Toeplitz C ∗ -algebra has a 2-step ﬁltration K I2 T (T2 ). Here T (T2 )/I2 = C(T2 ) is the commutator ideal, and I2 /K is stably isomorphic to AΘ1 ⊕ AΘ2 , where AΘ = C ∗ (u, v : uv = e2πiΘ vu)

unitary:

denotes the irrational rotation algebra (non-commutative torus) induced by the Kronecker foliation T2 /RΘ (cf. Figure 3). Figure 3. The Kronecker foliation T2 /RΘ

Θ

Based on this C ∗ -algebra ﬁltration, one obtains a real-valued index theorem: tr

Z2 ≈ K 1 (T2 ) −→ K0 (AΘ ) −→ R, ≈

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89

corresponding to the dense embedding (m, n) → m + n Θ. These results have also applications to complex analysis, for example the existence of smooth (Reinhardt) domains with non-compact ∂-Neumann operator, and a characterization of proper holomorphic maps between Reinhardt domains in terms of their associated Toeplitz C ∗ -algebras. It is interesting to note that Reinhardt domains are closely related to multi-variable Wiener-Hopf operators [MR], [U12], by embedding higher-dimensional cones in compact spaces via the Cayley transform.

6. Quantization of Hermitian symmetric domains A diﬀerent area of interaction between analysis on symmetric domains and Hilbert space operator theory concerns quantization methods in the sense of mathematical physics. Let D = G/K be a hermitian symmetric space of non-compact type, realized as the open unit ball of a Jordan triple Z = Cd . The group G = {biholomorphic automorphisms of D} = Aut(D) is a semi-simple Lie group, with maximal compact subgroup K = {Jordan triple automorphisms} = Aut(Z) ⊂ GL(Z). Example. In case D = {z ∈ Cp×q : z < 1} is the matrix ball for the operator norm, Z = Cp×q has the Jordan triple product uv ∗ w + wv ∗ u , 2 and we obtain the Moebius transformations ab G = SU (p, q) (z) = (az + b)(cz + d)−1 , cd whereas K = S(U (p) × U (q)) consists of the linear isometries. Here r = min (p, q) is the rank of D. {uv ∗ w} =

Example. In case D = {z ∈ Cr×r : z < 1, z t = z} consists of symmetric matrices, the tangent space Z = {z ∈ Cr×r : z t = z} becomes a Jordan sub-triple of Cr×r , and G = Sp (2r, R) ⊃ K = U (r). In the following we will also consider the ﬂat case D = Cn ≈ T ∗ (Rn ), identiﬁed with the cotangent bundle. Here G = U (n) Cn is a semi-direct product.

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The quantization Hilbert spaces are the weighted Bergman spaces described in more detail in Section 3. Consider the G-invariant measure dµ0 (z) = N (z, z)−p dz, where N (z, w) denotes the “Jordan triple determinant” and p is the genus. Example. For the matrix ball, N (z, w) = Det (I − zw∗ ) and the genus is p + q. Deﬁne the scalar holomorphic discrete series as follows. For ν > p − 1, ΓΩ (ν) N (z, z)ν−p dz π d ΓΩ (ν − d/r) is a probability measure and the associated weighted Bergman space dµν (z) =

Hν2 (D) := {ψ ∈ L2 (D, µν ) : ψ holomorphic} has the projective unitary representation (Uν (g −1 )ψ)(z) = (Det g (z))ν/p ψ(g(z)) and the reproducing kernel ν Kν (z, w) = N (z, w)−ν = Kw (z).

Accordingly, we have

ψ(z) =

dµν (w) Kν (z, w) ψ(w)

∀ ψ ∈ Hν2 (D).

D

For the ﬂat case,

ν d

e−ν(z|z) dz π denotes the Gauss measure, with corresponding Segal-Bargmann-Fock space dµ (z) =

H2 (Cn ) := {ψ ∈ L2 (Cn , dµ ) : ψ holomorphic} and its unitary representation (π (tb ) ψ)(z) = eν(z|b)−ν(b|b)/2 ψ(z − b) for the translations tb induced by b ∈ Cn . Here = ν1 plays the role of Planck’s constant. By deﬁnition [AU4] a covariant functional calculus (or “quantization”) is determined by a densely deﬁned ∗-linear mapping L2 (D, dµ0 )

A

−→

∗−linear

L2 (Hν2 (D)) (Hilbert-Schmidt operators),

associating to f the operator Af with active symbol f . The covariance condition can be expressed as Af ◦g−1 = Uν (g) Af Uν (g)−1 .

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91

The adjoint mapping A∗ : L2 (Hν2 (D)) −→ L2 (D, dµ0 ), associating to an operator T the passive symbol A∗T of T , is formally deﬁned by duality dµ0 (w) A∗T (w) f (w) = tr T ∗ Af . D

Composing both mappings, we obtain the link transform (or generalized Berezin transform) B = A∗ A : L2 (D, dµ0 ) → L2 (D, dµ0 ), which is a G-invariant pseudo-diﬀerential operator. It is often convenient to express the quantization by an integral representation Af = dµ0 (w) f (w) Aw , D

where w → Aw is a covariant ﬁeld of (bounded) operators, satisfying the covariance condition Ag(w) = Uν (g) Aw Uν (g)−1 . For any (non-compact) Hermitian symmetric space, we have the Plancherel decomposition dλ 2 Gλ L (G/K) = |c(λ)|2 Rr

into a direct integral of principal series representations πλ, where c(λ) is HarishChandra’s c-function [HE]. Here, by deﬁnition, the representation space Gλ contains a spherical vector φλ, explicitly given by φλ = dk πλ(k) eλ. K

For symmetric domains of tube type, eλ(z) = N (z + z ∗ )λ+ρ denotes the conical functions for λ = (λ1 , . . . , λr ), where as before we put N (x)λ = N1 (x)λ1 −λ2 N2 (x)λ2 −λ3 · · · Nr (x)λr and N1 , N2 , . . . , Nr = N are the Jordan algebraic minors. The half-sum ρ of positive roots has the components 1+b a (j − 1) + . 2 2 Now let A be a G-covariant calculus. The Berezin transform ρj =

σ ◦ A = A∗ ◦ A

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H. Upmeier

is G-invariant and hence, by diagonalization, ∗ A(λ) · id σ ◦ A|Gλ = A Gλ , ∗ A(λ) = σ(A where A eλ )(0) is the λ-eigenvalue. For the spectral resolution of Berezin-type transforms it suﬃces to determine these eigenvalues or, equivalently, to express σ ◦ A in terms of some set of generators ∆1 , ∆2 , . . . , ∆r of the algebra of G-invariant diﬀerential operators on D, where ∆1 is the Laplace-Beltrami operator. The primary example of a covariant quantization is the Toeplitz-Berezin calculus

T ν : L∞ (D) −→ L(Hν2 (D))

(bounded operators),

associating to f the Toeplitz operator Tfν deﬁned via Tfν ψ = Pν (f ψ), where Pν : L2 (D, dµν ) → Hν2 (D) is the orthogonal projection [B2], [B3]. Its integral representation has the form ν (Tf ψ)(z) = dµν (w) Kν (z, w) f (w) ψ(w). D

Therefore the adjoint (TT∗ )(w) =

ν ν (Kw |T Kw ) Kν (w, w)

coincides with the Berezin symbol and the deﬁning operator ﬁeld Twν is given by ν the rank 1 projection onto C · Kw . It follows that the Berezin transform f → Bf = ∗ T T f has the integral representation Kν (z, w) Kν (w, z) f (w), (Bf )(z) = cν dµ0 (w) Kν (z, z) Kν (w, w) D

where cν is chosen so that B1 = 1. In the ﬂat case of the Segal-Bargmann-Fock space H2 (Cn ), it has been shown in [G] that Bf = e−∆ f coincides with the heat semi-group. In the curved setting, we have instead [BLU], [E]: Theorem 6.1. For any bounded symmetric domain, the Toeplitz quantization satisﬁes the correspondence principle in the following sense (i) Tfν Tgν − Tfνg → 0 ν → 0. (ii) ν[Tfν , Tgν ] − i T{f,g} Here {f, g} denotes the G-invariant Poisson bracket.

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For the explicit calculation of the eigenvalues of the Toeplitz-Berezin transform, we pass to the Siegel domain realization Z = U ⊕ V , via the biholomorphic Cayley transform D −→ {(u, v) ∈ Z : u + u∗ − {ev ∗ v} ∈ Ω} C

≈

onto a symmetric Siegel domain. Here U = X C is the complexiﬁcation of a Euclidean Jordan algebra, x ◦ y denotes the Jordan product and Ω = {x2 : x ∈ X invertible} is the positive cone of X. In general, V is a complex representation space of X, with V = {0} corresponding to the tube type. The Cayley transform is given by √ C(u, v) = ((e + u) ◦ (e − u)−1 , 2 v ◦ (e − u)−1 ), where e is the unit element of X. By the classiﬁcation, we may identify X with the algebra Hr (K) of all self-adjoint (r×r)-matrices over K, under the anti-commutator . The associated matrix entries belong to product x ◦ y = xy+yx 2 ⎧ r≥4 ⎨ R, C, H R, C, H, O r = 3 exceptional K= ⎩ Euclidean space r = 2 spin factors, and we call a = dimR K the characteristic multiplicity. Using the conical functions N s (x) = N1 (x)s1 −s2 N2 (x)s2 −s3 · · · Nr (x)sr ,

the Euler integral r d1 −r . a ΓΩ (s) = dx N (x)−d1 /r N s (x) e−(x|e) = (2π) 2 Γ (sj − (j − 1)) 2 j=1 Ω

is called the Koecher-Gindikin Γ-function and d1 = d − rb is the dimension of X. Using this fundamental concept, the Toeplitz-Berezin eigenvalues have been computed in [UU], yielding the beautiful formula T ∗ T (λ) =

ΓΩ (ρ + ν −

d r

+ λ) ΓΩ (ρ + ν −

ΓΩ (ν −

d r ) ΓΩ

(ν)

d r

− λ)

,

where ρ denotes the half-sum of positive roots. A second example of a covariant calculus is the Weyl quantization W : L2 (D, dµ0 ) −→ L2 (Hν2 (D)), associating to f the Weyl operator Wf deﬁned via its integral representation Wf = dµν (w) f (w) Uν (sw ) D

[UU1], [UU2]. Here sw ∈ G denotes the geodesic symmetry, uniquely determined by sw (w) = w, sw (w) = −Id.

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H. Upmeier Figure 4. The geodesic reﬂection

sw (z) w z

By duality, we obtain the Weyl symbol WT∗ (w) = tr T Uν (sw ). In the ﬂat case, we have the formulas sw (z) = 2w − z, s0 = −id, sw = tw ◦ s0 ◦ t−w which induce for the complex wave Hilbert space H2 (Cn ) the representations = e(z|b)−(b|b)/2 ψ(z − b) and

(π(tb )ψ)(z)

= ψ(2w − z) e2(z−w|w).

(π(sw )ψ)(z) This yields the explicit form (WfC ψ)(z) =

dw f ( Cn

z + w (z−w|z+w)/2 )e ψ(w). 2

Remark 6.2. Both in the setting of bounded symmetric domains and for the ﬂat case, the quantization Hilbert spaces have also a real representation via a socalled Segal-Bargmann transform. For the real wave Hilbert space L2 (Rn ), the representations (π(ty,η ) φ)(x)

=

ei(y−2x)η φ(x − y),

π(sy,η φ)(x)

=

e4i(y−x)η φ(2y − x)

lead to (WfR

ψ)(x) =

dy Rn

dη f

Rn

x+y ,η 2

e2i(y−x)η φ(y).

It should be noted that the well-known Moyal ∗-product arises this way. In order to determine the eigenvalues of the link transform for the Weyl calculus, let G = KAN be the Iwasawa decomposition, and for λ ∈ a let eλ denote the conical function of G/K, with φλ = dk k eλ K

the associated spherical function. Realized as a Siegel domain G/K = {(u, v) : u + u∗ − 2{ev ∗ v} ∈ Ω},

Toeplitz Operator Algebras and Complex Analysis

95

the conical functions have the form eλ(u, v) = N λ+ρ (u + u∗ − 2{ev ∗ v}). Taking o = (e, 0) as a base point, we ﬁrst apply the so called product formula [AU4], [AU6], which asserts that whenever A, B are two covariant operator calculi on Hν2 (D), the eigenvalues of the link transform can be expressed as ∗ B (λ) = A

(Aeλ Ko |Ko )(Ko |Beλ Ko ) ∗ T (λ) T

.

Hence it suﬃces to compute (Aeλ Ko |Ko ) = (Aφλ Ko |Ko ). After this simpliﬁcation, the Weyl transform eigenvalues (at least for domains of rank 1) have been determined in [AU6]: Theorem 6.3. Let D = {z ∈ Cd : (z|z) < 1} be the unit ball of rank 1. Then the Weyl calculus satisﬁes Γ(ν − d2 + λ)Γ(ν − d2 − λ) d−ρ+λ d−ρ−λ α , (Ko |Weλ Ko ) = 2 F1 Γ(ν)Γ(ν − d) α−1 ν where 2 F1 is the Gauss hypergeometric function. Note that the product of Γ-factors, analogous to the Toeplitz-Berezin case, is still present, but has to be supplemented by the hypergeometric function. It is expected that a similar phenomenon holds for higher rank and also for more general functional calculi. A common generalization of both the Toeplitz-Berezin calculus and the Weyl calculus is the so-called interpolating calculus [AU7] which depends on an additional parameter α. For the Fock space we assume α ∈ C, whereas for bounded symmetric domains we assume that α ∈ B, the closed unit disk. Consider the map z → sα (z) := αz on the domain D. By conjugation, we deﬁne for any w ∈ D, α −1 sα w := g s g where g ∈ G satisﬁes g(0) = w. The interpolating calculus for parameter α is deﬁned by the local operator ﬁeld α α (Aα w ψ)(z) := ψ(sw (z)) j(sw (z))

where j denotes the Jacobian cocycle. In the ﬂat case D = Cd we have sα w (z) = αz + (1 − α) w and therefore ν(1−α)(z−w|w) . (Aα w ψ)(z) = ψ(αz + (1 − α) w) e

If α = 0, (A0w ψ)(z) = ψ(w) eν (z−w|w)

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and one obtains the Toeplitz calculus ν d (A0f ψ)(z) = dw f (w) ψ(w) e−ν(w|w) eν(z|w) . π Cd

If α = −1,

(z) = 2w − z s(−1) w is the geodesic symmetry and one obtains the Weyl calculus ν (−1) (Af ψ)(z) = ( )d dw f (w) ψ(2w − z) e−2ν(w|w) e2ν(z|w) . π Cd

It is shown in [AU4] that for α → ∞, the Wick calculus, formally given by hk → Th Tk∗ , is obtained via a suitable limit procedure. Theorem 6.4. For the interpolating α-calculus, in the ﬂat case, the link transform is given by 1 − |α|2 ∆ , A∗α Aα f = exp |1 − α|2 ν where ∆ is the Laplacian. The general set-up for Moyal products of quantized operators (also called ∗-products) is as follows: Let A, B, C be covariant calculi and f, g ∈ C ∞ (D) be symbol functions. Then one has the quantized operators Af , Bg and deﬁnes the “weak” Moyal product by f • g := C ∗ (Af Bg ). This is in general not associative. In contrast, the “strong” Moyal product f g ∈ C ∞ (D), uniquely deﬁned by Cf g = Af Bg , is associative on the formal level, but in general there is no such function f g which yields the required operator product. In the Toeplitz operator calculus, one has instead an asymptotic expansion [E] 1 Tf Tg = TCk (f,g) , νk k≥0

where Ck (f, g) are uniquely determined G-covariant bidiﬀerential operators. Therefore, in this case, 1 Ck (f, g). f g = νk k≥0

The Correspondence Principle [BLU] gives the following lower-order terms (i) Tfν Tgν − Tfνg → 0 implies that C0 (f, g) = f g, ν → 0 implies that C1 (f, g) is related to the Poisson (ii) ν [Tfν , Tgν ] − i T{f,g} bracket.

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In the ﬂat case on the Segal-Bargmann-Fock space one has a more or less complete theory of Moyal products, even in the general setting of the interpolating calculus. Let f, g, f g ∈ C ∞ (Cd , C) be symbol functions such that γ β Aα f Ag = Af g .

Then the Moyal product f g can be expressed as an integral (f g)(ζ) = dξ dη f (ξ) g(η) M(ξ, η; ζ). Cd

Cd

Equivalently,

Aα ξ

Aβη

dζ Aγζ M(ξ, η; ζ)

= Cd

for the corresponding operator ﬁelds. The following result appears in [AU7]. Theorem 6.5. For α, β, γ in the unit disk, subject to the restrictions αβ = γ and Re

(1 − γ)2 (1 + αβ) > 0, (αβ − γ)(1 + γ)

the 3 × 3 matrix ⎡

(1 − α)(β − γ) (1 − α)(1 − β) γ Φ = ⎣ (1 − α)(1 − β) (1 − β)(α − γ) (γ − 1)(1 − α) (γ − 1) α (1 − β)

⎤ (γ − 1)(1 − α) β (γ − 1)(1 − β) ⎦ (1 − γ)(1 − αβ)

has all rows and columns summing up to 0, and the integral kernel for the Moyal product is given as follows ⎛ ⎞ ξ ν (ξ, η, ζ) Φ ⎝ η ⎠ . M(ξ, η; ζ) = exp αβ − γ ζ Note that M is invariant under U (d) × Cd , acting jointly on ξ, η, ζ.

7. Deformation of real symmetric domains It turns out that similar results hold for the more general class of real symmetric domains, which are deﬁned as follows. Let D = G/K be a hermitian symmetric domain, with G a semi-simple Lie group of hermitian type, and let z → z be an antiholomorphic involution of D. Deﬁne the real form DR = {z ∈ D : z = z} = GR /KR , where GR = {g ∈ G : g(z) = g(z)} is a reductive Lie group, with maximal compact subgroup KR = K ∩ GR .

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Example. If D = T (Ω) = Ω + iX is a tube domain, the involution (x + iy)∗ = x − iy is the hermitian adjoint, and DR = Ω considered as a real symmetric domain. In this case, GR is only reductive with a non-trivial center. Example. If D = {z ∈ Cp×q |z ∗ z < I} = U (p, q)/U (p) × U (q) is the complex matrix ball, and z → z = (z ij ) is the conjugation, then DR = {x ∈ Rp×q |x∗ x < I} = O(p, q)/O(p) × O(q) coincides with the real matrix ball. Figure 5. Complex and real symmetric domains DC DC = T (Ω)

DR

DR = Ω

− d −c , one Example. For D = {z ∈ C2×2 : z ∗ z < I} and the involution ac db := −b a can show that DR ⊂ H is realized as the quaternion unit ball. Example. (product case) In case DR = GR /KR is itself hermitian symmetric, the complexiﬁcation D = DR × D R is endowed with the ﬂip involution (z1 , z 2 )− := (z2 , z 1 ) and G = GR × GR contains GR as the “diagonal” subgroup. We now introduce the “real” version of a covariant calculus. Let D ⊃ DR be a real symmetric domain, irreducible of rank r, with complexiﬁcation D not necessarily irreducible. Let Hν2 (D) denote the νth Bergman space over D, in case D is irreducible, whereas in the product case we consider Hν2 (DR × DR ) = Op Hν2 (DR ). Here one could also take pairs (ν1 , ν2 ) not necessarily equal. In both cases we obtain the restricted representation Uν |GR which is not irreducible but has a Plancherel decomposition which is multiplicity free. By deﬁnition [AU3], a covariant calculus in the “real” setting is a densely deﬁned linear map A : C ∞ (DR ) → Hν2 (D)

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associating to f a holomorphic function Af , subject to the covariance condition Af ◦g−1 = Uν (g) Af ∀ g ∈ GR . The inﬁnitesimal version corresponds to an integral representation Af = dµR 0 (ζ) f (ζ) Aζ , DR

where Aζ ∈ O(D) is a ﬁeld of “states” satisfying the covariance condition Ag(ζ) = Uν (g) Aζ , g ∈ GR . µR 0

Here is a (suitably normalized) GR -invariant measure on DR . Similar to the complex case, the adjoint transformation A∗ : Hν2 (D) → L2 (DR , dµR 0) is deﬁned by duality ∗ ∗ dµR 0 (ζ) f (ζ) AT (ζ) = (f |AT ) = (Af |T )Hν2 (D) . DR

By composition we obtain the link transform B = A∗ A acting on C ∞ (DR ) which commutes with GR -translations. As in the complex setting, the primary example of a real deformation is the Toeplitz calculus [AU3], [Z] T : C ∞ (DR ) → Hν2 (D), which associates to f the holomorphic function Tf = dµR 0 (ζ) f (ζ) Tζ DR

explicitly given by Tζ (z) =

Kν (z, ζ) ∈ Hν2 (D), ζ ∈ D. Kν (ζ, ζ)1/2

The normalization of µR 0 is chosen so that T1 = I is the identity operator, and hence depends on ν and also on the type of covariant calculus. The adjoint transformation T ∗ : Hν2 (D) → L2 (DR , µR 0) coincides with the so-called weighted restriction (T ∗ h)(ζ) = Kν (ζ, ζ)1/2 h(ζ). In the product case D = DR × DR , one computes easily that Tζ,ζ (z, w) =

Kν (z, w; ζ, ζ) Kν (z, ζ) Kν (ζ, w) = 1/2 Kν (ζ, ζ) Kν (ζ, ζ; ζ, ζ)

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is just the integral kernel of kζ ⊗ kζ∗ . Hence we recover the “complex” Toeplitz calculus. In order to determine the eigenvalues of the Toeplitz-Berezin transform in the real setting consider, as before, the exponentials λ+ρ

eλ (x + y, v) = NX

(x − {ev ∗ v})

and the corresponding Euler type integral Kν (e, ξ) ∗ T (λ) = T (e) = eλ (ξ) dµR T eλ 0 (ξ) Kν (ξ, ξ)1/2 =

dx

dy

BR

λ+ρ (x − {ev ∗ v}) dv NX

−ν NX⊕Y (e + x − y) −ν/2

NX⊕Y (x − {ev ∗ v})

.

The following result [DP], [N], [Z], [AU3] is the real version of the UnterbergerUpmeier formula: Theorem 7.1. Let DR = {x + y + v ∈ X ⊕ Y ⊕ V : x − {ev ∗ v} ∈ Ω} be a real symmetric domain, realized as a real Siegel domain, where U =X ⊕Y is a semi-simple real Jordan ∗-algebra, the self-adjoint part X = {x ∈ U : x∗ = x} is a Euclidean irreducible Jordan algebra and Ω ⊂ X denotes the symmetric cone. Moreover, Y = {y ∈ U : y ∗ = −y}. In this context, the Toeplitz deformation eigenvalues have the form ∗ T (λ) = T

ΓΩ (κ + λ) ΓΩ (κ − λ) ΓΩ (κ + ρ) ΓΩ (κ − ρ)

where ρ is the half-sum of positive roots and κ =

ν 2r

rank (U ) + ρ +

d r

− p.

In order to treat more general deformations, we need the “product formula” for real symbolic calculi [AU3], [AU6]: Theorem 7.2. Let A, B : C ∞ (DR ) → Hν2 (D) be two covariant symbolic calculi, acting on Hν2 (D). The link transform 2 R A∗ B : L2 (DR , µR 0 ) → L (DR , µ0 )

has the Plancherel eigenvalues Aeλ (0) Beλ (0) ∗ B (λ) = 4 A ∗ T (λ) T ∗ T (λ) > 0 is independent of A, B. In particular for all λ ∈ a = Rr , where T ∗ A (λ) = A

1 |Aeλ (0)|2 . ∗ T T (λ)

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Note that Aλ (0) =

101

∼ dµR 0 (ζ) φλ (ζ) σK0 (ζ) = σK0 (λ)

DR

coincides with the spherical Fourier transform of σK0 , K0 ∈ Hν2 (D) being the symbol of the kernel vector at the origin, which is a KR -invariant function. On a somewhat deeper level one can also deﬁne a “real” Weyl calculus W : C ∞ (DR ) → Hν2 (D), which associates to f the holomorphic function Wf = dµR 0 (ζ) f (ζ) Wζ DR

given by Kν (sζ (z), z)1/2 Kν (z, ζ)1/2 , Kν (sζ (z), ζ)1/2 where sζ ∈ GR is the symmetry about ζ ∈ DR ⊂ D. In the ﬂat case D = Cn , DR = Rn , the analogous mapping Wζ (z) =

Wν : L2 (Rn ) → Hν2 (Cn ) is the (unitary) Bargmann transform, given explicitly by ν n/2 ν (Wν f )(z) = dζ f (ζ) exp (2ν(z|ζ) − (z|z) − ν(ζ|ζ)). π 2 Rn

In order to compute the Weyl transform eigenvalues, let m = (m1 ≥ m2 ≥ · · · ≥ mr ≥ 0) be an integer partition, and deﬁne the Pochhammer symbol ΓΩ (α + m) ΓΩ (α) and the multivariable hypergeometric function (α)m (β)m αβ (x1 , . . . , xr ) = Jm (x1 , . . . , xr ). 2 F1 (γ)m (1)m γ m (α)m =

Here Jm are the well-known Jack polynomials [ST], which are proportional to the spherical functions on the symmetric cone Ω regarded as a real symmetric domain of root type (A), restricted to the “diagonal” C e1 + · · · + C er for a Jordan algebra frame e1 , · · · , er of X. The reason why only “discrete” parameters m occur lies in the fact that one really considers the “compact dual” S of Ω. For r = 1, the symmetric domains in K = R, C, H, O (m = 2, Cayley plane) can be expressed as a hyperbolic space m D = {x ∈ Km | xi x∗i < 1}. i=1

102

H. Upmeier Figure 6. Rank 1 real symmetric domains ZR ZC rC ρ

Rm Cm 1

Cm m Cm × C 2

m−1 4

m 2

Hm C2×2m 2 m + 12

O2 C16 V 2 11 2

In this case 2ρ + 1 = a+d 2 , where a = dim K. It follows that ZR is uniquely determined by d = dim ZR and ρ. Generalizing both the Toeplitz and the Weyl calculus, we may also consider two covariant calculi of interpolating type Aα , Aβ : C ∞ (DR ) → Hν2 (DC ) and the associated link transform A∗ α Aβ : C ∞ (DR ) → C ∞ (DR ) ∗ α Aβ (λ) for λ ∈ a . The which is uniquely determined by the eigenvalues A R product formula gives ∗ α Aβ (λ) = A

where

Aeλ (0) = DR

β Aα eλ (0) Aeλ (0) A0eλ (0)

dµR 0 (ζ) eλ(ζ) Aζ (0) =

dµR 0 (ζ) φλ (ζ) Aζ (0)

DR

and φλ is the spherical function. In view of the product formula, the following result [AU7], [AU9] solves the eigenvalue problem for rank 1 domains. Theorem 7.3. For real rank 1 domains DR we have d/2−ρ+λ d/2−ρ−λ α Γ (ν − ρ + λ) Γ (ν − ρ − λ) 2 F1 α−1 d/2+ν−2ρ R α Aeλ (0) = . d/2 d/2−2ρ α Γ (ν) Γ(ν − 2ρ) 2 F1 α−1 d/2+ν−2ρ For the special case of the Weyl calculus (α = −1) considered in [AU6] we have Theorem 7.4. For real symmetric domains of rank 1, the Weyl transform eigenvalues are determined by the formula d/2−ρ+λ d/2−ρ−λ 1 (2) Γ (ν − ρ + λ) Γ (ν − ρ − λ) 2 F1 d/2+ν−2ρ Weλ (0) = . d/2 d/2−2ρ 1 Γ (ν) Γ(ν − 2ρ) (2) 2 F1 d/2+ν−2ρ

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Remark 7.5. In the complex setting ZR = Cm we have d/2 = 2ρ = m and obtain the simpler formula m Γ (ν − m m−ρ+λ m−ρ−λ α R α 2 + λ) Γ (ν − 2 − λ) Aeλ (0) = 2 F1 Γ (ν) Γ(ν − m) α−1 ν which generalizes Theorem 6.3 (for α = −1). Analogous to the complex setting, the real symmetric domains give rise to a so-called Moyal restriction [AU7]: For the Fock space and F ∈ C ∞ (Cd , C), the Moyal restriction F ∈ C ∞ (Rd , C) is deﬁned by C

R γ 2 d Aα F I = AF ∈ Hν (C ).

Writing

dw M(w, ζ) F (w),

(F )(ζ) = Cd

we have C

dζ R Aγζ M(w, ζ)

Aα wI = Rd

where I(ζ) = Kν (ζ, ζ)1/2 . Theorem 7.6. For α, γ in the unit disk, subject to the restrictions α2 = γ and Re the symmetric 3 × 3 matrix ⎡ ⎢ Φ=⎣

(1 − γ)2 (1 + α2 ) > 0, (α2 − γ)(1 + γ)

(1−α) γ 2 α−γ 2 α(γ−1) 2

α−γ 2 1−α 2 γ−1 2

⎤

α(γ−1) 2 γ−1 2

⎥ ⎦

(1 − γ)(1 +

α 2)

has all rows and columns summing up to 0, and ⎞ w ν(1 − α) M(w, ζ) = exp (w w ζ) Φ ⎝ w ⎠ α2 − γ ζ ⎛

for all w ∈ Cd , ζ ∈ Rd .

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8. Vector-valued Bergman spaces and intertwining operators A current area of research in harmonic and complex analysis on symmetric domains concerns vector-valued holomorphic functions. Let D = G/K be a hermitian symmetric domain, and let G = Aut (D) be its holomorphic automorphism group. G is a semi-simple Lie group of non-compact type, and K = {g ∈ G : g(0) = 0} is its maximal compact subgroup. As always, we realize D ⊂ Z as the unit ball of a Jordan triple with triple product {uv ∗ w}. Then K = Aut (Z) = {k ∈ GL (Z) : k {uv ∗ w} = {ku(kv)∗ kw}}. Example. If Z = Cp×q is the matrix triple, uv ∗ w + wv ∗ u {uv ∗ w} = 2 and D = {z ∈ Z : zz ∗ < I} is the matrix ball. Here G = SU (p, q) consists of all Moebius transformations ab (z) = (az + b)(cz + d)−1 cd 6 5 and K = a0 d0 : a ∈ U (p), d ∈ U (q) consists of all linear isometries z → a z d−1 . Example. In case Z = {z ∈ Cp×p : z t = z} is the Jordan subtriple of all symmetric matrices, G = Sp (2p, R) is the symplectic group and K = U (p). One frequently 0} called Siegel’s uses the unbounded realization D = {z = z t ∈ Cp×p : z + z ∗ half-space. Not let E be a unitary representation space for K, with dimC E < ∞. There π is a homomorphism K −→ U (E) into the corresponding unitary group. Example. For K = SU (2), putting E = z1m z2n−m : 0 ≤ m ≤ n = SU (2)n yields all irreducible representations. Example. If K = U (k) and E = Ck is the deﬁning representation, the exterior powers E = Λj Ck , 0 ≤ j ≤ k, are called the fundamental representations. For the group SU (k), the well-known Schur functor construction Em for integer partitions m = m1 ≥ m2 ≥ · · · ≥ m k ≥ 0 of length ≤ k yields all irreducible representations. Example. If K = U (p) × U (q) is the direct product, we may consider the tensor product E = Ep ⊗ Eq∗ = L(Eq , Ep ) of all Hilbert-Schmidt operators, with its canonical U (p) × U (q)-representation, given by a0 (T ) = a T d∗ = πp (a) T πq (d)∗ 0d for T ∈ L(Eq , Ep ).

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In order to make contact with symmetric domains, let N : Z × Z → C be the Jordan triple determinant, which is given by N (z, w) = Det (I − zw∗ ) for matrices Z = Cp×q , and let B : Z × Z → End (Z) be the Bergman operator, denoted by z, w → B(z, w), and deﬁned as B(z, w) x = x − 2{zw∗ x} + {z{wx∗ w}∗ z}. Example. In the matrix case Z = Cp×q , it has been shown above that B(z, w) x = (1 − zw∗ ) x (1 − w∗ z) and hence B(z, w) = L1−zw∗ R1−w∗ z . Proposition 8.1. If z, w ∈ D, then B(z, w) ∈ K C ⊂ GL (Z). In particular, we obtain for matrices: 0 1 − zw∗ ∈ K C, B(z, w) = 0 (1 − w∗ z)−1 whereas in case of the unit disk, B(z, w) x = (1 − zw)2 x, and for the unit ball, B(z, w) x = (1 − (z|w))(x − (x|w) z). Recall that N (z, w) = det B(z, w)1/p . Now let π : K → U (E) be a unitary ﬁnite-dimensional representation, which is not necessarily irreducible. Then, for z ∈ D, π(B(z, z)) =: B(z, z)π is positive deﬁnite and we deﬁne as an induced representation the vector-valued Bergman space Hν2 (D, E) = {ψ : D → E holomorphic: dµ0 (z)N (z, z)ν (ψ(z)|B(z, z)π ψ(z)) < ∞}, D

where dµ0 (z) = N (z, z)−p dz = det B(z, z)−1 dz. If E = C, we recover the scalar-valued Bergman spaces. In case of the unit ball D ⊂ Cd , we have K = U (d) and the inner product has the form dz (1 − (z|z))ν−d−1 ((1 − (z|z))ψ(z)| (I − z ∗ z)π ψ(z)). ! " D

∈GL (d)=K C

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If ψ ∈ Hν2 (D, E) and φ ∈ E (corresponding to the constants), one has the reproducing property (φ|ψ(z))E

= (Bz−π φ|ψ)H 2 (D,E) = dµ0 (w) N (z, z)ν (B(w, z)−π φ|B(w, w)π ψ(w))E . D

This shows that the mapping D × D → GL (E), given by z, w → B(z, w)−π , plays the role of a matrix-valued reproducing kernel. The scalar-valued reproducing kernel N (z, w)−ν = det B(z, w)−ν/p arises as a special case, either by keeping ν and letting π be the trivial representation or, alternatively, by setting ν = 0 and considering the 1-dimensional representation K → U (1) given by k → (det k)ν/p . The full holomorphic discrete series, not necessarily scalar-valued, is related to the following construction. Lemma 8.2. For g ∈ G = Aut (D), z ∈ D the holomorphic derivative satisﬁes (i) g (z) ∈ K C ⊂ GL (Z) and the covariance property (ii) g (z) B(z, w) g (w)∗ = B(g(z), g(w)). Thus we may deﬁne the holomorphically induced representation πν,E : G → U (Hν2 (D, E)) via the formula (πν,E (g −1 ) ψ)(z) = det g (z)ν/p g (z)π [ψ(g(z))] ∈ E or, equivalently, as (πν,E (g −1 ) ψ)(z) = j(g, z)π ψ (g, z), where j : G × D → K C is the group-valued holomorphic cocycle j(g, z) = g (z). The cocycle property means j(g1 g2 , z) = j(g1 , g2 (z)) j(g2 , z). It is well known that for symmetric domains, the group K has a rich and interesting representation theory related to the Hua-Schmid decomposition introduced in Section 3. Let Z be an irreducible hermitian Jordan 7 triple, and consider the algebra P(Z) of all polynomials on Z, identiﬁed with sym Z , where Z is the dual space. Then K = Aut (Z) acts on P(Z) via (π(k) p)(z) = p(k −1 z). As shown in [S] there is a multiplicity-free K-decomposition P(Z) = Pm (Z) m

labelled by integer partitions m = (m1 ≥ m2 ≥ · · · ≥ mr ≥ 0)

Toeplitz Operator Algebras and Complex Analysis

107

of length ≤ r = rank (Z). Moreover, in [U3] it has been shown that Pm (Z) has the highest weight vector Nm (z) = N1 (P1 z)m1 −m2 N2 (P2 z)m2 −m3 · · · Nr (Pr z)mr , where Nk is the kth Jordan algebra minor, and Pk is the kth Peirce projection. Example. We have Nm,0,...0 (z) = (z|e1 )m and Nm,m,...m (z) = N (z)m (for tube type domains). Combining the previous concepts, for the compact group U = K, and the holomorphic cocycle j : G × D → K C given by the holomorphic derivative j(g, z) = g (z) ∈ K C , we obtain a unitary irreducible representation πm : K → U (Pm (Z)) via translation (πm (k) p)(ζ) = p(k −1 ζ). The induced K-homogeneous multiplier representations are realized on the vectorvalued Bergman space Hλ2 (D, Pm (Z)) = {ψ : D → Pm (Z) holomorphic: dz N (z, z)λ−p (ψ(z, ζ)|ψ(z, B(z, z) ζ))Pm (Z) < ∞} D

where, as above, B(z, w) ζ = (1 − zw∗ ) ζ(1 − w∗ z) is the Bergman operator acting on Z. This space carries a projective unitary G-action, which is irreducible and belongs to the holomorphic discrete series λ (g −1 ) ψ)(z, ζ) = det g (z)λ/p ψ(g(z), g (z) ζ). (Um

More generally, let U be a compact Lie group such that K ⊂ U and let J : G × D → UC be a holomorphic multiplier (cocycle), denoted by g, z → J (g, z), which is holomorphic in z ∈ D and satisﬁes J (g1 g2 , z) = J (g1 , g2 (z)) · J (g2 , z). Let π : U → U (E) be a unitary irreducible representation with dimC E < ∞, and denote by π C : U C → GL (E) its holomorphic extension. Let o be the base point of D and choose a real analytic cross-section γz ∈ G, γz (o) = z. Then K(z, z) = J (γz∗ , o)π J (γz , o)π belongs to GL (E) via π C . We let K(z, w) ∈ GL (E) be its sesqui-holomorphic extension. The induced U -homogeneous multiplier representation is realized on

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H. Upmeier

the vector-valued Bergman spaces

H (D, E) = {Ψ : D → E holomorphic: 2

dµ0 (z)(Ψ(z)|K(z, z) Ψ(z))E D

dµ0 (z)(J (γz , o)π Ψ(z)|J (γz , o)π Ψ(z))E < +∞}

= D

endowed with the G-action (Uπ (g −1 ) Ψ)(z) = J (g, z)π Ψ(g(z)). An important special case is given by the symmetric Grassmann manifolds. It is well known [LO2] that GC = Aut (M ), where D ⊂ Z ⊂ M is the compact symmetric dual space. Consider the ﬁnite-dimensional space P m (Z), P n (Z) = m1 ≤n

with its reproducing kernel K n (ω, ζ) = Kωn (ζ) = N (ζ, −ω)n =

(−n)m (−1)|m | Km (ζ, ω).

m1 ≤n

Figure 7. The partitions occurring in P n (Z) n

r

Deﬁne a GC -action on P n (Z) via the kernel vectors γ −1 Kωn = det γ (ω)−n/p Kγn∗ (ω) . Here GC × M → M is the Moebius type action extending the action of G on D. Let tz ∈ GC be the translation, deﬁned by tz (ζ) = z + ζ and consider the “adjoint” (relative to the compact form U ⊂ GC ) t∗−z ζ = ζ z = B(ζ, z)−1 (ζ − {ζz ∗ ζ}), which is called the quasi-inverse. Example. For matrices Z = Cp×q , we have ζ z = (1 − ζ z ∗ )−1 ζ.

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109

A holomorphic cocycle G × D → GC , denoted by g, z → J (g, z), is deﬁned via C J (g, z) = t−1 g(z) g tz ∈ G .

The cocycle property J (g1 g2 , z) = J (g1 , g2 (z)) J (g2 , z) is easily checked. In this situation, we obtain the “Grassmann homogeneous” vector-valued Bergman space Ψ

Hν2 (D, P n (Z)) = {D −→ P n (Z) : D × Z z, ζ → Ψ(z, ζ), hol ν−p dz N (z, z) Ψ(z, B(z, z)1/2 ζ z )2 < +∞} D

since, according to [LO2], γz (ζ) = z + B(z, z)1/2 ζ z ∈ G deﬁnes a real-analytic cross-section, i.e., γz (0) = z, with the property J (γz , 0) ζ = B(z, z)1/2 ζ z . The unitary G-action has the form (Unν (g −1 ) Ψ)(z, ζ) = det g (z)ν/p det g (z + ζ)−n/p Ψ(g(z), g(z + ζ) − g(z)). Theorem 8.3. There is a multiplicity-free G-decomposition ⊕ Hν2 (D, P n (Z)) = Hν2 (D, Pm (Z)), n≥m1 ≥···≥mr ≥0

showing that the representation is not irreducible. Nevertheless, the Toeplitz C ∗ -algebra T generated by coordinate multiplications acts irreducibly on Hν2 (D, P n (Z)). Conversely, one may ask whether every irreducible T -module is of this kind? This is closely related to the problem of classiﬁcation of homogeneous holomorphic vector bundles [KM]. The explicit intertwining operators use the Faraut-Koranyi binomial formula N (z, w)−λ = det (1 − zw∗ )−λ = (λ)m K m (z, w) m

where e(z|w) =

K m (z, w)

m

is expanded into the series of reproducing kernels for Pm (Z). Here r .

a (i − 1))mi 2 i=1 partitions having the is the multi-Pochhammer symbol, and we consider the n+r r form n ≥ m1 ≥ m2 ≥ · · · ≥ mr ≥ 0. (λ)m =

(λ −

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In the simplest (scalar-valued) case the intertwiner is given by (−n)m (I f )(z, ζ) = Km (ζ, ∂z ) f (z). (λ)m m

9. Analytic continuation of Wallach parameters Another recent generalization of Bergman spaces concerns the analytic continuation of the Wallach set. Let Pm (Z) be the space of all polynomials of type m = m1 ≥ · · · mr ≥ 0, and let K C Pm (Z) be the irreducible action deﬁned by (h p)(z) = p(h−1 z). Consider the Bergman operator B : Z × Z → End (Z) deﬁned by B(z, w) = id − 2 z w∗ + Qz Qw where Qz ζ := {zζ ∗ z}. Then B(z, w) ∈ K C if z, w ∈ D. Example. For matrices Z = Cp×q the Bergman operator is given by B(z, w) x = (1 − zw∗ ) x (1 − w∗ z). As discussed already, the vector-valued Bergman spaces of type m Hν2 (D, Pm (Z))

F

= {D −→ Pm (Z) : D × Z z, ζ → F (z, ζ) hol dz N (z, z)ν−p (F (z, ζ)|F (z, B(z, z) ζ)ζ < ∞} D

carry the irreducible projective G-representation ν (Um (g −1 ) F )(z, ζ) = det g (z)ν/p F (g(z), g (z) ζ)

induced by the holomorphic cocycle G × D → K C coming from the complex derivative g (z) and satisfying the chain rule (g1 g2 ) (z) = g1 (g2 (z)) g2 (z). In Jordan theoretic terms, a real analytic cross-section gz ∈ G, satisfying gz (0) = z, can be realized in exponential form gz = exp ((v − zv ∗ z)

∂ ). ∂z

Then gz (0) = B(z, z)1/2 is a positive operator. The classical concept of Grassmann manifolds has the following generalization. Let Z be a hermitian Jordan triple, which is irreducible of rank r, and consider the compact manifold S of all tripotents of rank ≤ r. As a special case, Sr is the

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111

Shilov boundary. Two tripotents u ∼ v are called Peirce-equivalent iﬀ they have the same Peirce spaces 1 Zk (u) = Zk (v), k = 0, , 1. 2 The quotient space M = S / ∼ is called the th Jordan Grassmann manifold [LO2]. In general, it is a compact hermitian symmetric space under the semi-simple Lie group K := K/center. Example. If Z = Cr×s is the matrix triple for r ≤ s, S consists of all partial isometries, satisfying uu∗ u = u, of rank u = ≤ r. One can show that M ≈ Gr (Cr ) × Grs− (Cs ) is the product of “classical” Grassmannians. In general, let U = Z1 (u) ∈ M be the Peirce 1-space. Then the tangent space TU (M ) = Z1/2 (u) coincides with the Peirce

1 2 -space.

Therefore M carries the tautological bundle Z1 ↓ M

U ↓ U

and the tangent bundle Z1/2 ↓ M

Z1/2 (u) ↓ U = Z1 (u).

On the analytic side, let 1 (p|q) = (∂p q)(0) = d π

dz e−(z|z) p(z) q(z)

Z

be the Fischer-Fock inner product on P(Z), with reproducing kernel e(z|w) = K m (z, w) m

expanded into the kernels for Pm (Z). Example. We have P10··0 (Z) = Z (dual space) and K10··0 (z, w) = (z|w) coincides with the normalized K-invariant scalar product. On the other hand, if Z = X C is of tube type, then Pm···m (Z) = C · N m and hence m 1 Km···m (z, w) = N (z)m N (w) . (d/r)m···m

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The Faraut-Kor´ anyi binomial series expansion N (z, w)−ν = (ν)m Km (z, w), m

expressed via the Pochhammer symbol (ν)m =

r . ΓΩ (ν + m ) a = (ν − (k − 1))mk , ΓΩ (ν) 2 k=1

can be used to deﬁne the discrete part of the Wallach set, the so-called discrete Wallach spaces H2a/2 (D) = Pm1 ,...,m , 0,...,0 (Z) m+1 =0

as the space of “-harmonic” functions, with inner product (φm |ψm ) (φ|ψ) a/2 = . ( a/2)m m =0 +1

2 Figure 8. The partitions occurring in Ha/2 (D) n

r It is an important problem to obtain an integral expression for the inner product [AU1], [AU2], [AU5]. Deﬁne GL (Ω)-invariant diﬀerential operators on the symmetric cone Ω ⊂ X by ΓΩ (α) n N (x)d/r−α ∂N Tαn = N (x)α+n−d/r ΓΩ (α + n) and consider the unique extension to a holomorphic diﬀerential operator T˜αn on X C . By construction [AU5, Section 3.1], we have the eigenvalues (α + n)m T˜αn pm = pm . (α)m Theorem 9.1. For φ, ψ ∈ H2a/2 (D) the inner product can be expressed as an integral (φ|ψ) a/2 = dU (φ|S(U ) |T˜U ψ|S(U ) )Hardy M

over the th Jordan Grassmann manifold M . Here S(U ) denotes the Shilov boundary of U ∩ D, so that · S = U × S(U ) U

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as a disjoint union. Moreover, (T˜U )U ∈M is a K-covariant family of holomorphic (pseudo) diﬀerential operators on U , which can be constructed (in the sense of spectral theory) from the building blocks T˜αn (restricted to U ) by requiring that the eigenvalues are given by (ra/2)m (d/r)m , ( a/2)m where m runs over all partitions such that m+1 = 0. Note that in contrast to the situation in Proposition 4.3, the union of the respective Shilov boundaries is in fact disjoint since everything is “centered at the origin”, whereas in Proposition 4.3 one considers the translated Shilov boundaries as subsets of S ⊂ ∂D. Another part of the analytic continuation are the generalized Dirichlet spaces [A], deﬁned for 1 ≤ n ∈ N. Let d1 /r = 1 + a(r − 1)/2 correspond to the dimension d1 of the Peirce 1-space of Z. Taking all partitions m1 ≥ m2 ≥ · · · ≥ mr ≥ n, one obtains as a completion ˜ d /r−n = Pm (Z), H 1 mr ≥n

which is a unitarizable G-space called the Dirichlet space, endowed with the inner product ∼ (fm |gm )F (f |g)n = . (1 + a2 (r − 1) − n)m mr ≥n

Figure 9. The partitions occurring in the Dirichlet space n

r

It is proved in [FK1] that, in case Z = X C is of tube type, there is a unitary isomorphism ∂n

N 2 ˜ d/r−n ≈ Hd/r+n (D) H

sending the parameter sense that

d r

− n = λ to p − λ =

d r

+ n, which is G-equivariant in the

j ∂N (Ud/r−n (g) ψ) = Ud/r+n (g) ψ. This is false for non-tube type domains, as pointed out by G. Zhang, who has found a vector-valued generalization using the so-called transvectants and Shimura operators [PZ], [HLZ]. Using the Jordan Grassmann manifolds the intertwiner can

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also be realized in more geometric terms [AU8]. For 1 ≤ ≤ r consider the homogeneous line bundle Ln = S × C = {[u, ξ] = [v, η] : u u∗ = v v ∗ , η = Nu (v)n ξ} ∼

over M , i.e., the nth power of the determinant bundle, which is the basic holomorphic line bundle over M . Here Nu is the Jordan determinant of Z1 (u). Theorem 9.2. There exists a unitary G-intertwiner into the vector-valued Bergman space ˜ d /r−n ∼ H (D, Pn···n (Z)). = H2 a 1+ 2 (r−1)+n

1

In case Z if of tube type, Pn···n (Z) = C · N n is 1-dimensional, but if Z is not of tube type, e.g., if D is the unit ball in Cd , d > 1, then dimC Pn···n (Z) > 1. The geometric framework for the intertwiner, using the Jordan-Grassmann manifolds, can be described as follows [AU8]: Proposition 9.3. As a special case of the Borel-Weil-Bott theorem, one may realize K-equi Pn··n (Z) −→ Γhol (Mr , Lnr ) ≈

via holomorphic sections over Mr , using the restriction p → p|Sr . More generally, we have an identiﬁcation K-equi Pn·· n 0···0 (Z) −→ Γhol (M , Ln ) !" ≈

via holomorphic sections over M . Theorem 9.4. Using the Jordan-Grassmann manifold, we obtain the G-equivariant intertwiner I ˜ d /r−n −→ H Hd21 /r+n (D, Pn···n (Z)) ≈ Hd21 /r+n (D, Γhol (Mr , Lnr )) 1 ≈

in the form n (I f )(z, U ) = (∂N f |U )(PU z), u

where U = Z1 (u) for some tripotent u. More generally, for 1 ≤ ≤ r, there are similar intertwiners ˜ 2a H (−1)+1−n (D) 2

I

−→ H 2a (−1)+1+n (D, Γhol (Mk , Ln )) ≈

≈

2

˜ 2a H (−1)+1+n (D, Pn··n 0··0 (Z)). 2

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H. Upmeier H. Upmeier, Jordan Algebras in Analysis, Operator Theory and Quantum Mechanics, CBMS Ser. Math. 67, Amer. Math. Soc., Providence RI, 1987. H. Upmeier, Toeplitz C ∗ -algebras and non-commutative duality, J. Oper. Th. 26 (1991), 407–432. H. Upmeier, Symmetric Banach Manifolds and Jordan C ∗ -Algebras, NorthHolland 1985. H. Upmeier, Multivariable Toeplitz Operators and Index Theory, Birkh¨ auser 1996. H. Upmeier, Toeplitz operators and solvable C ∗ -algebras on hermitian symmetric spaces, Bull. Amer. Math. Soc. 11 (1984), 329–332. H. Upmeier, Toeplitz operators on symmetric Siegel domains, Math. Ann. 271 (1985), 401–414. H. Upmeier, Index theory for Toeplitz operators on bounded symmetric domains, Bull. Amer. Math. Soc. 16 (1987), 109–112. H. Upmeier, Fredholm indices for Toeplitz operators on bounded symmetric domains, Amer. J. Math. 110 (1988), 811–832. H. Upmeier, Index theory for multivariable Wiener-Hopf operators, J. reine angew. Math. 384 (1988), 57–79. M. Vergne, H. Rossi, Analytic continuation of holomorphic discrete series of a semi-simple Lie group, Acta Math. 136 (1975), 1–59. N. Wallach, The analytic continuation of the discrete series I, II, Trans. Amer. Math. Soc. 251 (1979), 1–17, 19–37. A. Wassermann, Alg`ebres d’op´erateurs de Toeplitz sur les groupes unitaires, C.R. Acad. Sci. 299 (1984), 871–874. G. Zhang, Berezin transform on real bounded symmetric domains, Trans. Amer. Math. Soc. 353 (2001), 3769–3787.

Harald Upmeier Fachbereich Mathematik University of Marburg 35032 Marburg Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 121–142 c 2008 Birkh¨ auser Verlag Basel/Switzerland

Rotation Algebras and Continued Fractions Florin P. Boca Abstract. This paper discusses two problems related with the approximation of rotation algebras: (i) estimating the norm of almost Mathieu operators and (ii) studying a certain AF algebra associated with the continued fraction algorithm. The Eﬀros-Shen AF algebras naturally arise as primitive quotients of this algebra. Mathematics Subject Classiﬁcation (2000). Primary 46L05; Secondary 11A55, 47A30, 47B36. Keywords. Rotation algebras, almost Mathieu operators, continued fractions, AF algebras.

1. Introduction Let Γ be a ﬁnitely generated discrete group. Let λ denote the left regular repre2 sentation of Γ on the Hilbert space (Γ). For 2any ﬁnite set S ⊂ Γ, consider the 1 Markov operator µS = |S| s∈S λs acting on (Γ) as (µS f )(t) =

1 f (s−1 t). |S| s∈S

Suppose S is a symmetric generating set for Γ. A classical result of Kesten [23] shows that the group Γ is amenable if and only if the spectral radius σ(µS ) of µS is equal to 1. The natural example where Γ is the (non-amenable) free group Fn with generators g1 , . . . , gn and S = {g1±1 , . . . , gn±1√}, n 2, has also been considered by Kesten [22]. In this case σ(µS ) = µS = n1 2n − 1 and the spectrum of µS is the whole interval [−µS , µS ]. Things are diﬀerent when the representation λ is twisted by a cocycle. Here we are interested in the case where Γ is the abelian group Z2 . Given θ ∈ [0, 1), the mapping β(m, n) = eπiθm∧n deﬁnes a skew-symmetric bicharacter of Z2 × Z2 . The (left) regular representation of Z2 is twisted by β to (πs f )(t) = β(s, t)f (t − s),

s, t ∈ Z2 , f ∈ 2 (Z2 ).

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This formula deﬁnes a projective unitary representation π of Z2 on 2 (Z2 ) such that πs∗ = π−s and πs1 πs2 = β(s1 , s2 )πs1 +s2 = β(s1 , s2 )β(s2 , s1 )πs2 πs1 ,

s1 , s2 ∈ Z2 .

The unitaries Uθ = π(1,0) and Vθ = π(0,1) acting on the orthonormal basis {δ(m,n) } of 2 (Z2 ) consisting of Dirac functions as Uθ δ(m,n) = eπinθ δ(m+1,n) , Vθ δ(m,n) = e−πimθ δ(m,n+1) , commute as Uθ Vθ = e2πiθ Vθ Uθ . The rotation algebra Aθ is deﬁned as the universal C ∗ -algebra generated by two unitary operators uθ and vθ such that uθ vθ = e2πiθ vθ uθ . Using the simplicity of Aθ when θ is irrational and the canonical isomorphism between C ∗ (Uθq , Vθq ) and C(T2 ) when θ = pq in lowest terms, the mapping uθ → Uθ , vθ → Vθ , is seen to extend to an isomorphism ρθ : Aθ → C ∗ (Uθ , Vθ ) ⊂ B(2 (Z2 )). The C ∗ -algebra A θ is endowed with the tracial state τθ m n deﬁned on polynomials in uθ and vθ by τθ α u v m,n (m,n) θ θ = α(0,0) . It is easily seen that ρθ and the GNS representation of Aθ deﬁned by the state τθ are unitarily equivalent. Another representation, σθ : Aθ → B(2 (Z)), is obtained by mapping $θ , V$θ ), where the unitaries U $θ and V$θ act on an orthonormal basis (uθ , vθ ) → (U 2 2πinθ $ $ δn . Every irreducible representation {δn }n of (Z) by Uθ δn = δn−1 , Vθ δn = e of Ap/q has dimension q and is unitarily equivalent with one of the representations πz1 ,z2 deﬁned as πz1 ,z2 (uθ ) = z1 U0 , πz1 ,z2 (vθ ) = z2 V0 , with ⎤ ⎡ ⎡ ⎤ 0 1 0 ... 0 0 1 ⎥ ⎢0 0 1 . . . 0 0⎥ ⎢ ω ⎥ ⎢ ⎢ ⎥ 2 ⎥ ⎢ .. .. .. . . ⎢ ⎥ . . ω . . = U0 = ⎢ . . . , V ⎥ , ω = e2πip/q . ⎢ ⎥ 0 . . .⎥ ⎥ ⎢ ⎢ . .. ⎦ ⎣0 0 0 . . . 0 1⎦ ⎣ q−1 ω 1 0 0 ... 0 0 Rotation algebras are closely connected with the three-dimensional discrete Heisenberg group H3 (Z) generated by the set S = {x±1 , y ±1 , z ±1 }, with ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 1 0 1 0 0 1 0 1 x = ⎣0 1 0⎦ , y = ⎣0 1 1⎦ , z = ⎣0 1 0⎦ , 0 0 1 0 0 1 0 0 1 satisfying the commutation relations z = xyx−1 y −1 , zx = xz, and zy = yz. Since H3 (Z) is amenable, its reduced and full C ∗ -algebras coincide, being described as the universal C ∗ -algebra generated by three unitary elements u, v, w (corresponding to λx , λy , λz ) satisfying the commutation relations uv = vuw, wu = uw, and wv = vw. For every θ the mapping u → uθ , v → vθ , w → e2πiθ 1, extends to a representation βθ of C ∗ (H3 (Z)) onto Aθ . Conversely, for any irreducible unitary representation π of H3 (Z) there is some number θ ∈ [0, 1) such that πz = e2πiθ 1. Consequently πx and πy are unitaries such that πx πy = e2πiθ πy πx . When θ is irrational any two such representations are approximately unitarily equivalent as a consequence of Voiculescu’s noncommutative Weyl-von Neumann theorem. When θ = pq in lowest terms, πx and πy generate a copy of the C ∗ -algebra Mq of q × q matrices with complex entries, and there are z1 , z2 ∈ T such that πx = z1 U0

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and πy = z2 V0 (see, e.g., [14, Thm.VII.5.1]). These considerations show that the spectrum of any self-adjoint element h ∈ C ∗ (H3 (Z)) coincide with the closure of the union of the spectra of the elements βθ (h) ∈ Aθ , θ ∈ [0, 1). The C ∗ algebra of any ﬁnitely generated torsion-free nilpotent group contains no nontrivial projections (cf., e.g., [2, Prop.1]), hence spec(h) is always a compact interval. Using the equality spec(β√ θ (6µS )) = 2 cos(2πθ) + spec(Hθ ), it was proved in [2] that spec(µS ) = [− 13 (1 + 2), 1] for the set of generators S = {x±1 , y ±1 , z ±1 } of Γ = H3 (Z). The “twisted” Markov operator hθ = uθ + u∗θ + vθ + vθ∗ ∈ Aθ corresponding to the set S = {±(1, 0), ±(0, 1)} of generators of Z2 is called the Harper operator. More generally, one considers almost Mathieu operators hθ,λ = uθ +u∗θ + λ2 (vθ +vθ∗ ), λ ∈ R∗ . The norm of hp/q,λ is seen to be given by 8 8 8 8 λ ∗ ∗ 8 (V U + U + + V ) (1.1) hp/q,λ = π1,1 (hp/q,λ ) = 8 0 0 0 8. 8 0 2

1 0.8

Θ

0.6 0.4 0.2 0 -4

-2

0 spec hΘ

2

4

Figure 1. The Hofstadter butterﬂy (1 q 30) The spectrum of hp/q,λ is the union of q − 1, respectively q, disjoint intervals, according to whether q is even or odd. The union of the sets spec(hθ ), θ ∈ Q, forms the Hofstadter butterﬂy (see Figure 1). During the last three decades much eﬀort has been put in the spectral analysis of almost Mathieu operators. One of the crowning achievements is the recent result of Avila and Jitomirskaya [1] which shows that spec(hθ,λ ) is a Cantor set for every irrational θ whenever λ = 0, thus

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answering the ‘Ten Martini problem’ of Kac and Simon. In the next section we will discuss some results concerning the norm of almost Mathieu operators. Rotation algebras provide a noticeable role in noncommutative geometry [13]. A classical result of Pimsner and Voiculescu [26] shows that any irrational rotation algebra Aθ can be embedded into the Eﬀros-Shen AF algebra Fθ [15], which encodes the continued fraction expansion of θ. Combined with Rieﬀel’s construction [29] of projections of trace {nθ} in Aθ this led, via the (ordered) dimension group of Fθ , to the classiﬁcation of irrational rotation algebras: two irrational rotation algebras Aθ1 and Aθ2 are isomorphic if and only if θ2 = ±θ1 (mod Z). The order on the dimension group of Fθ also plays a role in the Elliott-Evans decomposition [16] of irrational rotation algebras as inductive limits of circle algebras. The results of Rieﬀel, Pimsner and Voiculescu yield {τθ (p) : p projection in Aθ , p = 1} = {{nθ} : n ∈ Z} for any irrational θ, thus gaps in the spectrum of almost Mathieu operators are canonically labelled by integers. The ‘dry’ Ten Martini problem asks / Q, is it whether all these canonical gap labels occur, i.e., given λ ∈ R∗ and θ ∈ true that for any n ∈ Z∗ there is p spectral projection of hθ,λ in Aθ such that τθ (p) = {nθ}? The answer is known to be positive for θ of Liouville type [12], but the problem is open in general (see also [1] and [28] for some recent thoughts). In Section 3 we will review some results from [6] concerning a certain AF algebra A associated with the continued fraction algorithm. This algebra is related with both the Eﬀros-Shen algebras and the GICAR AF algebra having the Pascal triangle as Bratteli diagram.

2. Norm estimates for almost Mathieu operators The problem of estimating the norm of Harper operators (i.e., to approximate the external boundary of the Hofstadter butterﬂy) has been ﬁrst considered in [2] in connection with the study of the spectrum of the Markov operator µS on the group C ∗ -algebra of the discrete Heisenberg group H3 (Z), generated by the set S = {x±1 , y ±1 , z ±1 }. Explicit estimates for the norm of almost Mathieu operators are provided by Theorem 2.1 ([7]). (i) For every λ ∈ R and θ ∈ [0, 12 ], hθ,λ Mλ (θ),

(2.1)

with

⎧9 2 ⎪ ⎨ 4 + λ + 4|λ|(cos πθ − sin πθ) cos πθ : ; Mλ (θ) = ⎪ 4 + λ2 − 1 − 1 1 − 1+cos2 4πθ min{4, λ2 } ⎩ tan πθ 2

In particular for every θ ∈ [ 14 , 34 ], hθ,λ

9 4 + λ2

if θ ∈ [0, 14 ], if θ ∈ [ 14 , 12 ].

Rotation Algebras and Continued Fractions

125

(ii) For every θ ∈ [0, 12 ], hθ m(θ) = max{f1 (θ), f2 (θ), f3 (θ)},

(2.2)

where f1 , f2 , f3 0 are the elementary functions given by f1 (θ)2 = 6 − 1+

1 sin πθ

4 ; + 1+

+ 1 sin πθ

8 cos2 πθ 2 + , 2 1 + 4 sin πθ cos πθ (1 + sin πθ)3/2

2 2 cos2 2πθ + f2 (θ)2 = 4 + 9 1 + | sin 4πθ| 1 + | sin 4πθ| < 2 = 2 = cos 2πθ 16 cos4 πθ sin2 2πθ > 1+ 9 + , +2 1 + | sin 4πθ| (2 + | tan 2πθ|)2 1 + | sin 4πθ| f3 (θ)2 = 4 + ? +

4(cos 2πθ + 2 cos2 2πθ + 2 cos4 2πθ) 5

4(cos 2πθ + 2 cos2 2πθ + 2 cos4 2πθ) 2− 5

2

√ 8 cos 2πθ + 10 + √ 10

2 .

Figure 2. The graphs of the functions M2 , h· and m on [0, 1/2] The map [0, 1] θ → hθ,λ was proved to be Lipschitz by Bellissard [3]. Figure 2 and the inequalities sup M2 (θ) − m(θ) 0.18962, M2 (θ) − m(θ) 0.16183, sup 0θ1/4

1/4θ1/2

126

F.P. Boca

show that Theorem 2.1 provides a reasonably accurate numerical approximation for the boundary of the Hofstadter butterﬂy. It also shows that Kesten’s result does not extend to the case of twisted Markov operators, even in the case of the abelian group Γ = Z2 . Actually the norm of 14 (uθ + u∗θ + vθ + vθ∗ ) is much smaller √ than the norm 23 of 14 (λg1 + λ∗g1 + λg2 + λ∗g2 ) ∈ Cr∗ (F2 ). To prove hθ fj (θ), one simply uses the trivial inequalities hθ σθ (hθ )

σθ (hθ )ξ2 , ξ2

where ξ = {ξn }n ∈ 2 (Z) is given, according to j = 1, 2, 3, by ⎧ sin √ β ⎪ ⎪ ⎪ % √ 10 ⎪ ⎨ 2√sin β s|k| cos α if n = 2k, 5 ξ ξn = r|n| , ξn = = n ⎪ s|k| sin α if n = 2k + 1, cos β ⎪ ⎪ ⎪ ⎩ 0

if n = ±2, if n = ±1, if n = 0, else,

for appropriate choices of r = r(θ), s = s(θ) ∈ (0, 1), α = α(θ), β = β(θ) ∈ [0, 2π). The continuity of the map θ → hθ,λ and (1.1) reduce (2.1) to the situation where θ = pq is a rational number in lowest terms, when hθ,λ = Hθ,λ with Hθ,λ = π1,1 (hθ,λ ) = U0 + U0∗ + λ2 (V0 + V0∗ ). Let E be an eigenvalue of the selfadjoint q × q matrix Hθ,λ and ξ = {ξm }m∈Zq ∈ Cq = 2 (Zq ) be a unit eigenvector for E. The equality Hθ,λ ξ = Eξ can be written as a three-term recurrence relation ξm+1 = (E − λ cos 2πmθ)ξm − ξm−1 , or in matrix form as

E − λ cos 2πmθ ξm+1 = 1 ξm

m ∈ Zq ,

(2.3)

−1 ξm . 0 ξm−1

Set S = m∈Zq ξm ξm−1 cos π(2m − 1)θ. After some heavy trigonometry and intensive use of the recurrence relation (2.3), one can express E 2 in two ways, as (ξm+1 − ξm−1 + λξm sin 2πmθ)2 E 2 =4 + λ2 + 4λS(cos πθ − sin πθ) − m∈Zq (2.4) 2 4 + λ + 4λS(cos πθ − sin πθ), and respectively as λ2 2 (4 − sin2 2πθ) ξm sin2 2πmθ 4 m∈Zq 2 λ ξm sin 2πθ sin 2πmθ + ξm+1 − ξm−1 − 2

E 2 =4 + 4λ2 + 4λS cos3 πθ −

m∈Zq

4 + 4λ2 + 4λS cos3 πθ.

(2.5)

Rotation Algebras and Continued Fractions

127

Some more trigonometry and use of Cauchy-Schwarz inequality lead to |S| cos πθ for every θ ∈ [0, 14 ]. In conjunction with (2.4) this completes the proof of (2.1) in this range of θ. In the range θ ∈ [ 14 , 12 ] we have cos3 πθ 0 cos πθ − sin πθ. So regardless of the actual√sign of S, inequalities (2.4) and (2.5) yield E 2 4 + λ2 , and therefore hθ,λ 4 + λ2 . This upper bound can be improved to that given by Mλ (θ) by further enhancing this kind of arguments. Remark 2.2. (i) The lower bound estimate (2.2) gives in particular √ min hθ min f1 (θ) = 6.59303 . . . = 2.56769 . . . 1/4θ1/2

(2.6)

1/4θ1/2

Numerical computations suggest that the left-hand side in (2.6) is only fractionally larger than 2.59. It would be interesting to improve these estimates. (ii) Estimate (2.2) also gives min0θ1/4 hθ 2 7.82387 (which is pretty √ close to 8) and hθ 2 2 for all θ ∈ [0, √ 0.23441]. Numerical computations seem to suggest that the inequality hθ 2 2 may hold in the whole range θ ∈ [0, 14 ]. It would be interesting to prove/disprove this. (iii) The spectrum of the non-self-adjoint operator uθ + u∗θ + λvθ , λ ∈ C, was computed for irrational θ in [4, 30]. It would be interesting to ﬁnd accurate estimates for its norm.

3. An AF algebra associated with the continued fraction algorithm 3.1. Continued fractions and the Farey tessellation The regular continued fraction representation of real numbers establishes a oneto-one correspondence NN a = (a1 , a2 , . . .) ←→ θ ∈ (0, 1] \ Q.

(3.1)

In one direction deﬁne, for given a, the rational numbers [a1 , . . . , an ] =

1 a1 +

1 a2 +...+

= 1 an

pn . qn

(3.2)

The representation (3.2) is not unique because [a1 , . . . , an ] = [a1 , . . . , an − 1, 1]. However, the Euclidean algorithm shows that any rational number in (0, 1] can be uniquely represented as in (3.2), for some n and a1 , . . . , an ∈ N with an 2. Moreover, pn and qn are obtained by plain matrix multiplication as an 1 qn qn−1 a1 1 a2 1 ··· = . (3.3) 1 0 1 0 1 0 pn pn−1 Applying the determinant on both sides of (3.3) one gets the familiar relation pn−1 qn − pn qn−1 = (−1)n .

(3.4)

128

F.P. Boca

It is easy to infer from (3.3) that qn is greater or equal than the nth Fibonacci number Fn . In conjunction with (3.4) this yields # # # pn+1 1 pn ## 1 # = − , # qn+1 qn # qn qn+1 Fn Fn+1 showing that { pqnn }n is a Cauchy sequence of rational numbers. Its limit is irrational and deﬁnes the number θ from the right-hand side of (3.1). In the opposite direction the digits an of θ are obtained by plain iterations of the Gauss map 1 1 1 = − , x = 0, G : [0, 1] → [0, 1], G(0) = 0, G(x) = x x x as 1 an = n−1 − Gn (θ), n 1. G (θ) The Gauss map acts as G[a1 , a2 , a3 , . . .] = [a2 , a3 , . . .], hence its periodic points are exactly the reduced (purely periodic) quadratic irrational numbers [a1 , . . . , an0 ].

F

B −1 F

BF

AF

A2 F 3

A F

A2 BF

1 4

0 1

ABF

1 3

AB 2 F

ABAF

2 5

1 2

3 5

3 4

2 3

1 1

Figure 3. The Farey tessellation Consider the matrices 1 0 1 1 A= , B= , 1 1 0 1 The equalities (3.3) and a 1 b M (a)M (b) = 1 0 1

J=

0 1 , 1 0

1 a 1 1 = 0 1 0 0

yield

a1

a2

B A

···B

a2m−1

a2m

A

B a1 Aa2 · · · Aa2m B a2m+1

M (a) =

a 1

b = B a Ab , 1

q = 2m p2m q = 2m p2m

q2m−1 , p2m−1 q2m+1 . p2m+1

1 ∈ GL2 (Z). 0

a, b ∈ Z,

(3.5)

(3.6)

Rotation Algebras and Continued Fractions

129

It is now plain to prove that the semigroup G generated by A and B in SL2 (Z) is canonically isomorphic to the free semigroup on two generators F+ 2 . This can be seen geometrically considering the Farey tessellation {gF : g ∈ G} of the upper half-plane H, where F = {0 Re z 1, |z − 12 | 12 } (see Figure 3). However, the group generated by A and B is not free, as one can see from the equality (BA−1 B)4 = I2 . Note also that conjugating by J in (3.6) one gets a b ∈ SL2 (Z) : 0 a c, 0 b d = AG. c d A ﬁrst estimate [21] on the number Ψ(N ) of elements C ∈ G such that 2 < Tr C N as N → ∞ was furthered in [5] to 3 ζ (2) N2 ln N + γ − − + Oε (N 7/4+ε ). (3.7) Ψ(N ) = ζ(2) 2 ζ(2) Interestingly, the contribution to the main term of even length words is which turns out to be much smaller than the contribution ζ (2) ζ(2) )

N2 ζ(2) (ln N

N 2 ln 2 ζ(2) ,

+ γ − ln 2 −

− of odd length words. The estimate on the number of even length words leads to good estimates of the number of reduced quadratic irrationals according to their natural length [18, 5]. 3 2

3.2. The mediant construction and AF algebras The mediant construction, pictured in Figure 4, associates to every two consecutive rational numbers pq < pq with p q − pq = 1 the new rational number p+p q+q . Clearly p q

Editor: I. Gohberg (GLWRULDO2I½FH School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel Editorial Board: D. Alpay (Beer-Sheva) J. Arazy (Haifa) A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Böttcher (Chemnitz) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) R. E. Curto (Iowa City) K. R. Davidson (Waterloo, Ontario) R. G. Douglas (College Station) A. Dijksma (Groningen) H. Dym (Rehovot) P. A. Fuhrmann (Beer Sheva) B. Gramsch (Mainz) J. A. Helton (La Jolla) M. A. Kaashoek (Amsterdam)

H. G. Kaper (Argonne) S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) B. Mityagin (Columbus) V. Olshevsky (Storrs) M. Putinar (Santa Barbara) L. Rodman (Williamsburg) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) I. M. Spitkovsky (Williamsburg) S. Treil (Providence) H. Upmeier (Marburg) S. M. Verduyn Lunel (Leiden) D. Voiculescu (Berkeley) D. Xia (Nashville) D. Yafaev (Rennes) Honorary and Advisory Editorial Board: C. Foias (Bloomington) T. Kailath (Stanford) H. Langer (Vienna) P. D. Lax (New York) H. Widom (Santa Cruz)

Operator Algebras, Operator Theory and Applications Maria Amélia Bastos Israel Gohberg Amarino Brites Lebre Frank-Olme Speck Editors

Birkhäuser Basel · Boston · Berlin

Editors: Maria Amélia Bastos Amarino Brites Lebre Frank-Olme Speck Departamento de Matemática Instituto Superior Técnico, U.T.L. $YHQLGD5RYLVFR3DLV /LVERD3RUWXJDO HPDLODEDVWRV#PDWKLVWXWOSW DOHEUH#PDWKLVWXWOSW IVSHFN#PDWKLVWXWOSW

Israel Gohberg School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University 5DPDW$YLY,VUDHO HPDLOJRKEHUJ#PDWKWDXDFLO

0DWKHPDWLFDO6XEMHFW&ODVVL½FDWLRQ''' /LEUDU\RI&RQJUHVV&RQWURO1XPEHU Bibliographic information published by Die Deutsche Bibliothek. 'LH'HXWVFKH%LEOLRWKHNOLVWVWKLVSXEOLFDWLRQLQWKH'HXWVFKH1DWLRQDOELEOLRJUD½H detailed bibliographic data is available in the Internet at http://dnb.ddb.de

,6%1%LUNKlXVHU9HUODJ$*%DVHO- Boston - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of WKHPDWHULDOLVFRQFHUQHGVSHFL½FDOO\WKHULJKWVRIWUDQVODWLRQUHSULQWLQJUHXVHRI LOOXVWUDWLRQVUHFLWDWLRQEURDGFDVWLQJUHSURGXFWLRQRQPLFUR½OPVRULQRWKHUZD\VDQG storage in data banks. For any kind of use permission of the copyright owner must be obtained.

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Contents Editorial Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

List of Participants of WOAT 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Summer School: Lecture Notes S.C. Power Subalgebras of Graph C*-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

B. Silbermann C ∗ -algebras and Asymptotic Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . .

33

H. Upmeier Toeplitz Operator Algebras and Complex Analysis . . . . . . . . . . . . . . . . . . .

67

Workshop: Contributed Articles F.P. Boca Rotation Algebras and Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 121 G. Bogveradze and L.P. Castro On the Fredholm Index of Matrix Wiener-Hopf plus/minus Hankel Operators with Semi-almost Periodic Symbols . . . . . . . . . . . . . . . . 143 L.P. Castro and D. Kapanadze Diﬀraction by a Strip and by a Half-plane with Variable Face Impedances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159

A.C. Concei¸c˜ ao and V.G. Kravchenko Factorization Algorithm for Some Special Matrix Functions . . . . . . . . . . 173 E. Gots and L. Lyakhov On a Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 R. El Harti Extensions of σ-C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201

A.Y. Karlovich Higher-order Asymptotic Formulas for Toeplitz Matrices with Symbols in Generalized H¨older Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 207

vi

Contents

Yu.I. Karlovich Nonlocal Singular Integral Operators with Slowly Oscillating Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Y.I. Karlovich and L.V. Pessoa Poly-Bergman Projections and Orthogonal Decompositions of L2 -spaces Over Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 V. Kokilashvili and S. Samko Vekua’s Generalized Singular Integral on Carleson Curves in Weighted Variable Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 V. Manuilov On Homotopical Non-invertibility of C ∗ -extensions . . . . . . . . . . . . . . . . . . . 295 S. Mendes Galois-ﬁxed Points and K-theory for GL(n) . . . . . . . . . . . . . . . . . . . . . . . . . . 309 K.M. Mikkola and I.M. Spitkovsky Spectral Factorization, Unstable Canonical Factorization, and Open Factorization Problems in Control Theory . . . . . . . . . . . . . . . . . 321 K. Nourouzi Compact Linear Operators Between Probabilistic Normed Spaces . . . . 347 V.S. Rabinovich and S. Roch Essential Spectra of Pseudodiﬀerential Operators and Exponential Decay of Their Solutions. Applications to Schr¨ odinger Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 V.S. Rabinovich, S. Roch and B. Silbermann On Finite Sections of Band-dominated Operators . . . . . . . . . . . . . . . . . . . . 385 H. Rafeiro and S. Samko Characterization of the Range of One-dimensional Fractional Integration in the Space with Variable Exponent . . . . . . . . . . . . . . . . . . . . . 393 C.C. Ramos, N. Martins and P.R. Pinto Orbit Representations and Circle Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 N. Samko and B. Vakulov On Generalized Spherical Fractional Integration Operators in Weighted Generalized H¨older Spaces on the Unit Sphere . . . . . . . . . .

429

Editorial Introduction This volume is devoted to the International Summer School and Workshop on Operator Algebras, Operator Theory and Applications, WOAT 2006, held at Instituto Superior T´ecnico in Lisbon, Portugal on 1–5 September 2006. WOAT 2006 was a satellite conference of the International Congress of Mathematicians 2006 that was held in Madrid, Spain. Operator Algebras and Operator Theory are important areas of Mathematics that play an important role in diﬀerent mathematics areas and its applications, particularly in Mathematical Physics and Numerical Analysis. The main aim of WOAT 2006 was to bring together researchers in the Operator Algebras and Operator Theory areas. This volume contains three lecture notes of the Summer School courses and nineteen articles, contributions to the workshop of the WOAT 2006. The lecture notes, written by leading experts in the ﬁelds, are focused on: • Subalgebras of Graph C ∗ -Algebras (S. Power) A self contained introduction to two novel classes of non self-adjoint operator algebras, namely the generalized analytic Toeplitz algebras associated with the Fock spaces of a directed graph and subalgebras of graph C ∗ algebras, are given. The topics are independent but in both cases the focus is on techniques and problems related to classifying isomorphism types and to the recovery of underlying foundational structures, be they graphs or groupoids. • C ∗ -Algebras and Asymptotic Spectral Theory (B. Silbermann) An introduction to asymptotic spectral theory is presented using the elementary theory of C ∗ algebras. Given a bounded sequence of matrices with increasing size the spectra, ε-pseudospectra and the singular values of theses matrices are characterized. Three fundamental notions are discussed: stability, fractality and Fredholm sequences. The theory is applied to ﬁnite sections of quasidiagonal operators, Toeplitz operators, and operators with almost periodic diagonals. • Toeplitz Operator Algebras and Complex Analysis (H. Upmeier) Recent investigations are presented concerning Hilbert spaces of holomorphic functions on hermitian symmetric domains of arbitrary rank and dimension, in relation to operator theory (Toeplitz C ∗ -algebras and their representations), harmonic analysis (discrete series of semi-simple Lie groups) and quantization (covariant functional calculi and Berezin transformation).

viii

Editorial Introduction

The articles were based contributions to the workshop, the majority of them being centered on the main topics of the workshop: • Crossed product C ∗ -algebras. C ∗ -algebras of operators on Hardy and Bergman spaces. Invertibility theory for non-local C ∗ -algebras. Von Neumann algebras. • Approximate methods in operator algebras. Asymptotic properties of approximation operators. • Toeplitz, Hankel, and convolution type operators and algebras. Symbol calculi. Invertibility and index theory. • Operator theoretical methods in diﬀraction theory. Factorization theory and integrable systems. Applications to Mathematical Physics. The organizers gratefully acknowledge the support of the WOAT 2006 sponsors: the Portuguese Foundation for Science and Technology, Center for Mathematics and its Applications, Center for Mathematical Analysis, Geometry and Dynamical Systems, Research Project FCT/FEDER/POCTI/MAT/59972/2004, as well as Caixa Geral de Dep´ositos, Cˆamara Municipal de Lisboa, Funda¸c˜ao Calouste Gulbenkian, Embassy of Germany in Portugal, Funda¸c˜ao Luso-Americana, and Reitoria da Universidade T´ecnica de Lisboa. Lisbon, September 2007 The Editorial Board

WOAT 2006 – Program Friday, 1 September 2006 Workshop – Room ACI Registration Opening Session Session W1 Nikolai Nikolski Coﬀee break David Evans Lewis Coburn Lunch

09:00–09:30 09:30–09:50 09:55–10:45 10:50–11:10 11:10–12:00 12:05–12:55 13:00–14:00 14:00–15:30 15:30–15:45 15:45–17:15 17:15–17:30 17:30–19:00

Summer School – Room ACI Stephen Power Break Course B/Lecture 1 Konrad Schm¨ udgen Coﬀee break Course C/Lecture 1 Bernd Silbermann Course A/Lecture 1

Saturday, 2 September 2006

09:00–09:50 09:55–10:45 10:50–11:05

11:05–11:30 11:35–12:00 12:05–12:30 12:35–13:00 13:00–14:00 14:00–15:30 15:30–15:45 15:45–17:15 17:15–17:30 17:30–19:00

Course Course Course Course

Session W3 Room QA1.1 D. Mushtari I. Todorov R. Popescu V. Strauss

Workshop Session W2 – Room QA02.3 Ilya Spitkovsky Vladimir Manuilov Coﬀee break Session W4 Session W5 Room QA1.2 Room QA1.3 M. Ptak E. Gots H. Kaptanoglu P. Lopes M.C. Cˆ amara A. Montes-Rodr´ıguez A. Karlovich H. Rafeiro Lunch

Summer School – Room ACI Harald Upmeier Break Course A/Lecture 2 Stephen Power Coﬀee break Course B/Lecture 2 Konrad Schm¨ udgen Course D/Lecture 1

A: Subalgebras of Graph C ∗ -algebras B: C ∗ -algebras – Selected topics C: C ∗ -algebras and Asymptotic Spectral Theory D: Toeplitz Operator Algebras and Multivariable Complex Analysis

WOAT 2006 – Program Monday, 4 September 2006

09:00–09:50 09:55–10:20 10:25–10:50 10:50–11:05

11:05–11:30 11:35–12:00 12:05–12:30 12:35–13:00 13:00–14:00 14:00–15:30 15:30–15:45 15:45–17:15 17:15–17:30 17:30–19:00

Workshop Session W6 – Room QA1.1 Session W7 – Room QA1.2 Florin Boca Yuri Karlovich Session W8 – Room QA1.1 Session W9 – Room QA1.2 A. Katavolos R. Duduchava L. Marcoux L. Castro Coﬀee break Session W10 Session W11 Session W12 Room QA1.1 Room QA1.2 Room QA1.3 M. Dritschel J. Rodriguez A. Concei¸ca ˜o H. Tandra R. Marreiros C. Diogo E. Lopushanskaya K. Nourouzi T. Malheiro H. Mascarenhas M.C. Martins Lunch Summer School – Room ACI Bernd Silbermann Break Course D/Lecture 2 Harald Upmeier Coﬀee break Course A/Lecture 3 Stephen Power Course C/Lecture 2

Tuesday, 5 September 2006

09:00–09:50 09:55–10:45 10:50–11:05 11:05–11:30 11:35–12:00 12:05–12:30 12:35–13:00 13:00–14:00 14:00–15:30 15:30–15:45 15:45–17:15 17:15–17:30 17:30–19:00

Workshop Session W13 – Room QA1.1 Session W14 – Room QA1.2 Mikhail Agranovich Steﬀen Roch Nikolai Rabinovich Stefan Samko Coﬀee break Session W15 – Room QA1.1 Session W16 – Room QA1.2 C. Fernandes N. Samko R. El Harti L. Pessoa C. Ramos G. Bogveradze S. Mendes A. Nolasco Lunch Summer School – Room ACI Konrad Schm¨ udgen Break Course C/Lecture 3 Bernd Silbermann Coﬀee break Course D/Lecture 3 Harald Upmeier Course B/Lecture 3

List of Participants of WOAT 2006 Agranovich, Mikhail Moscow Institute of Electronics and Mathematics, Russia Al-Rashed, Maryam Imperial College London, United Kingdom Bastos, M. Am´elia Universidade T´ecnica de Lisboa/IST, Portugal Becher, Florian University of Freiburg, Germany Boca, Florin University of Illinois-Urbana-Champaign, USA Bogveradze, Giorgi Universidade de Aveiro, Portugal Bravo, Ant´ onio Universidade T´ecnica de Lisboa/IST, Portugal Cˆ amara, M. Cristina Universidade T´ecnica de Lisboa/IST, Portugal Campos, Hugo Universidade do Algarve/FCT, Portugal Campos, Lina Universidade do Algarve/FCT, Portugal Carvalho, Catarina Universidade T´ecnica de Lisboa/IST, Portugal Castro, Lu´ıs Universidade de Aveiro, Portugal Coburn, Lewis The State University of New York at Buﬀalo, USA Concei¸ca ˜o, Ana Universidade do Algarve/FCT, Portugal Diogo, Cristina Instituto Superior de Ciˆencias do Trabalho e da Empresa, Portugal Dritschel, Michael University of Newcastle, United Kingdom Duduchava, Roland A. Razmadze Mathematical Institute, Georgia

xii

List of Participants of WOAT 2006

Efendiev, Messoud GSF/TUM-Munich, Germany El Harti, Rachid University Hassan I, Morocco Erkursun, Nazife Middle East Technical University, Turkey Esteves, Ana Universidade de Aveiro, Portugal Evans, David Cardiﬀ University, United Kingdom Fernandes, Cl´ audio Universidade Nova de Lisboa/FCT, Portugal Ferreira dos Santos, Ant´ onio Universidade T´ecnica de Lisboa/IST, Portugal Gots, Ekaterina Voronezh State Technological Academy, Russia Habgood, Joe Queens University Belfast, United Kingdom Kaptanoglu, H. Turgay Bilkent University, Turkey Karlovich, Yuri Universidad Aut´ onoma del Estado de Morelos, Mexico Karlovych, Oleksiy Universidade do Minho, Portugal Katavolos, Aristides University of Athens, Greece Kravchenko, Viktor Universidade do Algarve/FCT, Portugal Lazar, Aldo Tel Aviv University, Israel Lebre, Amarino Universidade T´ecnica de Lisboa/IST, Portugal Lopes, Paulo Universidade T´ecnica de Lisboa/IST, Portugal Lopushanskaya, Ekaterina Voronezh State University, Russia Malheiro, Teresa Universidade do Minho, Portugal Manuilov, Vladimir Moscow State University, Russia Marcoux, Laurent W. University of Waterloo, Canada Marreiros, Rui Universidade do Algarve/FCT, Portugal

List of Participants of WOAT 2006 Martins, Maria do Carmo Universidade dos A¸cores, Portugal Mascarenhas, Helena Universidade T´ecnica de Lisboa/IST, Portugal Mendes, S´ergio University of Manchester, United Kingdom Montes-Rodr´ıguez, Alfonso Universidad de Sevilla, Spain Moura Santos, Ana Universidade T´ecnica de Lisboa/IST, Portugal Mushtari, Daniar Kazan State University, Russia Nikolski, Nikolai Universit´e Bordeaux, France and Steklov Institute of Mathematics, Russia Nolasco, Ana Universidade de Aveiro, Portugal Nourouzi, Kourosh K.N.Toosi University of Technology, Iran Oliveira, Isabel Universidade T´ecnica de Lisboa/IST, Portugal Oliveira, Lina Universidade T´ecnica de Lisboa/IST, Portugal Pereira, Paulo Universidade de Aveiro, Portugal Pessoa, Lu´ıs Universidade T´ecnica de Lisboa/IST, Portugal Pinto, Paulo Universidade T´ecnica de Lisboa/IST, Portugal Popescu, Radu Universidade T´ecnica de Lisboa/IST, Portugal Power, Stephen Lancaster University, United Kingdom Ptak, Marek University of Agriculture of Krakow, Poland Quint˜ ao Braga, Maria Jo˜ ao Universidade Cat´ olica Portuguesa, Portugal Rabinovich, Vladimir Instituto Politecnico Nacional, ESIME, Mexico Rafeiro, Humberto Universidade do Algarve/FCT, Portugal Ramos, Carlos ´ Universidade de Evora, Portugal Roch, Steﬀen Technical University Darmstadt, Germany

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List of Participants of WOAT 2006

Rodrigues, C´ atia Universidade de Aveiro, Portugal Rodr´ıguez, Juan Universidade do Algarve/FCT, Portugal Rodr´ıguez Mart´ınez, Alejandro Universidad de Sevilla, Spain Samko, Natasha Universidade do Algarve/FCT, Portugal Samko, Stefan Universidade do Algarve/FCT, Portugal Sangha, Amandip University of Oslo, Norway Santos, Pedro Universidade T´ecnica de Lisboa/IST, Portugal Schm¨ udgen, Konrad University of Leipzig, Germany Silbermann, Bernd Technical University of Chemnitz, Germany Skill, Thomas Philipps-University Marburg, Germany Speck, Frank-Olme Universidade T´ecnica de Lisboa/IST, Portugal Spitkovsky, Ilya College of William & Mary, USA Strauss, Vladimir Simon Bolivar University, Venezuela Tandra, Haryono Bandung Institute of Technology, Indonesia Teixeira, Francisco Universidade T´ecnica de Lisboa/IST, Portugal Todorov, Ivan Queen’s University Belfast, United Kingdom Upmeier, Harald Marburg University, Germany

Operator Theory: Advances and Applications, Vol. 181, 3–32 c 2008 Birkh¨ auser Verlag Basel/Switzerland

Subalgebras of Graph C*-algebras Stephen C. Power Abstract. I give a self-contained introduction to two novel classes of nonselfadjoint operator algebras, namely the generalised analytic Toeplitz algebras LG , associated with the “Fock space” of a graph G, and subalgebras of graph C*algebras. These two topics are somewhat independent but in both cases I shall focus on fundamental techniques and problems related to classifying isomorphism types and to the recovery of underlying foundational structures, be they graphs or groupoids. Mathematics Subject Classiﬁcation (2000). Primary 47L40. Keywords. Operator algebras, directed graphs.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2 The Cuntz algebras, intuitively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3 Toeplitz algebras for countable directed graphs . . . . . . . . . . . . . . . . . . . . . . . . .

17

4 Subalgebras of On . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

5 Appendix: Digraph algebras and limit algebras . . . . . . . . . . . . . . . . . . . . . . . . .

27

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

1. Introduction I give a self-contained introduction to two novel classes of nonselfadjoint operator algebras, namely the generalised analytic Toeplitz algebras LG , associated with the “Fock space” of a graph G, and subalgebras of graph C*-algebras. These two topics are somewhat independent but in both cases I shall focus on fundamental techniques and problems related to classifying isomorphism types and to the recovery of underlying foundational structures, be they graphs or groupoids.

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S.C. Power

1.1. Generalities on operator algebras Let us set the scene with a bird’s eye view of how operator algebras “come about” and comment on their morphisms. I shall take the term operator algebra to mean a complex algebra of bounded linear operators on a separable complex Hilbert space. For example, the operator algebra A could be the set of all complex single variable polynomials in a given operator T ; that is A = {p(T ) = a0 I + a1 T + · · · + an T n : p(z) = a0 1 + a1 z + · · · + an z n }. Often, the operator algebras of interest are manufactured by specifying a set of generators (such as the set {I, T } in the example) on a Hilbert space both of which (set and space) arise from a “foundational” mathematical structure, such as a group, or a graph, or a dynamical system. We might call this a spatial setting since the Hilbert space is in place at the outset and the operator algebra is taken to be the algebra generated by the given generators. The term “generated” may mean simply the unclosed complex algebra or it may refer to the closure of this algebra in some topology, usually either the operator norm topology or the weak operator topology. We do not assume that the generated algebra is self-adjoint, that is, closed under the conjugate transpose operation X → X ∗ , although of course that will follow if the set of generators is a self-adjoint set. Operator algebras are also constructed in a Hilbert space-free way, for example, as a particular operator algebra, within some huge general class of operator algebras, satisfying a universal property for (perhaps) a set of generators and relations. Alternatively the algebra A might be constructed in the category of normed algebras with the expectation that A is isometrically isomorphic to an operator algebra by virtue of the fact that the ingredients for A are operator algebras. For example A might be a Banach algebra direct limit of operator algebras, or, again, simply a quotient of operator algebras. We shall focus on spatial viewpoints. However, let us recall that the celebrated Gelfand–Naimark theorem can bring us back to a spatial context from a space free one. In truth, there are usually more ready-to-hand ways of providing a Hilbert spaces on which an indirectly constructed operator algebra can sit. Theorem 1.1. Let A be a C*-algebra (an involutive complete normed algebra with ab ≤ ab and a∗ a = a2 for all a, b ∈ A). Then there is a Hilbert space H (a separable one is possible if A is separable) and an isometric star homomorphism A → B(H). A fundamental question for a class of operator algebras is: When are two operator algebras A1 and A2 isomorphic? The strongest sense of isomorphism is undoubtedly unitary equivalence, that is, A1 = U A2 U ∗ for some isometric onto map U from the Hilbert space of A2 to that of A1 . Here the operator algebras really are the same if only we would identify the Hilbert spaces. A weaker form of isomorphism which also takes account of how the operator algebra sits on the underlying Hilbert space is star-extendible isomorphism. This requires that there is a map φ : A1 → A2 which is the restriction

Subalgebras of Graph C*-algebras

5

of an adjoint respecting algebra isomorphism φ : C ∗ (A1 ) → C ∗ (A2 ) between the generated C*-algebras. Weaker still, and now ignoring how the operator algebras are manifested, an isometric algebra isomorphism is simply an algebra isomorphism which is an isometric linear map. For nonselfadjoint operator algebras this form of isomorphism is usually the essential case to elucidate. In truth, while these forms of isomorphisms certainly are diﬀerent, in the case of operator algebras constructed from the same spatial scheme, as alluded to above, the resulting forms of isomorphism type are usually the same. By this I mean that if A1 and A2 are isomorphic in one of the three sense above then they are isomorphic in the other senses. Is there a metatheorem here I wonder? Let us also note a companion question to that above, which is generally deeper, concerning the symmetries of an operator algebra A. What is the isometric automorphism group of an operator algebra? Naturally one expects that when two instances of a foundational structure are isomorphic then this entails an isometric isomorphism between the associated operator algebras and indeed this is generally a routine veriﬁcation. (We might more formally realize the association as a functor.) But how about the converse direction? If A(S1 ) and A(S2 ) are the (norm closed say) operator algebras obtained from the foundation structures S1 and S2 , and if Φ : A1 → A2 is an isometric algebra isomorphism then does this somehow induce an isomorphism between S1 and S2 ? This would provide a satisfyingly deﬁnitive answer to the isomorphism question and it is in this connection that there is, as we say, clear blue water between the non-self-adjoint and the self-adjoint theory. Algebras of the former category seem to remember their foundations while the self-adjoint algebras need not.1 As an indication of the general landscape ahead here is a list of the ingredients of several operator algebra contexts concerning a hierarchy of Toeplitz algebras. A: The classical context : The Hardy–Hilbert space H 2 for the unit circle, the unilateral shift operator S, with dim(I − SS ∗ ) = 1, the disc algebra A(D), the function algebra H ∞ (D), realizations of A(D) and H ∞ (D) as “analytic” Toeplitz algebras, and the Toeplitz C*-algebra TZ+ . Let us outline some important classical facts. The Hilbert space 2 (Z+ ) is unitarily equivalent to the Hardy space H 2 of the Hilbert space L2 (T) of square integrable functions on the circle. Here H 2 denotes the closure in L2 (T) of the space of polynomials in z. The basis matching unitary U : 2 (Z+ ) → H 2 which does this (with U en := z n for n ≥ 0) eﬀects a unitary equivalence between the unilateral shift S (with Sen = en+1 , n ≥ 0) and the multiplication operator Tz : f → zf, f ∈ H 2 . That is U SU ∗ = Tz . C*-algebras it is K-theory and associated invariants that often lead to classifying invariants. In general such invariants are insuﬃcient as they are generally determined by the diagonal part A ∩ A∗ of the operator algebra A.

1 For

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S.C. Power

More generally, the Toeplitz operator Tφ with symbol function φ in C(T) or L∞ (T) is given by Tφ : f → PH 2 φf . When φ is analytic, that is, is the boundary function of an analytic function on the disc, then Tφ is simply the restriction of the multiplication operator Mφ to H 2 . In this way we have a representation of the so called disc algebra A(D) as an algebra of operators on H 2 . A good exercise to do at least once is to show that the Toeplitz algebra TZ+ = C ∗ (I, Tz ) (which is equal to U C ∗ (I, S)U ∗ ) contains every compact operator K on H 2 . The idea is to start with the rank one operator I − Tz Tz∗ and “move it around” with the shifts Tz , Tz∗ to obtain every rank one operator of the form f → f, z k z l . For example, show that Tzn (I − Tz Tz∗ ) is the matrix unit operator En,0 : f → f, z 0 z n . Then from these operators and their adjoints create all the matrix units Ei,j . Finally, take linear combinations to approximate any ﬁnite rank operator. Once this is done one a little more work leads to the following theorem. Theorem 1.2. (i) For each Toeplitz operator Tφ and compact operator K we have Tφ + K ≥ Tφ . (ii) The Toeplitz algebra TZ+ is equal to the set of operators Tφ +K with φ in C(T) and K compact and the quotient TZ+ /K is naturally isomorphic to C(T). On the other hand the norm closed operator algebra generated by Tz is abelian and isometrically isomorphic to the disc algebra. Indeed, it is the algebra {Tφ : φ ∈ A(D)}. The weak operator topology closed algebra is similarly a copy of H ∞ (D), namely, {Tφ : φ ∈ H ∞ (D)}. On occasion we simply write H ∞ for this operator algebra when the context is clear. Recall that the weak operator topology is the weakest topology for which the spatial linear functionals T → T f, g are continuous. There are a great many ways in which one can move on from the Toeplitz context above and below I discuss some aspects of the following operator algebra directions. B: The (spatial) free semigroup context (Section 3.): The Fock space 2 (F+ n) for the free semigroup on n generators, the freely noncommuting shifts L1 , . . . , Ln with dim(I − (L1 L∗1 + · · · + Ln L∗n )) = 1, the noncommutative disc algebra An and free semigroup algebra Ln , and the Cuntz–Toeplitz C*-algebra on 2 (F+ n ). C: The (spatial) graph context (Section 3.): The Fock space of a directed graph G = (V, E), the freely noncommuting partial isometries Le , e ∈ E, the tensor algebra AG , the free semigroupoid algebras LG , and the Cuntz–Krieger– Toeplitz C*-algebras TG = C ∗ (AG ). D: The (universal) free semigroup context (Sections 2, 4): The freely noncommuting isometries S1 , . . . , Sn with S1 S1∗ + · · · + Sn Sn∗ = I, the Cuntz algebras On = C ∗ (S1 , . . . , Sn ).

Subalgebras of Graph C*-algebras

7

E: The (universal) graph context (Section 4): The (universal) graph C*algebra C*(G) of a countable directed graph G = (V, E) with partial isometry generators Se , for e ∈ E, and relations Se Se∗ = Px , Se∗ Se = Ps(e) , r(e)=x

where {Px : x ∈ V } is a family of orthogonal projections and e = (r(e), s(e)).

2. The Cuntz algebras, intuitively The Cuntz algebra On is a certain C*-algebra generated by n isometries, S1 , . . . , Sn say, satisfying S1 S1∗ + · · · + Sn Sn∗ = 1. That is, their range projections Si Si∗ are orthogonal and sum to the identity operator. In fact I could have dropped the word “certain” because of the following remarkable uniqueness property. Theorem 2.1. If n ≥ 2 and s1 , . . . , sn is any family of n isometries in a unital C*algebra with s1 s∗1 + · · · + sn s∗n = I, then C ∗ ({s1 , . . . , sn }) is naturally isometrically isomorphic to On . Our main aim is to obtain tools and results that will help in understanding norm closed subalgebras of the Cuntz algebras. In this connection we are prepared to consider operator algebras generated by semigroups of words in the generators and to contemplate quite general subalgebras. Perhaps it is fair to say that a C*-algebraist is largely happy with the state of knowledge of the Cuntz algebras. He/she is more interested in looking for generalisations and wider classes to classify and understand (such as Graph C*-algebras, or C*-algebras allied to dynamical systems). Our view here is quite diﬀerent – we are intending to linger with On and look inside it with a view to understanding classes of nonselfadjoint operator algebras. Our orientation and motivation comes partly from the existing theory of limit algebras which are found as nonselfadjoint subalgebras of approximately ﬁnite C*-algebras. We shall focus on the Cuntz algebras for clarity but the methods we discuss do extend to more general graph C*-algebras. 2.1. Cuntz algebra basics One direct way to deﬁne On is to look into the next section, take the freely noncommuting shifts L1 , . . . , Ln on the Fock space for the free semigroup on n generators, take the generated C*-algebra and divide out by the ideal of compact operators. (This should sound familiar if n = 1!) In taking the quotient we lose the Hilbert space and gain equality in place of the one-dimensional defect dim(I − L1 L∗1 + · · · + Ln L∗n ) = 1. The uniqueness allows us to consider two other models for On which will in fact be our viewpoint. I call these models the interval picture and the Cantorised interval picture. The ﬁrst uses isometric operators S1 , . . . , Sn on L2 [0, 1] whose i ranges are the orthogonal subspaces L2 [ i−1 n , n ]. For deﬁniteness we may deﬁne √ i−1+x i Si ( n ) = nf (x), for x ∈ [0, 1], and Si f )(t) = 0 for t not in [ i−1 n , n ].

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S.C. Power

Notice that a product Si1 Si2 . . . Sik has range L2 [Ek ] where Ek is an interval with |Ek | = n−k . Write Sµ for this product where µ is the word i1 i2 . . . ik . These k-fold products have distinct ranges and so if the lengths of µ and ν are |µ| = |ν| = k then Sµ∗ Sν is the zero operator when µ and ν diﬀer. (Exercise: Prove this algebraically.) It follows in general that if µ and ν have diﬀering lengths and Sµ∗ Sν is nonzero then either Sµ∗ Sν = Sµ∗ where µ = νµ , or Sµ∗ Sν = Sν where ν = µν . On the other hand products with stars on the right are always non-zero. Indeed, from the interval picture we see that for |µ| = |ν| = k the operator Sµ Sν∗ acts as an isometry from L2 [Eν ] to L2 [Eµ ] for certain intervals Eµ , Eν with lengths |n|−k . Moreover, the set of operators {Sµ Sν∗ : |µ| = |ν| = k} satisﬁes the relations of a matrix unit system. The span of this set, Fkn say, is thus a copy of the matrix algebra Mnk , and we have the matrix algebra tower F1n ⊆ F2n ⊆ F3n · · · . n We see then that the generators of On provide a distinguished subalgebra F∞ = ∞ n ∞ n ∪k=1 Fk which we refer to as a matricial star algebra of type n . Write F for the closure of this subalgebra in On . Notice that the diagonal matrix units are those of the form Sµ Sµ∗ . The closed linear span of these is an abelain algebra, C say, in F n and On which plays a distinguished role.

0

1

1

0 1/4

1/2

Figure 1. Interval picture for the operator S1 S2 in O2 . Theorem 2.1 gives rise immediately to an important family of automorphisms of On , the so called gauge automorphisms γz , for z ∈ T, which satisfy γz (Si ) = zSi , 1 ≤ i ≤ n. (Alternatively, these automorphisms are inherited from easily deﬁned unitarily implemented automorphisms of the Cuntz-Toeplitz C*-algebra.)

Subalgebras of Graph C*-algebras

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Proposition 2.2. (i) Each operator a in the star algebra generated by S1 , . . . , Sn has a representation N N (S1∗ )i a−i + a0 + ai S1i a= i=1

i=1

n F∞

where ai ∈ for each i. This representation is unique if for each i ≥ 0 we have ai = ai Pi and a−i = Pi a−i where Pi is the ﬁnal projection of S1i . (ii) The linear maps Ei , deﬁned by Ei (a) = ai , extend to continuous, contractive, linear maps from On to F n . (iii) The generalised Cesaro sums Σk (a) =

N

1−

k=1

N |k| ∗ k |k| (S1 ) E−k (a) + Ek (a)sk1 1− N N k=0

converge to a as N → ∞. Proof. Our observations above show that the linear span of the operators Sµ Sν∗ is the algebra generated by the generators. The representation in (i) (with the dilation actions carried by S1 alone) follows from this by means of formulae such as Sµ = (Sµ (S1∗ )k )S1k = aS1k where k = |µ|. The key to uniqueness is to make use of “recovery formulae” such as 2π dθ a0 = γz (a) 2π 0 where the integral is a Riemann integral of a norm continuous function. The representation in (i) can be viewed as a generalised Fourier series representation for the operator a. In fact to any operator a in On one may assign generalised Fourier coeﬃcients ak in F n by means of the maps Ek (.). The operators ak S1k (k > 0) and (S1∗ )k a−k (k < 0) appear as the conventional Fourier series coeﬃcients for the norm continuous operator-valued function fa : z → γz (a). The Cesaro polynomials pn (z) for the continuous function fa converge uniformly in operator norm on |z| = 1 by classical theory. Finally, the specialisation z = 1 gives the desired norm convergence of the generalised Ces´aro polynomials. Exercises. (i) Show that E0 is faithful, that is, if a ≥ 0, a = 0 then E0 (a) = 0. (ii) Show that ∗k γz (aS1 )dz S1k , k > 0. ak = T

The Cantorised interval picture for On is a reﬁnement of the interval presentation. The beauty of this perspective is that it provides a context for deﬁning binary relation (and groupoid) invariants for subalgebras. The essence of the idea is captured in Figure 2, shown with 2-fold branching relevant to the n = 2 case.

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S.C. Power

0

1 01

00

11

10 101

100

0

x

1

X

Figure 2. Cantorised interval picture. Let X be the set of inﬁnite paths on the tree, starting at vertex 0 or vertex ∞ 1. This set is identiﬁable with the direct product k=1 {0, 1}, consisting of points x which are zero-one sequence x1 x2 . . . . Each vertex word w in the tree, such as 1001, gives rise to an “interval” Ew of points x whose product expansion starts with w. With the usual product topology the direct product is a Cantor space whose topology has the set of intervals as a base of closed-open sets. For each pair of vertex words w1 , w2 there is a partial homeomorphism αw2 ,w1 from Ew1 to Ew2 deﬁned by matching tails: if x = w1 xp xp+1 . . .

then

αw2 ,w1 (x) = w2 xp xp+1 . . .

Notice that for |w1 | = |w2 | = k the partial homeomorphism has an action on ∗ the set X that bears close analogy with Sw2 Sw and its interval picture, where we 1 have relabeled the generators as S0 and S1 . In fact we can add the natural product probability measure to X and present On on L2 (X) in terms of (newly labeled) generators S0 and S1 induced by the partial homeomorphisms α∅,0 : x → 0x, α∅,1 : x → 1x. That is, for i = 1, 2 we have (Si f )(α∅,0 (x)) =

√ 2f (x),

and (Si f )(y) = 0 if y is not in the range of α∅,i . Exercise. Obtain the partial homeomorphism that is associated with the partial ∗ . (Here we have the 0-1 subscript labeling as opposed to isometry S1 S0∗ + S00 S11 the 1-2 labeling.) 2.2. Normalising partial isometries We now come to an important class of partial isometries associated with the abelian diagonal algebra C.

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Deﬁnition 2.3. A partial isometry in F n (or, more generally, in On ) is C-normalising if vCv ∗ ⊆ C and v ∗ Cv ⊆ C. The obvious examples are the matrix units Sµ Sν∗ for |µ| = |ν| = k, and the sums v of these when the initial projections are orthogonal and the ﬁnal projections are orthogonal. Also we may multiply such a v by a unitary diagonal element d. The resulting C-normalising partial isometry has support which can be indicated pictorially, as shown. The coordinates have been arranged so that the support picture represents v intuitively as a continuous matrix (although the d information is now lost). The picture should be “Cantorised” and viewed as a subset of X × X. 0

1

0

1 ∗ Figure 3. The support the partial isometry S2 S1∗ + S11 S22 .

The next theorem is an extremely useful characterisation. It shows in particular that for subalgebras of F n an isometric isomorphism α : A1 → A2 with α(C) = C preserves the set of normalising partial isometries in the algebras. Theorem 2.4. Let v be an element of F n . Then the following assertions are equivalent: (i) v is a C-normalizing partial isometry. (ii) v is an orthogonal sum of a ﬁnite number of partial isometries of the form dSµ Sν∗ , where |µ| = |ν| and d ∈ C. (iii) For all projections p, q ∈ C, the norm qvp is equal to 0 or 1. First we note some general “recovery facts” about F n which show how operators may be approximated in an explicit manner. For b ∈ F n the “diagonal part” ∆(b) ∈ C may be deﬁned as the limit of the block diagonal operators bk := Σi ekii bekii

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S.C. Power

where ekii are the diagonal matrix units of Fkn . The map ∆ : F n → C is a projection and moreover is faithful in the sense that ∆(b∗ b) = 0 entails b = 0. Likewise one can use block diagonal maps (via matrix unit projections taken from the commutants n n n of Fm in Fkn , k = m, m + 1, . . . ) to deﬁne explicit maps ∆m : F n → F˜m where F˜m ∗ n n m m is the C*-algebra C (C, Fm ) (which in fact is identiﬁable with Fm ⊗ (e11 Ce11 ) and we have ∆m (b) → b as m → ∞ for all b ∈ F n . In this way (and analogously to Cesaro convergence) we can approximate a general element b in F n by explicitly n . constructed approximants ∆m (b) in F˜m Proof of Theorem 2.4: The directions (ii) =⇒ (i) =⇒ (iii) are elementary so assume that v is an element which satisﬁes the zero one norm condition. Choose m large so that v = ∆m (v) + v , v < 1. Since v satisﬁes the zero one norm condition this is also true of ∆m (v) and v . Indeed, this holds for operators in the matrix subalgebras and by approximation holds in general. The implication (iii) implies (ii) is straightforward for elements of F˜kn . Thus it remains to show that if v has norm less than one and satisﬁes the zero one norm condition then v = 0. This too follows from approximation. Exercise. Show that F n has no proper closed two-sided ideals. (Hint: E0 is faithful.) Remark. The map a → ∆(E0 (a)) is a positive faithful contraction onto the diagonal algebras but it is not (as above) deﬁned as a limit of block diagonal parts. (eg consider a = S1 ). (However, as we note in the Notes below, it may be deﬁned in a more subtle algebraic manner.) n 2.3. Subalgebras of F∞ With the two models for On above and generalised Fourier series we are almost ready to contemplate the following vague question. What are the natural subalgebras of On ?

Before turning to this we should of course look ﬁrst inside the C*-algebra n Mn and the algebras F∞ , F n. The most natural subalgebras of Mn are perhaps those unital subalgebras A which are spanned by a subset of the standard matrix unit system {eij : 1 ≤ i, j ≤ n}. In fact these subalgebras are precisely those that contain the diagonal algebra C spanned by {eii }. (It is this latter property that we essentially use to deﬁne inﬁnite-dimensional variants.) In this case the set E = {(i, j) : eij ∈ A} can be viewed as the set of edges of a directed graph G with n vertices. It follows that (i, i) ∈ E for all i and that (i, k) ∈ E if (i, j), (j, k) ∈ E. Conversely, if G is the graph (V, E), with E such a reﬂexive and transitive relation, then A(G) := span{eij : (i, j) ∈ E} is a complex unital subalgebra containing C. These algebras are the building block algebras of limit algebras. See the Appendix for a fuller discussion.

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It is an elementary but worthwhile exercise to show that the operator algebra A(G) remembers the graph G: Proposition 2.5. (Recovery theorem.) If A(G1 ) and A(G2 ) are isometrically isomorphic algebras then the graphs G1 , G2 are isomorphic. Sketch proof: (i) Projections must map to projections (since projections are the idempotents with norm one), (ii) minimal projections map to minimal projections, (iii) one can reduce (via unitary equivalence) to the case that diagonal projections map to diagonal projections and this gives the needed vertex map. We can think of E both as a “support set” for the algebra (should we view a matrix (aij ) as a function ij → aij ), and also, more usefully, as a binary relation that comes with the algebra. In these terms the proposition says that a digraph algebra has isometric isomorphism type determined by the isomorphism type of its binary relation. In Section 4 I give a generalisation of this fact for a wide class of subalgebras of On . First however, let us look inside the matricial star algebra n F∞ and its closure F n . In fact we may as well consider a more general class of approximately ﬁnite algebras. Deﬁnition 2.6. (a) A unital matricial star algebra is a complex algebra B for which there exists a spanning set {ekij : 1 ≤ i, j ≤ nk , k = 1, 2, . . . } such that (i) for each k the set {ekij : 1 ≤ i, j ≤ nk } is a matrix unit system for Mnk , (ii) for each k, Mnk ⊆ Mnk+1 and moreover the inclusion map is a C*algebra injection which maps each ekij to a sum of matrix units from {ek+1 : 1 ≤ i, j ≤ nk }. ij (b) A regular matricial algebra is a complex algebra A which is a unital subalgebra of a matricial star algebra containing the diagonal subalgebra C = span{ekij }. It is straightforward to see that the regular matricial algebra A in the matricial star algebra B is the union of the algebras Ak = A ∩ Mnk = span{ekij : ekij ∈ A} and that each Ak is a digraph algebra A(Gk ) relative to the given matrix unit sysn tem. Our subalgebra F∞ is a particular unital matricial star algebra in which each inclusion map has the same multiplicity n. Further examples of such algebras can be obtained with the technology of direct limits: ﬁrst take a tower of appropriate inclusions maps, A(G1 ) → A(G2 ) → . . . . Algebraic direct limits then provide the algebra A = lim→ A(Gk ) as well as B and C.

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2.4. Subalgebras of unital AF C*-algebras Suppose we have the purely algebraic setting C ⊆ A ⊆ B given in Deﬁnition 2.4. The matricial star algebra carries a unique C*-algebra norm and taking operator norm closures gives the triple inclusion C ⊆ A ⊆ B. Here B is a UHF C*-algebra, C is a particularly nice abelian subalgebra in B and A is an instance of a limit algebra A = lim(A(Gk ), φk ) →

where the inclusion maps φk : A(Gk ) → A(Gk+1 ) are particularly nice. (In the terminology of the Appendix, they are star-extendible and regular.) We give three key results for such limit algebras. The ﬁrst, rather surprisingly perhaps, shows that any norm closed algebra A with C ⊆ A ⊆ B is necessarily the closure of a regular matricial algebra. This is justiﬁcation for the opinion that the subalgebras of UHF C*-algebras which contain the distinguished masa are the natural generalisations of ﬁnite-dimensional digraph algebras. The following “local recovery” theorem gives a key step towards understanding the limit algebras A. Theorem 2.7. (Inductivity principle.) Let B be a unital matricial star algebra with subalgebra chain {Mnk } and diagonal C and let B, C be their norm closures. If A ⊆ B is a norm closed subalgebra containing C then A is the closed union of the digraph algebras Ak = A ∩ Mnk , k = 1, 2, . . . . The next two theorems (and that above) have more general forms for subalgebras of AF C*-algebras but we state them here for subalgebras of the UHF C*-algebra F n . We ﬁrst need to deﬁne the appropriate substitute for the graph that underlies a digraph algebra and for this the Cantorised interval picture provides what we need, both for F n and for On . In fact we are going to deﬁne an isometric isomorphism invariant for the algebra A together with its diagonal C which is in the category of topological binary relations. Often (always?!) the binary relation is a complete isometric isomorphism invariant for the algebra alone. ∞ Let X be the Cantor space k=1 {0, 1, . . . , n − 1}. For words µ, ν with the same length k recall that Sµ Sν∗ is a partial isometry on L2 (X) which is induced by the partial homeomorphisms αµ,ν : νxk+1 xk+2 · · · → µxk+1 xk+2 . . . . Deﬁne the topological space R to be the set in X × X which is the union of the graphs of these partial homeomorphism: R = {(α(x), x) : x ∈ dom(α), α = αµ,ν , |µ| = |ν| = k, k = 1, 2, . . . } (k)

k for the graph of the partial homeomorphism for the matrix unit eij Write Ei,j n in Fk and one can conceive of these sets as the “support set” of the matrix (k) units. The diagonal matrix units eij provide closed-open sets Eiik in the diagonal ∆ = {(x, x) : x ∈ X} and these give a base for the natural Cantor space topology. k We topologise R by taking the sets Eij as a base for the topology.

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Exercise. Show that R is an equivalence relation. Show that the topology is not the relative product topology. It follows from Theorem 2.7 that if C ⊆ A ⊆ F n then A, and the given subalgebra chain, determines a subset R(A) of R, namely k R(A) := R(A) := ∪{Eij : ekij ∈ Ak }.

With the relative topology, R(A) is known as the topological binary relation of A. In fact the topological binary relation R(A) is determined by the pair (A, C) and serves as the analogue of the graph of a digraph algebra. The following uniqueness theorem also follows from Theorem 2.7. Theorem 2.8. (Spectral theorem for subalgebras.) Let A1 , A2 be norm closed subalgebras of F n which contain the canonical diagonal algebra C. If R(A1 ) = R(A2 ) then A1 = A2 . That R(A) is intrinsic to the pair A, C is also revealed by the following proposition. We identify X with the Gelfand space of C. Proposition 2.9. As a set, R(A) is the set of points (x, y) in X × X for which there exists a ∈ A and δ > 0 such that paq ≥ δ for all projections p, q in C with x(p) = y(q) = 1. We can now state the following classiﬁcation theorem which, loosely paraphrased, asserts that a triangular subalgebra of F n remembers its topological binary relation. Theorem 2.10. Let A1 and A2 be norm-closed subalgebras of F n with Ai ∩ A∗i = C for i = 1, 2. (Such algebras are said to be triangular.) Then the following statements are equivalent n n and A2 ∩ F∞ are isometrically isomorphic normed algebras. (i) A1 ∩ F∞ (ii) A1 and A2 are isometrically isomorphic operator algebras. (iii) The topological binary relations R(A1 ), R(A2 ) are isomorphic, that is, there is a homeomorphism α : M (C) → M (C) such that α × α : R(A1 ) → R(A2 ) is a homeomorphism.

The key to the proofs of Theorem 2.7, Theorem 2.8, Proposition 2.9 and Theorem 2.10 is the structure of partial isometries given in Theorem 2.4. It is this which which makes the link between operator algebra entities and the underlying topological binary relation. Remarkably, perhaps, there is a close generalisation of Theorem 2.10 to gauge invariant subalgebras of On . (Theorem 4.3.) Sketch proof of Theorem 2.10. Let Φ : A1 → A2 be an isometric isomorphism. By triangularity, the set of projections p in Ai generate Ci . Since the projections are the norm one idempotents it follows that Φ(C1 ) = C2 . By the zero-one characterisation

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in Theorem 2.4 it follows that Φ maps the C1 -normalising partial isometries to C2 normalising partial isometries. Considering the support of a normalising partial isometry as a closed-open set in M (Ci ) × M (Ci ) it follows (from the ﬁniteness of Theorem 2.4 (ii)) that Φ maps a base of closed open sets to a base of closed-open sets. Moreover Φ induces a point bijection α : R(A1 ) → R(A2 ). (Take intersections of neighbourhoods.) The point bijection is a topological homeomorphism since its map on sets extends the original bijection of closed-open sets. Open problems. (See also the Notes below.) 1. For the algebraic context C ⊆ A ⊆ B given at the end of Section 2.3 is C unique in A up to automorphisms of A? If this is the case then R(A) becomes a well-deﬁned invariant for the algebra A. 2. Is R(A) an algebraic isomorphism invariant for a regular matricial algebra A? 2.5. Notes Theorem 1.2 is usually referred to as Coburn’s theorem. For more on this, Cuntz algebras and other C*-algebras, see, for example, Davidson [3] and the references therein. On is usually deﬁned as the universal C*-algebra generated by isometries S1 , . . . , Sn with S1 S1∗ + · · · + Sn Sn∗ = I. The direct sum of irreducible realisations of such relations gives generators for this algebra; the universal property is that to any isometric realisation s1 , . . . , sn of the relations there should exist a canonical homomorphism On → C ∗ ({s1 , . . . , sn }) and this is readily checked for this universal direct sum. The uniqueness theorem, Theorem 2.1, will follow now from the simplicity of On , and this in turn follows readily from an algebraic formulation of the map E0 : On → Fn . (See [3],[1].) For if J is an ideal and a ∈ J is nonzero, then a∗ a is a positive nonzero element in J and so it follows from the algebraic formulation that a0 = E0 (a) is in J. Since Fn is simple the simplicity of On follows. In fact more is true in that one can ﬁnd elements x, y ∈ On such that xab = I. To do this one uses the algebraic deﬁnitions of E0 and ∆ to get, in J, an operator d = pdp = x1 ay2 close to a diagonal matrix unit p. Then one ﬁnds the appropriate isometry Sµ to conjugate p to an operator close to the identity. The normalising partial isometry theorem, from which Theorem 2.7, Theorem 2.8 and Theorem 2.10 are easily obtained, are discussed further in [21]. Here one can also ﬁnd their natural extensions to AF C*-algebras. The open problems are essentially the problems 7.8, 7.9 of [21]. Those problems are stated for closed algebras but because of inductivity the problems really reside in pure algebra and are stated in these terms here. If A is self-adjoint then the masa C is unique up to automorphism. More is true: for any other matrix unit system for A, with subalgebra system {Ak } and masa C (as in the deﬁnition), there is an automorphism A → A which maps C to C . (See [21].) However in this case the automorphism, which is determined by an intertwining diagram A1 → An1 → Am1 → An2 → · · · , can be constructed in such a way so that there is an intertwining diagram with regular maps in the sense that the normaliser of each diagonal algebra maps into

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17

the normaliser of the diagonal of the next algebra. (See the Appendix.) Part of the diﬃculty of the general nonselfadjoint problem is that it is known that there are diagonal masas (of the type above) which although automorphically equivalent are not equivalent through an intertwining diagram of regular maps. See [9] for this subtlety, and see [10], [24] for further discussions. The importance of the problem is that all kinds of putative invariants can be associated with classes of regular direct system and one would like these constructs (partial isometry homology groups for example) to be invariants for the algebra rather than the pair (A, C).

3. Toeplitz algebras for countable directed graphs We now take up a diﬀerent topic and formally deﬁne the analytic Toeplitz algebras AG and LG . Let G be a ﬁnite or countable directed graph, with edge set E(G) and vertex set V (G). The free semigroupoid F+(G) determined by G is a set with partially deﬁned associative multiplication. The set consists of the vertices, which act as multiplicative units, and all ﬁnite directed paths in G. The partially deﬁned product is the natural operation of concatenation of paths, with a vertex considered as a path. Thus a nonunit element of F+(G) is a path (or word) w = e1 e2 . . . en where the initial vertex of each ei , for i < n, is equal to the ﬁnal vertex of ei+1 . Vertices may appear in a word of edges, redundantly, to indicate information. For example given a path w in F+(G) we have w = ywx when the initial and ﬁnal vertices of w are, respectively, x and y. Let HG = 2 (F+(G)) be the Hilbert space with orthonormal basis of vectors ξw indexed by elements w of F+(G). For each edge e ∈ E(G) and vertex x ∈ V (G) deﬁne partial isometries and projections on HG by the following left-sided actions on basis vectors: ξew if ew ∈ F+(G) Le ξw = 0 otherwise and ξxw = ξw if w = xw ∈ F+(G) Lx ξw = 0 otherwise. We also write Px for the projection Lx . If G has a single vertex x then ξx is referred to as the vacuum vector and the operators Lw are isometries. If there are, additionally, only ﬁnitely many edges e1 , . . . , en then each path w is a free word in these edges and the semigroupoid of paths is simply the free (unital) semigroup F+ n with n generators. For n = 2 one can visualise the action of the two isometries Le1 and Le2 as downward left and right shifts of basis vectors placed at the vertices of a downward branching tree. Deﬁnition 3.1. (i) The free semigroupoid algebra LG is the weak operator topology closed algebra generated by the projections Lx and the (partial) shift operators Le ; LG

= wot–Alg {Le , Lx : e ∈ E(G), x ∈ V (G)}.

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(ii) The algebra AG is the norm closed algebra generated by {Le , Lx : e ∈ E(G), x ∈ V (G)}. This Toeplitz algebra is also referred to as the tensor algebra for G. The algebra LG can also be thought of as being generated by the left “regular” representation λG : F+ G → B(HG ), λG (e) = Le , which faithfully represents the as partial isometries. In the case of a noncomposable elements semigroupoid F+ G w1 and w2 one can check that the corresponding partial isometries have zero product. 3.1. Examples and matrix function algebras It should be apparent that the algebra LG for the graph with a single vertex x and single loop edge e = xex is unitarily equivalent to the analytic Toeplitz algebra TH ∞ ; the Fock space naturally identiﬁes with the Hardy space H 2 , and Le is then unitarily equivalent to the unilateral shift Tz . More generally, the noncommutative analytic Toeplitz algebras Ln , n ≥ 2, also known as the free semigroup algebras, arise from the graphs with a single vertex and n distinct loop edges, while L∞ comes from the single vertex graph with countably many loops. (i) If G is a ﬁnite graph with no directed cycles, then the Fock space HG is ﬁnite-dimensional and so too is LG . As an example, consider the graph with three vertices and two edges, labelled x1 , x2 , x3 , e, f where e = x2 ex1 , f = x3 f x1 . Then the Fock space is spanned by the vectors {ξx1 , ξx2 , ξx3 , ξe , ξf } and with this basis the general operator X = αLx1 + βLx2 + γLx3 + λLe + µLf in LG is represented by the matrix ⎤ ⎡ α 0 0 0 0 ⎢ 0 β 0 0 0⎥ ⎥ ⎢ ⎥ X⎢ ⎢ 0 0 γ 0 0⎥ . ⎣λ 0 0 β 0 ⎦ µ 0 0 0 γ Here, LG is isometrically isomorphic to (but not unitarily equivalent to) the socalled digraph algebra (see later) in M3 (C) consisting of the matrices ⎡ ⎤ α 0 0 ⎣λ β 0⎦ . µ 0 γ (ii) Consider the graph G with two vertices x, y, a loop edge e = xex and the edge f = yex directed from vertex x to vertex y. The tree graph for Fock space takes the form shown in Figure 4. The semigroupoid algebra LG is generated by {Le , Lf , Px , Py }. If we make the natural identiﬁcations HG = Px HG ⊕ Py HG H 2 ⊕ H 2 , then Tz 0 0 0 I 0 0 0 , Px Le , Lf , Py . 0 0 Tz 0 0 0 0 I Thus, LG is seen to be unitarily equivalent to a matrix Toeplitz algebra, which we can also view as (isometrically and weak star – weak star isomorphic to) the

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x

19

y

e

f

ee

fe

eee

fee

Figure 4. Fock space graph. matrix function algebra

∞ H H0∞

0 CI

where H0∞ is the subalgebra of H ∞ formed by functions h with h(0) = 0. Exercise. Add to G a “returning” directed edge g = xgy to obtain a graph G . Show that LG contains a “copy” of L2 by virtue of the fact that it contains isometries with mutually orthogonal ranges. (iii) Let n ≥ 1 and consider the cycle graph Cn which has n vertices x1 , . . . , xn and n edges en = x1 en xn and ek = xk+1 ek xk for k = 0, . . . , n − 1. Identify Lxi HG with H 2 for each i in the natural way (respecting word length). Then HG = Lx1 HG ⊕ . . . ⊕ Lxn HG Cn ⊗ H 2 and the operator α1 Le1 + . . . + αn Len is identiﬁed with the operator matrix ⎡ ⎤ 0 αn Tz ⎢α1 Tz ⎥ 0 ⎢ ⎥ ⎢ ⎥ α2 Tz 0 ⎢ ⎥. ⎢ ⎥ .. .. ⎣ ⎦ . . αn−1 Tz Hn∞ ∞

∞

0

Write for the subalgebra of H arising from functions of the form h(z n ) with h in H . It follows that the algebra LCn is isomorphic to the matrix function algebra ⎡ ⎤ Hn∞ z n−1 Hn∞ . . . zHn∞ ⎢ .. ⎥ ⎢ zHn∞ Hn∞ . ⎥ ⎢ ⎥. ⎢ ⎥ .. .. ⎣ ⎦ . . ... Hn∞ z n−1 Hn∞

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3.2. Fourier series There is a companion algebra to LG coming from “right shifts”. Consider the right regular representation ρG : F+(G) → B(HG ) determined by a directed graph G. This yields partial isometries ρG (w) ≡ Rw for w ∈ F+(G) acting on HG deﬁned by the equations Rw ξv = ξvw , where w is the word w in reverse order. The corresponding algebra is RG

= wot–Alg {Re , Rx : e ∈ E(G), x ∈ V (G)}.

Given edges e, f ∈ E(G), observe that Le Rf ξw = ξewf = Rf Le ξw , for all w ∈ F+(G), so that Le Rf = Rf Le , and similarly for the vertex projections. In fact, we have Proposition 3.2. The commutant LG of LG is equal to RG . The proof of this proposition makes use of an important tool, namely the Fourier series expansion of elements of LG . Recall that Px is the projection for the subspace spanned by basis vectors ξw with w = xw. Write Qx for the projection onto the subspace spanned by basis vectors ξw with w = wx. (The projections Qx correspond to the “components” of the Fock space when basis elements are linked by the natural tree structure.) The proposition below takes a simple form when there is a single vertex, i.e., LG = Ln . Proposition 3.3. Let A ∈ LG , x ∈ V (G), and let aw ∈ C be the coeﬃcients for which Aξx =Qx Aξx = w=wx aw ξw . Then the Cesaro sums associated with the formal sum w∈F+(G),w=wx aw Lw , given by |w| aw L w 1− Σk (Qx A) = k w=wx;|w|≤k

converge in the strong operator topology to Qx A. 3.3. LG remembers the graph G Let us ﬁrst observe why Ln and Lm are not isomorphic when n = m. This can be seen from the following theorems the ﬁrst of which introduces another important tool, namely the eigenvectors for L∗G . By an eigenvector for L∗n we really mean a vector ν which is a joint eigenvector for the n-tuple of generators L∗e1 , . . . , L∗en , so that L∗ei ν = αi ν, , 1 ≤ i ≤ n, for some complex numbers αi ∈ C. The notation below is that w(λ) is the complex number obtained on substituting λi for ei in the word w. Theorem 3.4. The eigenvectors for L∗n are complex multiples of the unit vectors w(λ)ξw , νλ = (1 − λ2 )1/2 w∈F+ n

for λ = (λ1 , . . . , λn ) in the open unit ball Bn ⊆ Cn . Furthermore L∗ei νi = λi νλ , for each i.

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Note that for λ in the open unit ball n λi Lei 2 = |λi |2 = λ2 < 1 so that I −

1

n

1 λi Lei is invertible, with inverse −1 k I− λi Lei = λi Lei = w(λ)Lw .

e

k≥0

e

w

In particular w w(λ) is convergent, and a similar shows the norm to be (1 − λ2 )−1/2 . Eigenvectors are important since they are allied to characters, i.e., multiplicative linear functionals φ : Ln → C, φ : An → C. Indeed, the map φλ : An → C deﬁned by φλ (A) = Aνλ , νλ satisﬁes φλ (p(Le1 , . . . , Len )) = νλ , p(L∗e1 , . . . , L∗en )νλ = νλ , p(λ1 , . . . , λn )νλ = p(λ1 , . . . , λn ). It follows that the vector functional actually deﬁnes a character. One can go on to show that the character space of An is homeomorphic to closed unit ball. The dimension of the character space serves as a classifying invariant for the algebras An and Ln . More generally one has the following theorem, and an analogous result for the weakly closed free semigroup algebras. Theorem 3.5. Let G, G be directed graphs. Then the following assertions are equivalent. (i) G and G are isomorphic graphs. (ii) AG and AG are unitarily equivalent. (iii) AG and AG are isometrically isomorphic. 3.4. Notes There are a number of approaches to the classiﬁcation theorem above. In Kribs and Power [14], following the free semigroup algebra analysis of Davidson and Pitts [5], [6], wandering vectors are analysed to obtain a Beurling type theorem for invariant subspaces of the algebra. Using this one obtains unitarily implemented automorphisms of LG that act transitively on the set of eigenvectors. Now the eigenvectors are parametrised by the union of unit balls for each vertex with loop edges. In particular isomorphisms can be normalised to the special case where vacuum vectors map to vacuum vectors. Consequently the ideal A0G generated {Le : e ∈ E(G)} is preserved. Theorem 3.5 is straightforward in this case. See also Solel [29] and Katsoulis and Kribs [13]. Other topics that can be found in these papers, and others, are the determination of unitary automorphisms, the structure of partial isometries, the reﬂexivity and hyper-reﬂexivity of the algebras LG , and determination of the Jacobson radical and semisimplicity.

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S.C. Power The Hilbert space Hn is readily identiﬁable with the Fock space ⊕(Cn )⊗k Hn = C ⊕ k∈Z+

formed by the direct sum of multiple tensor products of Cn . With this formulation the operators Le are conveniently speciﬁed by the shift property Le (ξ1 ⊗ · · · ⊗ ξk ) = ξe ⊗ ξ1 ⊗ · · · ⊗ ξk where ξ1 ⊗ · · · ⊗ ξk is an elementary tensor in the k-fold tensor product summand. In general the generating operators Le are partial isometries acting on a natural generalized Fock space Hilbert space, in which not all tensors are admissible. Although we have not needed the tensor formalism this does provide a fundamental construction allowing for further generalisations, most notably for the tensor algebras of correspondences. See, for example, Muhly and Solel [17]. We remark that there is presently a theory of higher rank versions of such non-selfadjoint operator algebras being developed which is associated with higher rank graphs and with higher rank correspondences (product systems). For more on this see Kribs and Power [16], Power [25], Solel [30] and Power and Solel [26]. The following result from [15] gives a graph theoretic condition corresponding, roughly speaking, to the separation of the algebras LG into two classes, those which are “matrix function like” and those that are “free semigroup like”. The following notion parallels somewhat the requirement that a C∗ -algebra contain O2 , or that a discrete group contain a free group. Deﬁnition 3.6. A wot-closed algebra A is partly free if there is an inclusion map L2 → A which is the restriction of an injection between the generated von Neumann algebras. If the map can be chosen to be unital, then A is said to be unitally partly free. A directed graph G is said to have the double-cycle property if there are distinct minimal cycles w = xwx, w = xw x over some vertex x in G. Theorem 3.7. The following assertions are equivalent for a countable directed graph G with a ﬁnite number of vertices. (i) G has the double-cycle property. (ii) LG is partly free.

4. Subalgebras of On Returning to the themes of Section 2, we are now ready to look inside On . Our context is that of a norm closed subalgebra A ⊆ On which contains the canonical diagonal subalgebra C associated with the given generators of On . Let us ﬁrst note that there are a number of ways such algebras arise. (i) Generator constraints: (a) If S is a semigroup of operators of the form Sµ Sν∗ which contains all the projections Sµ Sµ∗ then the norm closed linear span is

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a subalgebra of On which contains the canonical diagonal subalgebra. Note that this algebra is left invariant by the gauge automorphisms of On . (b) Let A1 be the norm closed algebra generated by the diagonal algebra C and the single operator S1 . Let A2 be the (nonunital) subalgebra which is the ideal in A1 generated by 1 − S1 . The abelain algebra A1 /com(A1 ) can be naturally identiﬁed with the disc algebra A(D), while A2 /com(A2 ) identiﬁes with the ideal of functions h(z) with h(1) = 0. The gauge automorphisms of On rotate the ideals of A(D) and so the algebra A2 is not gauge invariant. (ii) Fourier series constraints: Let A ⊆ F n be a triangular subalgebra with A ∩ A∗ = C. Then A = {a ∈ On : E0 (a) ∈ A, Ek (a) = 0, k < 0} is a triangular subalgebra of On . Once again, A is gauge invariant. (iii) Extrinsic constraints: Let N ⊆ C be a totally ordered family of projections. For example, N could consist of the projections corresponding to the intervals [0, k/2n] in the interval picture of On . To such a nest of projections one can assign the nest subalgebra A = On ∩ AlgN = {a ∈ On : (1 − p)ap = 0, for all p ∈ N }. Once again, masa normalising partial isometries give a key tool for recovering the underlying “coordinates” from the algebra structure. Each partial isometry Sµ Sω∗ is C-normalising, as are ﬁnite sums of these when they have orthogonal ranges and orthogonal domains. Also we may multiply these sums by unitary elements of C to obtain further examples. These turn out to be all the normalising partial isometries and they may be characterised in intrinsic terms as in the following theorem. In terms of the interval picture, or the Cantor interval picture one can, once again, indicate pictorially the support of such a partial isometry, as shown. 0

1

0

1

Figure 5. The support a normalising partial isometry in O2 .

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Theorem 4.1. Let v be a contraction in On . Then the following assertions are equivalent: (i) v is a C-normalizing partial isometry. (ii) v is an orthogonal sum of a ﬁnite number of partial isometries of the form dSµ Sν∗ , where d ∈ C. (iii) For all projections p, q ∈ C, the norm qvp is equal to 0 or 1. Proof. The implications from (ii) to (i) to (iii) are routine and left to the reader. Let v be a contraction with the zero one norm condition. We claim ﬁrst that E0 (v) is a C-normalizing partial isometry in F n . The intuitive reason for this follows on contemplating the (Cantor space) support supp(v) of v which is deﬁned as the set of points (y, x) in X × X such that for some δ > 0 qvp = 1 for all projections p, q in C with y(q) = x(p) = 1. Considering continuous matrices we can see that the support supp(w) of a ﬁnite sum w of products of the generators and their adjoints will be the union of the sets supp(E0 (w)) and supp(w − E0 (w)), the former set consisting of the diagonal segments parallel to the main diagonal. Because of the essential disjointness of these sets it follows from (iii) that E0 (w) satisﬁes the zero-one condition and so by Theorem 2.10 is a normalising partial isometry. For a rigorous proof we can argue as follows. If E0 (v) were not a normalising partial isometry then by Theorem 2.10 we would be able to ﬁnd δ > 0 and projections p, q in C so that δ ≤ qE0 (v)p ≤ 1 − δ. Moreover, for each N it is possible to choose the projections in such a way so that qwp = 0 for any standard partial isometry w of the form Sµ Sν∗ with |µ| = |ν| and 0 ≤ |µ|, |ν| < N . Choose v in the star algebra generated by the generators with v − v < δ/3. Since v − E0 (v ) is a linear combination of Sµ Sν∗ with |µ| = |ν| we may choose p, q as above so that pvq − pv q = pvq − pE0 (v )q. It follows that pvq is not zero or one, a contradiction. Now suppose that m > 0. If |ν| = m and |λ| − |µ| = m, then the product Sν∗ Sλ Sµ∗ is either zero or of the form Sλ1 Sµ∗ with |λ1 | = |µ|. It follows that if Φm (v) is the mth term in the series expansion of v (Φm (v) = am S1m for m ≥ 0), then Sν∗ Φm (v) = E0 (Sν∗ v). Since v satisﬁes the zero one condition, so does Sν∗ v and the argument above shows that Sν∗ Φm (v) is C-normalizing and so has the desired form. This, in turn, implies that Sν Sν∗ Φm (v) is C-normalizing and has the required form for any word ν of length m. Consequently, Φm (v) has the desired form. In a similar fashion, we can show that when m < 0, Φm (v) also has the desired form (consider adjoints, for example). Finally, if w is a partial isometry and ww∗ xw∗ w = 0 then w + ww∗ xw∗ w > 1. From this observation and the Ces´ aro convergence of generalised Fourier series, it follows that the operators Φm (v) are non-zero for only ﬁnitely many values of m and that v is the orthogonal sum of these operators. Thus v itself has the desired form.

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25

Recall the Cantor interval picture and the partial homeomorphisms αµν . The dilation factor of αµν we deﬁne to be k = |ν| − |µ|. Previously we focused on the case k = 0 appropriate to F n . We now deﬁne the counterpart to R(F n ). The Cuntz groupoid R(On ) is, intuitively speaking, the support of the algebra in the Cantorised interval picture, with record taken of the dilation factors. More formally it is the set R(On ) = {(x, k, y) : x = αµν (y) for some αµν }, together with (i) the totally disconnected topology with (as before) the set of graphs Eµν , for the partial homeomorphisms αµν , as a base, (ii) the natural partially deﬁned multiplication coming from composition of appropriate partial homeomorphisms. It is natural now to seek to obtain for subalgebras of On results analogous to those in Section 2. It turns out that there is a complication in that “synthesis”, as expressed in Theorem 2.7, may fail. However, it is precisely the gauge invariant closed subalgebras containing C that are determined by their groupoid support: Theorem 4.2. Let A be a closed subalgebra of On containing the canonical diagonal masa C. Then A is generated by the partial isometries Sµ Sν∗ belonging to A if and only if A is invariant under the gauge automorphisms γz for |z| = 1. For a gauge invariant algebra as above we deﬁne an associated topological semigroupoid R(A). This is the set R(A) = {(x, k, y) : x = αµν (y) for µ, ν such that Sµ Sν∗ ∈ A} with the relative topology and partially deﬁned multiplication. With the characterisation of normalising partial isometries given above it is now possible to prove the following analogue of Theorem 2.10, and in a similar manner to the proof of that theorem. To paraphrase, gauge invariant triangular subalgebras of On remember their semigroupoids and are classiﬁed by them. Theorem 4.3. Let A1 and A2 be norm-closed subalgebras of On with Ai ∩ A∗i = C for i = 1, 2. Then the following statements are equivalent (i) A1 and A2 are isometrically isomorphic operator algebras. (ii) The semigroupoids R(A1 ), R(A2 ) are isomorphic, that is, there is a homeomorphism α : R(A1 ) → R(A2 ) which respects the partially deﬁned multiplication. Proof. The direction (i) =⇒ (ii) is similar to that of the proof of Theorem 2.10. The direction (ii) =⇒ (ii) is more straightforward. One lifts the map α to a map on the semigroup of normalising partial isometries generated by the Sµ Sν ∗ and this can be extended to an isometric algebra isomorphism. Suppose now that G is a countable directed graph (V, E) with range and source maps r, s : E → V . One can generalise the Cuntz relations as we indicated above in context E of the introduction. To each edge e there is a partial isometry

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S.C. Power

Se and to each vertex x a projection Px . The initial projection of Se is Ps(e) while the range projection is dominated by Pr(e) . Moreover, under a given Py the range projections sum to that projection: Se Se∗ = Py e:r(e)=y

with weak operator topology convergence if the edge incidence is inﬁnite. We see then that it is simply the graph that encodes partial isometry generators and relations. The graph C*-algebra C ∗ (G) is deﬁned to be the associated universal C*algebra. Much is known about the structure of this diverse class. See for example [27]. Once again, words in the generators and their adjoints have a reduced form Sα Sβ∗ where, in the graph case, α = α1 . . . αn is a directed path in G (directed from left to right in our convention) and there are natural counterparts to methods and results in Section 2.1. Moreover, if G has no source vertices, in the sense that r is onto, then the abelian C*-algebra generated by the projections Sα Sα∗ is a masa. In this setting, with the simplifying assumption of ﬁnite incidence at every vertex (so-called row ﬁniteness) one has exact counterparts to all the results of this section. 4.1. Notes Theorems 4.1, 4.2 and 4.3 are taken from Hopenwasser, Peters and Power [11] where one can also ﬁnd the more general variants for graph C*-algebras. The methods related to the AF classiﬁcation, Theorem 2.10, has assisted in the analysis of many particular families of subalgebras of AF C*-algebras and much is known of the structure of ideals and representations, for example. On the other hand subalgebras of graph C*-algebras have not received such attention but it may be timely to do so. In these lectures we have been led from operator algebra considerations to speciﬁc topological groupoids and semigroupoids, for F n and On . It is an important and natural consideration to complete the circle and construct operator algebras associated with general abstract topological groupoids. For this see Renault [28], Paterson [18] and Raeburn [29]. For further perspectives on non-self-adjoint operator algebras see the recent article of Donsig and Pitts [8] who also comment on variants of the open problems in Section 2.

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27

5. Appendix: Digraph algebras and limit algebras We give a brief self-contained account of digraph algebras A(G) and some examples of direct limit algebras. 5.1. Digraph Algebras A digraph is a directed graph G = (V, E) with no multiple directed edges, so that E ⊆ V × V and each edge e in E can be written as (x, y) with initial vertex y and ﬁnal vertex x. If V = {1, . . . , n} then the subspace A(G) ⊆ Mn (C) is deﬁned by A(G) := span{ei,j : (i, j) ∈ E}, where {eij } is the standard matrix unit system for Mn (C). Suppose further that E, when regarded as a binary relation, is reﬂexive. Thus (v, v) ∈ E for all v ∈ V . Then Dn ⊆ A(G), where Dn = span {eii }. Note that A(G) is a complex algebra if and only if G is transitive, that is, if (i, j) and (j, k) are edges then so is (i, k). We say that A(G) is a standard digraph algebra in this case. Examples. (i) If Km is the complete digraph on {1, . . . , m} then A(Km ) = Mm (C). (ii) Let D2m be a 2m-sided polygon with alternating directions on the edges and loops at each vertex. Then A(D2m ) is identiﬁable as a rather sparse algebra of matrices. (iii) Let Tn be the subalgebra of upper triangular matrices in Mn (C). Then Tn is a digraph algebra. (iv) Given the digraph G, we can construct G × Km , the relative product by replacing each vertex of G by Km , and replacing each proper edge of G by all the n2 edges between the new vertices. The digraph algebra A(G × Km ) is identiﬁable with A(G) ⊗ Mm (C) . Deﬁnition 5.1. A ⊆ Mn (C) is a digraph algebra if A is a complex algebra and A contains a maximal abelian self-adjoint algebra (masa) D. Proposition 5.2. If D ⊆ Mn (C) is a masa then there is a unitary matrix u ∈ Mn (C) such that uDu∗ = Dn , the standard masa. Proof. D being a masa means that if D properly contains D and D is also a self-adjoint algebra, then D is not abelian. Let {p1 , . . . , pt } ⊆ D be a maximal set of pairwise orthogonal projections. Then rank pi = 1, for all i, by maximality of D. For if not split the projection into a sum of two projections to obtain a larger abelian algebra. It follows, again by maximality, that t = n and so there is a unitary u such that for all i, upi u∗ = eii as required. Proposition 5.3. If D ⊆ A(G) is a masa then there exist a unitary u ∈ A(G) such that uDu∗ = Dn . Thus, maximal abelian self-adjoint subalgebras of digraph algebras are unique up to inner conjugacy (inner unitary equivalence). Proof. Use the previous proposition, combining unitaries in each block of the block diagonal self-adjoint subalgebra A(G) ∩ A(G)∗ . Since an algebra in Mn (C) which contain the standard masa is a standard digraph algebra we have the following corollary.

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Corollary 5.4. Every digraph algebra A ⊆ Mn (C) is inner conjugate to a standard digraph algebra. That is uAu∗ = A(G) for some digraph G and some unitary u in A. If standard digraph algebras are unitarily equivalent then by Proposition 5.2 we can assume that the unitary equivalence maps the standard masa to the standard masa and it follows readily that the graphs are isomorphic. 5.2. Maps between digraph algebras To begin to understand algebras of the form A = k A(Gk ), where the building block algebras are nested, i.e., A(Gk ) ⊆ A(Gk+1 ), we must consider the nature and m variety of the possible inclusion maps A(Gk ) → A(Gk+1 ). Let (fij )i,j=1 ⊆ Mn (C) be operators with the relations of an m × m matrix unit system, that is, ⎤ fij fjk = fik ∀ ijk ⎦ fij∗ = fji (∗) fij fkl = 0 if j = k Then the map φ : Mm → Mn which is deﬁned to be the linear extension of the correspondences eij → fij , is an injective C ∗ -algebra homomorphism. Conversely, it is straightforward to show that if φ is such a map, then {φ(eij )} satisfy the relations (∗). Deﬁnition 5.5. Let A(G1 ) ⊆ Mm , A(G2 ) ⊆ Mn be unital standard digraph algebras with connected graphs. Then an algebra homomorphism φ : A(G1 ) → A(G2 ) is said to be star-extendible if φ is the restriction of a C ∗ -algebra map between the ﬁnite-dimensional C*-algebras C ∗ (A(G1 )) and C ∗ (A(G2 )). Example (i) The map φ : T2 → T4 given by ⎛ a 0 ⎜ 0 a a b φ =⎜ ⎝ 0 0 0 c 0 0 is a star-extendible algebra injection. Example (ii) The map φ : T2 → T4 given by ⎛ a ⎜ a b φ: →⎜ ⎝ c

√ b/√2 b/ 2 c 0

b c

0 0 a

√ −b/√ 2 b/ 2 0 c

⎞ 0 0 ⎟ ⎟ 0 ⎠ 0

is not star-extendible (and is an isometric algebra injection). Example (iii) φ : T2 → T4 given by ⎡ ⎤ a b ⎢ ⎥ a b a b ⎥ φ: →⎢ ⎣ ⎦ c c c is a star-extendible algebra injection.

⎞ ⎟ ⎟ ⎠

Subalgebras of Graph C*-algebras

29

Remark 1 Star-extendible injective maps between digraph algebras are necessarily isometric: since they are restrictions of C*-algebra maps. This is an elementary fact from spectral theory. Indeed, note ﬁrst that if p ∈ Mm is a nonzero selfadjoint projection then ||p|| = 1, by the deﬁnition of the operator norm. If φ is a star homomorphism then φ(p) is a projection, so φ(p) = 1 = p. We claim that φ(a) = a if a is a self-adjoint operator in Mm . By the spectral theorem a = λ1 p1 + · · · + λm pm , and we can suppose a = |λ1 | ≥ |λk | for k = 2, . . . , m, where p1 , . . . , pm are pairwise orthogonal projections. Let qi = φ(pi ) 1 ≤ i ≤ m. Then φ(a) = λ1 q1 + · · · + λm qm , with q1 . . . qm pairwise orthogonal projections. Thus (exercise) φ(a) = |λ1 | = a. Finally if b ∈ Mm is a general element, then φ(b)2 = φ(b)∗ φ(b) = φ(b∗ )φ(b) = φ(b∗ b) = b∗ b = b2 . Deﬁnition 5.6. Let A(G1 ), A(G2 ) be digraph algebras with standard matrix unit systems {ekij : (ij) ∈ E(Gk )}, k = 1, 2, as usual. (i) An algebra injection φ : A(G1 ) → A(G2 ) is a standard regular injection, with respect to {e1ij }, and {e2ij }, if φ is star-extendible and maps each e1ij to a sum of matrix units in {e2kl } (ii) An algebra injection ψ : A(G1 ) → A(G2 ) is a regular (star-extendible) injection if there exists a unitary operator u in A(G2 ) such that ψ(a) = uφ(a)u∗ ∀a ∈ A(G1 ), where φ is a standard regular injection. We say that ψ is inner equivalent (or, less precisely, unitarily equivalent) to φ when such a relationship holds. Remark. The unitary u belongs to A2 ∩ A∗2 . Indeed, by the spectral theorem, u = λ1 p1 + · · · + λr pr where pi is the spectral projection for the eigenspace for λi and each pi lies in the self-adjoint subalgebra. Exercises. (i) Prove that there are uncountably many inner conjugacy classes of embedding φ : T2 → T4 which are star-extendible and unital. (ii) Prove that there are only ﬁnitely many inner conjugacy classes of regular embeddings between two digraph algebras. 5.3. Direct limits We leave it to the reader to recall the deﬁnition of the direct limit algebra of a direct system and we now give some standard direct systems of triangular matrix algebras. The examples are all determined by regular star extendible inclusion maps. Limits of ﬁnite-dimensional operator algebras with respect to isometric irregular embeddings are less well understood and have not received much attention. The interested reader can ﬁnd some classiﬁcations in this direction in [24]. Standard limits. Let n2 = rn1 and let σ : Mn1 → Mn2 be the inclusion map such that (1)

(2)

(2)

(2)

σ(ei,j ) = ei,j + ei+n1 ,j+n1 + · · · + ei+(r−1)n1 ,j+(r−1)n1

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or equivalently, identifying Mn2 with Mr ⊗ Mn1 , σ(a) = I ⊗ a, so that, in block matrix terms, ⎡ ⎤ a 0 ... 0 ⎢ 0 a ⎥ ⎥. σ(a) = ⎢ ⎣ ⎦ 0 a Then σ(Tn1 ) ⊆ Tn2 , and so, repeating, we may construct the regular matricial algebra Aσ = lim(Tnk , σ), when nk |nk+1 for all k. Reﬁnement limits. Let n2 = rn1 and let ρ : Mn1 → Mn2 be the inclusion map such that (1)

(2)

(2)

(2)

ρ(ei,j ) = e(i−1)n1 +1,(j−1)n1 +1 + e(i−1)n1 +2,(j−1)n1 +2 + · · · + e(i−1)n1 +r,(j−1)n1 +r or equivalently, identifying Mnk with Mn1 ⊗Mr , ρ(a) = a⊗I, so that, ρ((aij )) is the inﬂated matrix (aij Ir ). Again, ρ(Tn1 ) ⊆ Tn2 , and for a sequence with nk |nk+1 for all k we can deﬁne the regular matricial algebra Aρ = lim(Tnk , ρ). This algebra is not isometrically isomorphic to the Aσ algebra for the same sequence (nk ) despite the fact that their generated C*-algebras are isomorphic. Countable total order limits. The standard embeddings σ and the reﬁnement embeddings ρ can also be alternated in which case one obtains a more general class of algebras, the so called alternation algebras. The three classes described correspond to a Cantor space product coordinate indexing by Z− , Z+ , and Z respectively. One can generalise this further to obtain strange triangular algebras whose background Cantor product is indexed over an arbitrary countable order. Moreover this countable order is an algebra isomorphism invariant. For further detail see [23].

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References [1] J. Cuntz, Simple C*-algebras generated by isometries, Comm. Math. Phys. 57 (1977), 173–185. [2] K.R. Davidson, Free Semigroup Algebras: a survey. Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), 209–240, Oper. Theory Adv. Appl. 129, Birkh¨ auser, Basel, 2001. [3] K.R. Davidson, C*-algebras by Example, Fields Institute Monograph Series, vol. 6, American Mathematical Society, 1996. [4] K.R. Davidson, E. Katsoulis, Nest representations of directed graph algebras, Proc. London Math. Soc., to appear. [5] K.R. Davidson, D.R. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras, Math. Ann. 311 (1998), 275–303. [6] K.R. Davidson, D.R. Pitts, Invariant subspaces and hyper-reﬂexivity for free semigroup algebras, Proc. London Math. Soc., 78 (1999), 401–430. [7] K.R. Davidson, E. Katsoulis and J. Peters, Meet-irreducible ideals and representations of limit algebras. J. Funct. Anal. 200 (2003), no. 1, 23–30. [8] A.P. Donsig and D.R. Pitts Coordinate Systems and Bounded Isomorphisms for Triangular Algebras. math.OA/0506627 66 pages. [9] A.P. Donsig and S.C. Power, The failure of approximate inner conjugacy for standard diagonals in regular limit algebras. Canad. Math. Bull. 39 (1996), no. 4, 420–428. [10] P.A. Haworth and S.C. Power, The uniqueness of AF diagonals in regular limit algebras. J. Funct. Anal. 195 (2002), no. 2, 207–229. [11] A. Hopenwasser, J. Peters and S.C. Power, Subalgebras of Graph C*-Algebras, New York J. Math. 11 (2005), 1–36. [12] A. Hopenwasser and S.C. Power,Limits of ﬁnite dimensional nest algebras, Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 1, 77–108. [13] E. Katsoulis, D.W. Kribs, Isomorphisms of algebras associated with directed graphs, Math. Ann., 330 (2004), 709–728. [14] D.W. Kribs, S.C. Power, Free semigroupoid algebras, J. Ramanujan Math. Soc. 19 (2004), 75–117. [15] D.W. Kribs, S.C. Power, Partly free algebras, Operator Theory: Advances and Applications, Birkh¨ auser-Verlag Basel/Switzerland, 149 (2004), 381–393. [16] D.W. Kribs and S.C. Power, The H ∞ algebras of higher rank graphs, Math. Proc. of the Royal Irish Acad., 106 (2006), 199–218. [17] P. Muhly, B. Solel, Tensor algebras, induced representations, and the Wold decomposition, Can. J. Math. 51 (4), 1999, 850–880. [18] A. L. Paterson, Groupoids, inverse semigroups and their operator algebras, Progress in Mathematics, Vol. 170, Birkh¨ auser, Boston, 1999. [19] G. Popescu, Multi-analytic operators on Fock spaces, Math. Ann. 303 (1995), 31–46. [20] G. Popescu, Noncommuting disc algebras and their representations, Proc. Amer. Math. Soc. 124 (1996), 2137–2148.

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[21] S.C. Power, Limit algebras: an introduction to subalgebras of C ∗ -algebras, Pitman Research Notes in Mathematics Series, vol. 278, (Longman Scientiﬁc & Technical, Harlow, 1992) CRC Press ISBN: 0582087813. [22] S.C. Power, Lexicographic semigroupoids, Ergodic Theory Dynam. Systems 16 (1996), no. 2, 365–377. [23] S.C. Power, Inﬁnite lexicographic products of triangular algebras, Bull. London Math. Soc. 27 (1995), no. 3, 273–277. [24] S.C. Power, Approximately ﬁnitely acting operator algebras, J. Funct. Anal. 189 (2002), no. 2, 409–468. [25] S.C. Power, Classifying higher rank analytic Toeplitz algebras, preprint 2006, preprint Archive no., math.OA/0604630. [26] Operator algebras associated with unitary commutation relations, preprint March 2007. [27] I. Raeburn, Graph algebras C.B.M.S lecture notes, vol. 103, Amer. Math. Soc., 2006. [28] J. Renault, A groupoid approach to C ∗ -algebras, Springer, Berlin, 1980. [29] B. Solel, You can see the arrows in a quiver algebra, J. Australian Math. Soc., 77 (2004), 111–122. [30] B. Solel, Representations of product systems over semigroups and dilations of commuting CP maps, J. Funct. Anal.235 (2006), 593–618. Stephen C. Power Department of Mathematics and Statistics Lancaster University England LA1 4YF e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 33–66 c 2008 Birkh¨ auser Verlag Basel/Switzerland

C ∗ -algebras and Asymptotic Spectral Theory Bernd Silbermann Abstract. The presented material is a slightly polished and extended version of lectures given at Lisbon, WOAT 2006. Three basic topics of numerical functional analysis are discussed: stability, fractality, and Fredholmness. It is further shown that these notions are corner stones in order to understand a few topics in asymptotic spectral theory: asymptotic behavior of singular values, ε-pseudospectra, norms. Four important examples are discussed: Finite sections of quasidiagonal operators, Toeplitz operators, band-dominated operators with almost periodic coeﬃcients, and general band-dominated operators. The elementary theory of C ∗ -algebras serves as the natural background of these topics. Mathematics Subject Classiﬁcation (2000). 47B35. Keywords. C∗-algebras, operator sequences, asymptotics, ﬁnite sections.

1. Introduction One goal of functional analysis is to solve equations with “inﬁnitely” many variables, and that of linear algebra to solve equations in ﬁnitely many variables. Numerical analysis builds a bridge between these ﬁelds. Functional numerical analysis is concerned with the theoretical foundation of numerical analysis. Given a bounded linear operator A acting on some Hilbert space H, that is A ∈ B(H), consider the equation Ag = h ,

(1.1)

where h ∈ H is given and g is to ﬁnd if this equation is supposed to be uniquely solvable. Even if the operator A is continuously invertible (and this will be assumed in what follows), it is as a rule impossible to compute the solution A−1 h. Then one tries to solve (1) approximately. For, one chooses a sequence (hn ) ⊂ H of elements which approximates the right-hand side h, and a sequence (An ) of operators

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which approximates the operator A, and one replaces (1.1) by the approximation equations An gn = hn , n = 1, 2, . . . (1.2) the solutions gn of which are sought in H (or in certain subspaces Hn of H) again. Approximation of h by hn means that h − hn H → 0 as n → ∞. It is tempting to suppose that the operators An also approximate A in the norm, but this assumption does not work in practice. The point is that usually A acts on an inﬁnite-dimensional space, whereas one will, of course, try to choose the An as acting on spaces of ﬁnite dimension, i.e., as ﬁnite matrices. But the only operators which can be approximated in norm by ﬁnite rank operators, are the compact ones. The kind of approximation which ﬁts much better to the purpose of numerical analysis is that of pointwise or strong convergence: the sequence (An ) converges strongly to the operator A if Ah − An hH → 0 for every h ∈ H (notation: s-lim An = A). We write s∗ -lim An = A, if s-lim An = A and s-lim A∗n = A∗ . Basic question: suppose that An is invertible for all n ≥ n0 . Does the sequence (gn ) of solutions of (1.2) converge to the solution g of (1.1)? The answer is NO! Let l2 := {(x0 , . . . , xn , . . . ) :

|xk |2 < ∞},

k∈Z+

˜l2 := {(. . . , x−n , . . . , x0 , . . . , xn , . . . ) :

|xk |2 < ∞}.

k∈Z

Example 1. Let ε = (εn ) be a sequence with εn > 0 and lim εn = 0 and Aε an inﬁnite matrix given by ⎞ ⎛ ε0 1 ⎟ ⎜ 1 ε0 0 ⎟ ⎜ ⎟ ⎜ 0 ε 1 1 ⎟ ⎜ ⎟ ⎜ 1 ε 0 1 ⎟ ⎜ (with respect to the standard basis of l2 ). ⎟ ⎜ 1 0 ε 2 ⎟ ⎜ ⎜ .. ⎟ ⎜ . ⎟ 1 ε 2 ⎠ ⎝ .. .. . . Consider Pn Aε Pn , Pn : l2 → l2 , (x0 , . . . , xn , xn+1 , . . . ) → (x0 , . . . , xn , 0, 0, . . . ). Then: ≤2 if n is odd (Pn Aε Pn )−1 Pn = ≥ ε−1 if n is even. n Thus, if n is even, then (Pn Aε Pn )−1 Pn → ∞, that is there is an h ∈ H such that gn = (Pn Aε Pn )−1 Pn h A−1 h = g by the Steinhaus-Banach Theorem.

C ∗ -algebras and Asymptotic Spectral Theory

35

Let us turn back to the general situation: Suppose that (Pn ) is a sequence of orthogonal projections such that s-lim Pn = I, and (An ) a bounded sequence of operators An : imPn → imPn with the property that there is an n0 such that An is invertible for n ≥ n0 and sup A−1 n Pn < ∞. If s-lim An Pn = A, then n≥n0

−1 A−1 x → 0 for every x ∈ H : n Pn x − A −1 −1 A−1 x ≤ A−1 x + Pn A−1 x − A−1 x n Pn x − A n Pn x − Pn A −1 x + Pn A−1 x − A−1 x → 0 . ≤ A−1 n Pn x − An Pn A ∗ Remark. Suppose again that sup A−1 n Pn < ∞ and s -lim An Pn = A. Then A n≥n0

is invertible. Indeed, we have An Pn x ≥ CPn x

(n ≥ n0 , C > 0) .

Passing to limits gives Ax ≥ Cx . Thus im A = im A and ker A = {0}. Using A∗n Pn x ≥ CPn x we get in the same manner A∗ x ≥ Cx . Thus A∗ = im A∗ and ker A∗ = {0}. Deﬁnition 1. A sequence of operators An ∈ B(imPn ) is called stable if there exists a number n0 such that the operators An are invertible for every n ≥ n0 and if the norms of their inverses are uniformly bounded: sup A−1 n Pn < ∞ .

n≥n0

The above discussion shows the crucial role of stability in analysis. How to prove stability? There is no general idea. In most cases it is very complicated. Easy cases: • A = B + iS, B positive and S selfadjoint, • A = I + T , T compact, and An = Pn APn , where (Pn ) is a sequence of orthogonal projections with slim Pn = I. Exercise: prove that in both cases the sequence (An ) is stable.

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We will show that the stability problem can frequently be tackled by the help of C ∗ -algebra techniques. Recall that a complex Banach algebra is called C ∗ algebra if there is an involution a → a∗ such that aa∗ = a2 . Given two C ∗ algebras A and B, a ∗-homomorphism ϕ : A → B is a continuous homomorphism such that ϕ(a∗ ) = ϕ(a)∗ for all a ∈ A.

2. Algebraization of stability Let H be a (separable) Hilbert space and (Ln ) be a sequence of orthoprojections on H with s-lim Ln = I. Deﬁnition 2. Let F be the set of all sequences (An )∞ n=0 of operators An ∈ B (im Ln ) which are uniformly bounded: sup An Ln < ∞ . n≥0

The natural operations (An ) + (Bn ) := (An + Bn ), (An )(Bn ) := (An Bn ), λ(An ) := (λAn ) , (An )∗ := (A∗n ) make F to an algebra with involution. Proposition 1. F is a C ∗ -algebra (prove it). We are mainly interested in the asymptotic behavior of the sequences belonging to F . This means that sequences which diﬀer in a ﬁnite number of entries only will have the same asymptotic behavior, and therefore can be identiﬁed. For this goal we introduce the set G of all sequences (Gn ) in F with lim Gn Ln = 0. n→∞

Proposition 2. G is a closed ideal in F (prove it). The following theorem reveals a perfect frame to study stability problems in an algebraic way. Theorem 1. (A. Kozak) A sequence (An ) ∈ F is stable ⇔ the coset (An ) + G is invertible in the quotient algebra F /G. Proof. ⇒: If (An ) is stable, then (A−1 n )n≥n0 is bounded for some suﬃciently large n0 by deﬁnition. We make (A−1 ) n n≥n0 to a bounded sequence −1 , A (B0 , B1 , . . . , Bn0 −1 , A−1 n0 n0 +1 , . . . ) in F by freely choosing operators Bi ∈ B(imLi ). It is evident that this sequence is an inverse of (An ) modulo G. ⇐ Let conversely, (An )+ G be invertible in F /G. Then there are sequences (Bn ) ∈ F as well as (Gn ) and (Hn ) in G such that An Bn = Ln +Gn , Bn An = Ln +Hn . If n is large enough, then Gn < 12 , Hn < 12 , and a Neumann series argument yields the invertibility of Ln + Gn and Ln + Hn as well as the uniform boundedness of their inverses by 2. Hence, Bn (Ln +Gn )−1 , (Ln +Hn )−1 Bn are uniformly bounded. Thus, the operators An are invertible for all suﬃciently large n, and their inverses are uniformly bounded.

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37

Proposition 3. For all (An ) ∈ F, (An + G)F /G = lim sup An Ln n→∞

(2.1)

(where lim sup stands for lim superior).

Proof. Exercise.

Formula (2.1) gives raise to ask if there are interesting sequences in F for which lim sup in (2.1) can be replaced by lim. This question is important in order to prove that the condition numbers of a stable sequence converge. Recall, that the condition number (cond A) for an invertible matrix (operator) A is deﬁned by cond A := A A−1 (for computational purposes: cond A should be small). The right tool to study this and related questions is another fundamental notion of numerical analysis – that of a fractal sequence, which we are now going to discuss. It is not important in this place that the elements of the sequences under consideration are operators. So we will use slightly generalized deﬁnitions of the C ∗ -algebras F and G, namely, given unital C ∗ -algebras Cn , n = 0, 1, 2, . . . , with identity elements en , let F stand for the set of all bounded sequences (c0 , c1 , . . . ) with cn ∈ Cn , and let G refer to the set of all sequences (c0 , c1 , . . . ) in F with cn → 0 as n → ∞. Deﬁning elementwise algebraic operations and an elementwise involution, and taking the supremum norm, we make F to a C ∗ -algebra and G to a closed ideal of F . Thus, F is the product of the C ∗ -algebras Cn , and G-their restricted product. Given a strongly monotonically increasing sequence η : Z+ → Z+ , let Fη and Gη denote the product and the restricted product of the C ∗ -algebras Cη(0) , Cη(1) , . . . , respectively, and let Rη stand for the restriction mapping Rη : F → Fη , (an ) → (aη(n) ). The mapping Rη is a ∗-homomorphism from F onto Fη . Further, given a C ∗ -subalgebra A of F , let Aη refer to the image of A under Rη . By the ﬁrst isomorphy theorem for C ∗ -algebras ([7], Theorem 1.45), Aη actually is a C ∗ -algebra. Deﬁnition 3. Let A be a C ∗ -subalgebra of F . (a) A ∗-homomorphism W : A → B of A into a C ∗ -algebra B is fractal if for every strongly monotonically increasing sequence η, there is a ∗-homomorphism Wη : Aη → B such that W = Wη Rη . (b) The algebra A is fractal, if the canonical homomorphism π : A → A/A ∩ G is fractal. (c) A sequence (an ) ∈ F is fractal, if the smallest C ∗ -subalgebra of F containing (an ), is fractal. Roughly spoken: given a subsequence (aη(n) ) of a sequence (an ) which belongs to a fractal algebra A, it is possible to reconstruct the original sequence (an ) from its subsequence modulo sequences in A ∩ G.

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Consequences: • (aη(n) ) ∈ Gη ⇒ (an ) ∈ G ([7], Theorem 1.66),

(2.2)

• (aη(n) ) stable ⇒ (an ) stable

(2.3)

(see Theorem 4 below). ∗

Theorem 2. ([7], Theorem 1.7.1) Let A be a fractal C -subalgebra of F . If (an ) ∈ A, then the limit lim an exists and is equal to (an ) + G. |an |2 < ∞} and the bounded Example 2. Consider again l2 := {(an )n∈Z+ : n∈Z+

linear operators Pn , Rn : l2 → l2 given by (ak ) → (a0 , a1 , . . . , an , 0, 0, 0, . . . ) , (ak ) → (an , an−1 , . . . a0 , 0, 0, 0, · · · ) , respectively. Let Cn = B(imPn ), and let F W refer to the set of all sequences (An ) ∈ F ˜ (An ) := s-lim Rn An Rn for which the strong limits W (An ) := s-lim An Pn and W ∗ ∗ ˜ as well as the strong limits W (An ) and W (An ) exist. The set F W actually forms a C ∗ -subalgebra of F (prove it or compare the proof of Theorem 1.18 (a) in [7]). ˜ : F W → B(l2 ) turn out to be fractal: given a strongly The ∗-homomorphism W, W ˜ η : Fη → B(l2 ) via monotonically increasing sequence η, we can deﬁne Wη , W Wη (Aη(n) ) := s- lim Aη(n) Pη(n) and ˜ η (Aη(n) ) := s- lim Rη(n) Aη(n) Rη(n) . W ˜ =W ˜ η Rη . The algebra F W is not fractal: consider Then, obviously, W = Wη Rη , W the sequence (An ) ∈ F, where A2n+1 = 0 and A2n = diag (0, . . . , 0, 1, 0, . . . , 0), where the 1 stands in the center of this diagonal matrix. It is easily seen, that ˜ (An ) = 0, but (An ) ∈ (An ) ∈ F W , W (An ) = 0, W / G(⊂ F W ). For the special choice η(n) = 2n + 1 one obtains Rη (An ) = (A2n+1 ) ∈ Gη . By (2.2) F W cannot be fractal. One the other hand, F W contains interesting fractal subalgebras as we will see later on.

3. Asymptotic behavior Given a sequence (An ) ∈ F one can ask how the spectra (ε-pseudospectra) of the entries develop. We need some deﬁnitions. Let (Mn )∞ n=1 be a set sequence with values in the set of all subsets of the complex plane. For instance, if (An ) ∈ F, then the mapping n → sp An is a set sequence in this sense. Deﬁnition 4. Let (Mn )∞ n=1 be a set sequence. The partial limiting set or limes superior lim sup Mn (resp. the uniform limiting set or limes inferior lim inf Mn ) of the sequence (Mn ) consists of all points m ∈ C which are a partial limit (resp. limit) of a sequence (mn ) of points mn ∈ Mn (partial limit of a sequence (mn ) is by deﬁnition a limit of some subsequence of (mn )).

C ∗ -algebras and Asymptotic Spectral Theory

39

Observe that the partial limiting set lim sup Mn is non-empty if inﬁnitely many of the Mn are non-empty and if Mn is bounded, whereas the uniform n

limiting set can be empty even under these restrictions as the trivial example Mn = {(−1)n } shows. Let CC denote the set of all non-empty and compact subsets of C. The Hausdorﬀ distance of two elements A and B of CC is deﬁned by h(A, B) := max max dist (a, B), max dist (b, A) , a∈A

b∈B

where dist (a, B) = min |a − b|. The function h is actually a metric on CC . We b∈B

denote limits with respect to this metric by h-lim. Proposition 4. ([7], Proposition 3.6) Let (Mn ) be a set sequence taking values in CC . Then lim sup Mn and lim inf Mn coincide if and only if the sequence (Mn ) is h-convergent. In that case lim sup Mn = lim inf Mn = h- lim Mn . Example 3. Let V : l2 → l2 be the shift operator acting by (a0 , a1 , a2 , . . . ) → (0, a0 , a1 , a2 , . . . ) and consider (Pn V Pn ). It is easy to see that the matrix representation of Pn V Pn with respect to the standard basis of im Pn equals ⎞ ⎛ 0 ⎟ ⎜ 1 0 ⎟ ⎜ ⎟ ⎜ 1 0 0 ⎟. ⎜ ⎟ ⎜ 0 1 0 ⎟ ⎜ ⎠ ⎝ 1 0 1 0 Hence, sp Pn V Pn = {0} for all n and lim inf sp Pn V Pn = lim sup sp Pn V Pn = {0}, but sp V = {z ∈ C : z ≤ 1} ⊂ spF /G ((Pn V Pn ) + G). Therefore h-lim Pn V Pn exists but this limits has almost nothing to do with sp V . What is the reason for this unpleasant fact? One can prove that for (an ) ∈ F a point s ∈ C belongs to the partial limiting set lim sup sp an if and only if the sequence (an −sen ) is not spectrally stable (Theorem 3.17 in [7]). (A sequence (an ) is spectrally stable if its entries an are invertible for suﬃciently large n and if the spectral radii ρ(a−1 n ) of their inverses are uniformly bounded.) Spectral stability is a very involved notion and not much is known. We accomplish this discussion with Theorem 3. ([7], Theorem 3.19) Let Cn = Cn×n and (An ) ∈ F. Then lim sup sp (An + Cn ) = spF /G ((An ) + G). (Cn )∈G

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One conclusion: Spectral stability is very sensitive with respect to perturbations from G (contrary to stability). These diﬃculties disappear if we restrict our attention to sequences for which stability and spectral stability coincide. Corollary 1. ([7], Corollary 3.18) If (an ) ∈ F is a sequence of normal elements, then lim sup sp an = spF /G ((an ) + G). For fractal algebras we get reﬁnements. Theorem 4. ([7], Theorem 3.20) Let A be a fractal C ∗ -subalgebra of F which contains the identity. (a) A sequence (an ) ∈ A is stable if and only if it possesses a stable (inﬁnite) subsequence. (b) If (an ) ∈ A is normal, then lim sup sp an = lim inf sp an = h-lim sp an . (c) If (an ) ∈ A is normal, then the limit lim ρ(an ) exists and is equal to ρ((an ) + G) (ρ-spectral radius). Let us shortly discuss limiting sets of singular values (because of their importance in numerical analysis). Let B be a unital C ∗ -algebra and a ∈ B. The set (a) of the singular values of a is deﬁned to be {λ ∈ R+ : λ2 ∈ sp (a∗ a)}. Since the determination of the singular values is equivalent to the determination of the spectrum of a self-adjoint element, the previous results have the following evident analogues for singular value sets. Theorem 5. If (an ) ∈ F, then lim sup (an ) = ((an ) + G). Theorem 6. If A is a fractal C ∗ -subalgebra of F containing the identity and if (an ) ∈ A, then lim sup (an ) = lim inf (an ) = h- lim (an ) . The last topic in this section is ε-pseudospectra. A computer working with ﬁnite accuracy cannot distinguish between a noninvertible matrix and an invertible matrix the inverse of which has a very large norm. This suggests the following deﬁnition reﬂecting ﬁnite accuracy. Deﬁnition 5. Let B be a C ∗ -algebra with identity e and let ε be a positive constant. An element a ∈ B is ε-invertible if it is invertible and a−1 < 1ε . The ε-pseudospectrum spε (a) of a consists of all λ ∈ C for which a − λe is not εinvertible. It is easily seen that ε-invertible elements of a C ∗ -algebra form an open set, and that ε-pseudospectra are compact and non-empty subsets of C. The following theorem provides an equivalent description of the ε-pseudospectrum which oﬀers a way for numerical computations at least for (ﬁnite) matrices.

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41

Theorem 7. ([7], Theorem 3.27) Let B be a unital C ∗ -algebra and ε > 0. Then, for every a ∈ B, the ε-pseudospectrum is equal to spε (a) = sp (a + p). p∈B p≤ε

Let us still remark that (unital) C ∗ -algebras are also inverse closed with respect to ε-invertibility. What about limiting sets of ε-pseudospectra? Theorem 8. ([7], Theorem 3.31) Let (an ) ∈ F and ε > 0. Then /G ((an ) + G). lim sup spCε n (an ) = spF ε

The proof of Theorem 8 is based on the following result. Proposition 5. (Daniluk) ([3], Theorem 3.14) Let B be a C ∗ -algebra with identity e, let a ∈ B, and suppose a − λe is invertible for all λ in some open subset U of the complex plane. If (a − λe)−1 ≤ C for all λ ∈ U, then (a − λe)−1 < C for all λ ∈ U. In other words: the analytic function U → B, λ → (a − λe)−1 satisﬁes the maximum principle. This is a surprising fact since – in contrast to complex-valued analytic functions – the maximum principlefails in general for operator-valued λ 0 analytic functions (consider C → C2×2 , λ → ). 0 1 It is an open question for which Banach algebras Daniluk’s result is true (one particular answer is in [3], Theorem 7.15). In case A is a fractal C ∗ -subalgebra of F we have the following reﬁnement of Theorem 8: /G ((an ) + G) . h- lim spCε n (an ) = spF ε

4. First applications I. Quasidiagonal operators and their ﬁnite sections Recall that a bounded linear operator T on a separable (complex) Hilbert space is said to be quasidiagonal if there exists a sequence (Pn )n∈N of ﬁnite rank orthogonal projections such that s-lim Pn = I and which asymptotically commute with T , that is [T, Pn ] := T Pn − Pn T → 0 as n → ∞ . In particular, every selfadjoint or even normal operator is quasidiagonal as well as their perturbations by compact operators. However it is by no means trivial to single out a related sequence (Pn ). For instance, for multiplication operators in periodic Sobolev spaces H λ related sequences can explicitly be given: these are orthogonal projections on some spline spaces. Let T be quasidiagonal with respect to (Pn ) = (Pn )n∈N . Consider the C ∗-subalgebra F l of F , the last one deﬁned by help of (Pn ), consisting of all sequences of

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F for which s∗ - lim An Pn exist. It is not hard to prove that J := {(Pn KPn )+(Cn ): K-compact, (Cn ) ∈ G} forms a two-sided closed ideal in F l (but not in F !) Let C(Pn ) (T ) denote the smallest C ∗ -subalgebra of F l containing the sequences (Pn T Pn ), (Pn ), and the ideal J. Proposition 6. The quotient algebra C(Pn ) (T )/G is isometrically isomorphic to the smallest C ∗ -subalgebra C(T ) of B(H) containing T, I, and all compact operators. This isomorphism is given by the quotient map induced via s-lim An Pn ((An ) ∈ C(Pn ) (T )). Sketch of the proof. Suppose s-lim An Pn =: A is invertible. Then A−1 ∈ C(T ), and since every element in C(T ) is quasidiagonal, A−1 also owns this property, and Pn − Pn APn A−1 Pn = Pn AA−1 Pn − Pn APn A−1 Pn = Pn (Pn A − APn )A−1 Pn → 0 . Hence, (Pn APn ) is stable. This means that (Pn APn ) + G is invertible if and only if s-lim Pn APn is invertible. Now it is suﬃcient to prove that Pn APn − An ∈ G. For, it is suﬃcient to show this for the special case An = Pn B1 Pn B2 Pn . We have Pn B1 B2 Pn − Pn B1 Pn B2 Pn = Pn (Pn B1 − B1 Pn )B2 Pn → 0 as n → ∞. Corollary 2. A sequence (An ) ∈ C(Pn ) (T ) is stable if and only if s-lim An is invertible. Moreover, C(Pn ) (T ) is fractal. Now it is evident that the theory of Section 3 applies. Proposition 7. Let (An ) ∈ C(Pn ) (T ) and A = s-lim An . Then (a) lim An = s-lim An Pn . (b) lim inf spε An = lim sup spε An = spε (s-lim An ) (ε > 0). (c) If (An ) is normal, then (d) lim inf

lim inf sp An = lim sup sp An = sp (s- lim An ) . (An ) = lim sup (An ) = (s-lim An ).

In the papers [5], [6] Nathaniel Brown proposed further reﬁnements into two directions: speed of convergence and how to choose the sequence (Pn ) of orthoprojections in some special cases such as quasidiagonal unilateral band operators, bilateral band operators or operators in irrational rotation algebras. Remark. If (an ) ∈ F l is stable and s∗ -lim an = A , A+K invertible and K compact, then (an + Pn KPn ) is stable (this sequence equals (an )(Pn + Pn a−1 n Pn KPn ) for n large enough). II. Toeplitz operators and their ﬁnite sections Let a ∈ L∞ (T) and denote by ak the kth Fourier coeﬃcient of a: ak =

1 2π

2π 0

a(eiθ )e−ikθ dθ , k ∈ Z ,

where

T := {z ∈ C : |z| = 1}.

C ∗ -algebras and Asymptotic Spectral Theory

43

Then the Laurent operator L(a) on ˜ l2 , the Toeplitz operator T (a) on l2 , and 2 the Hankel operator H(a) on l are given via their matrix representation with respect to the standard bases of ˜l2 and l2 by L(a) = T (a) =

(ak−j )∞ k,j=−∞ , ∞ (ak−j )∞ k,j=0 , H(a) = (aj+k+1 )k,j=0 .

Here is a list of elementary properties of these operators (see any textbook on Toeplitz operators). • If a ∈ L∞ (T), then the Laurent operator L(a) is bounded on ˜l2 . • (Brown/Halmos) If a ∈ L∞ (T), then the Toeplitz operator T (a) is bounded on l2 , and T (a) = a∞ . • (Nehari) If a ∈ L∞ (T), then H(a) is bounded on l2 , and ∞

H(a) = distL∞ (T) (a, H ). • T (ab) = T (a)T (b) + H(a)H(˜b), where ˜b(t) := b( 1t ). • T (a)∗ = T (a). • (Coburn) Let a ∈ L∞ (T) \ {0}. Then at least one of the spaces ker T (a) and l2 /imT (a) consists of the zero element only. Proposition 8. ([3], Chapter 1) (i) Let a ∈ C(T). The Toeplitz operator T (a) is Fredholm on l2 if and only if 0 ∈ / a(T). In this case, ind T (a) = −wind a, where wind a refers to the winding number of the curve a(T), provided with the orientation inherited by the usual counter-clockwise orientation of the unit circle, around the origin. / a(T) (ii) Let a ∈ C(T). The Toeplitz operator is invertible on l2 if and only if 0 ∈ and wind a = 0. (iii) Let a ∈ C(T). Then H(a) is compact on l2 . (iv) The smallest C ∗ -subalgebra T (C) of B(l2 ) containing all Toeplitz operators with continuous generating functions, decomposes as 2 ˙ T (C) = {T (a) : a ∈ C(T)}+K(l ),

where K(l2 ) stands for the (closed) ideal of all compact operators. Now let us turn to the ﬁnite section method for Toeplitz operators (with continuous generating function). The ﬁrst question is about the stability of the sequence (Pn T (a)Pn ), where Pn : l2 → l2 is the projection deﬁned by (a0 , a1 , . . . , an , an+1 , . . . ) → (a0 , . . . , an , 0, 0, . . . ) . This problem was investigated by many people. G. Baxter, 63 : (Pn T (a)Pn ) stable in l1 if and only if T (a) is invertible (a ∈ W, 0 ∈ / a(T), wind a = 0, where W stands for the Wiener algebra). I. Gohberg, I. Feldmann, 65 : (Pn T (a)Pn ) stable in l2 if and only if T (a) is invertible.

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Later on related results for classes of discontinuous generating functions were achieved: QC, C +H ∞ , P C, P QC. For instance QC stands for the class of all quasi continuous functions, P QC for the smallest closed subalgebra of L∞ (T) containing the algebra P C of all piecewise continuous functions and QC. Treil, 87 : There are generating functions a with only one point of discontinuity such that T (a) is invertible but (Pn T (a)Pn ) is not stable. Recall the deﬁnition of the algebra F W (Example 2): A sequence (An ) ∈ F belongs to F W , if and only if the strong limits W (An ) := s˜ (An ) = s-lim Rn An Rn as well as W (A∗n ) , W ˜ (A∗n ) exist. Because of lim An Pn , W ∞ Rn T (a)Rn = Pn T (˜ a)Pn (a ∈ L (T)) it is easy to see that (Pn T (a)Pn ) ∈ F W . Moreover, Rn KRn tends strongly to zero for every compact operator due to the weak convergence of (Rn ) to zero. Hence, the smallest C ∗ -subalgebra S(C) in F containing all sequences (Pn T (a)Pn ), a ∈ C(T), is actually contained in F W . Our next goal is to describe the structure of the algebra S(C). For, we need Widom’s identity Pn T (ab)Pn = Pn T (a)Pn T (b)Pn + Pn H(a)H(˜b)Pn + Rn H(˜ a)H(b)Rn (prove it). The collection JW := {(Pn KPn + Rn LRn + Cn ) : K, L compact, (Cn ) ∈ G} forms a closed two-sided ideal in F W . Theorem 9. ([7], Theorem 1.5.3) JW ⊂ S(C). Moreover, each element (An ) ∈ S(C) can uniquely be written as An = Pn T (a)Pn + Pn KPn + Rn LRn + Cn , where K, L are compact operators, (Cn ) ∈ G. Using this representation, it is evident that ˜ (An ) = T (˜ a) + L , W (An ) = T (a) + K , W ˜ = G. and ker W ∩ ker W ˜ : S(C) → B(l2 ) and glue them toNow take the ∗-homomorphisms W, W 0 gether to obtain a ∗-homomorphism smb : S(C) → B(l2 ) × B(l2 ), ˜ (An )) . (An ) → (W (An ), W Furthermore, it is clear that ker smb0 = G. Thus, the quotient homomorphism smb : S(C)/G → B(l2 ) × B(l2 ) is correctly deﬁned and is injective. Notice that an injective ∗-homomorphism is isometric.

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Theorem 10. (i) The map smb is a ∗-isomorphism from S(C)/G onto the C ∗ -subalgebra of ˜ (An )) with (An ) running B(l2 ) × B(l2 ) which consists of all pairs (W (An ), W through S(C). ˜ (An ) are invertible oper(ii) (An ) ∈ S(C) is stable if and only if W (An ) and W ators. (iii) S(C) is fractal. Now it is clear that the theory of Section 3 applies. Theorem 11. ˜ (An )}. (a) lim An = max{W (An ) , W ˜ (An ). (b) lim inf spε An = lim sup spε An = spε W (An ) ∪ spε W ˜ (An ). a) = spε W If (An ) = (Pn T (a)Pn ), then spε W (An ) = spε T (a) = spε T (˜ ˜ (An ). (c) If (An ) is normal, then lim inf spAn = lim sup sp An = sp W (An )∪sp W ˜ (d) lim inf (An ) = lim sup (An ) = (W (An )) ∪ (W (An )).

5. Fredholm sequences Now we are going to introduce a third fundamental notion, namely that one of Fredholm sequences. First we introduce Fredholm sequences in some restricted form and ﬁnally in full generality. We introduce C ∗ -subalgebras of F which are generalizations of the algebras l F and F W and which give raise to consider Fredholm sequences. Let H be an inﬁnite-dimensional Hilbert space and (Ln ) be a sequence of orthogonal projections such that Ln → I strongly as n → ∞. The related C ∗ -algebra of all bounded sequences is again denoted by F . We shall assume that all projections Ln are ﬁnite rank operators. Let T be a (possibly inﬁnite) index set and suppose that, for every t ∈ T , we are given an inﬁnite-dimensional Hilbert space H t with identity operator I t as well as a sequence (Ent ) of partial isometries Ent : H t → H such that • the initial projections Ltn of Ent converge strongly to I t as n → ∞, • the range projection of Ent is Ln , • the separation condition ∗

(Ens ) Ent → 0 weakly as n → ∞

(5.1)

holds for every s, t ∈ T with s = t. (Recall that an operator E : H → H is a partial isometry if EE ∗ E = E and that E ∗ E and EE ∗ are orthogonal projections which are called the initial t and the range projections of E, respectively). For brevity, write E−n instead t ∗ t t of (En ) , and set Hn := im Ln and Hn := im Ln . Let F T stand for the set of all sequences (An ) ∈ F for which the strong limits t t s-limn→∞ E−n An Ent and s- lim(E−n An Ent )∗

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exist for every t ∈ T , and deﬁne mappings W t : F T → B(H t ) by W t (An ) := t s- limn→∞ E−n An Ent . It is easily seen that F T is a C ∗ -subalgebra of F which contains the identity, and that the W t are ∗-homomorphisms. The separation condition (5.1) ensures that, for every t ∈ T and every comt ) belongs to the algebra F T , pact operator K t ∈ K(H t ), the sequence (Ent K t E−n and that for all s ∈ T t if s = t K t W s (Ent K t E−n )= . (5.2) 0 if s = t Conversely, (5.2) implies (5.1). Moreover, the ideal G belongs to F T . So we t can introduce the smallest closed ideal J T which contains all sequences (Ent K t E−n ) t t with t ∈ T and K ∈ K(H ) as well as all sequences (Gn ) ∈ G. Remark. The algebra F W provides an example of this type. Indeed T consists only ˜ , and of two points, say 1 and 2. Then W 1 = W, W 2 = W J1 J2

= {(Pn KPn + Cn ) : K compact, (Cn ) ∈ G} , = {(Rn LRn + Cn ) : L compact, (Cn ) ∈ G} .

The ideal JT is exactly the earlier introduced ideal JW . The separation condition (5.1) is obviously fulﬁlled (recall that Rn tends weakly to zero). There are examples which show that indeed inﬁnite index sets T are needed ([7], 4.5.1–4.5.2, for instance). Theorem 12. ([7], Theorem 6.1) (a) A sequence (An ) ∈ F T is stable if and only if the operators W t (An ) are invertible in B(H t ) for every t ∈ T and if the coset (An ) + J T is invertible in the quotient algebra F T /J T . (b) If (An ) ∈ F T is a sequence with invertible coset (An ) + J T , then all operators W T (An ) are Fredholm on H t , and the number of the non-invertible operators among the W t (An ) is ﬁnite. Notice that this theorem can be used to give a diﬀerent proof of Theorem 10, (ii). Deﬁnition 6. (a) A sequence (An ) ∈ F T is called Fredholm if the coset (An ) + J T is invertible. (b) If the sequence (An ) ∈ F T is Fredholm, then its nullity α(An ), deﬁciency β(An ) and index ind (An ) are deﬁned by α(An ) := dim ker W t (An ), β(An ) := dim coker W t (An ), t∈T

t∈T

and ind (An ) := α(An ) − β(An ). It is a triviality to carry over well-known properties of Fredholm operators to Fredholm sequences.

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47

Remark. As we will see later on, this notion of Fredholm sequence depends on the underlying algebra F T . We shall also see that Fredholmness of a sequence in the sense of Deﬁnition 6 implies its Fredholmness in a general sense which has still to be deﬁned. Let (An ) ∈ F be arbitrary. We order the singular values of An as follows (ln = rank Ln ): 0 ≤ s1 (An ) ≤ · · · ≤ sln (An )(= An ) . For the sake of convenience let us also put s0 (An ) = 0. Recall that usually the singular values are ordered in the reverse manner. Deﬁnition 7. We say that (An ) ∈ F has the k-splitting property if there is a k ∈ Z+ such that lim sk (An ) = 0 , n→∞

while the remaining ln − k singular values stay away from zero, that is sk+1 (An ) ≥ δ > 0 for n large enough. The number k is also called the splitting number. Notice if (An ) has the 0-splitting property then (An ) is stable (hint: if s1 (An ) = 0 then An is invertible and A−1 = s1 (An )−1 ). Theorem 13. (a) Let (An ) ∈ F T be Fredholm. Then (An ) is subject to the k-splitting property t Ht t En Pker W t (An ) E−n ). with k = α(An ) and sα(An ) (An ) ≤ An ( t∈T

(b) If for (An ) ∈ F T there is at least one t1 ∈ T such that W t1 (An ) is not Fredholm, then lim sl (An ) = 0 for all l ∈ Z+ . n→∞

Assertions (a) and (b) can be proved slightly modifying the idea of the proof of Theorem 6.11 and using Theorem 6.67 in [7]. A complete proof of Theorem 13 is contained in [17]. We present here the proof of Theorem 13, (a), for the special case F l (we use here (Pn ) instead of (Ln ) in accordance with the notations of 4.I.). Proof. We shall make use of the following alternative description of the singular values (as approximation numbers): sj (An ) :=

min

B∈Flln

An − B ,

n−j

ln where Fm denotes the collection of all ln × ln -matrices of rank at most m. Let Rn be the orthoprojection onto im (Pn Pker A Pn ), where A = s-lim An Pn . It is easy to check that im Rn = im Pn Pker A Pn ,

rank Rn = rank Pn Pker A Pn = rank Pker A = dim ker A =: k for n large enough, and Rn − Pn Pker A Pn → 0 as n → ∞ .

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Consequently, ||An Rn || → 0 as n → ∞, and (An Rn ) ∈ G. Consider the sequence (Bn ) ∈ F l , Bn := A∗n An (Pn −Rn )+Pn Pker A Pn . Obviously, this sequence is also Fredholm and s-lim Bn Pn = A∗ A + Pker A is invertible. Then (Bn ) is stable by Theorem 12, (a). Since rank (Pn − Rn ) = ln − k for n large enough we get for those n sk (An ) ≤ || An − An A∗n An (Pn − Rn )Bn−1 Pn || ≤ || (An Bn − An A∗n An (Pn − Rn )) Pn || ||Bn−1 Pn || ≤ ||Bn−1 Pn || ||An Pn Pker A Pn || . Since (Bn ) is stable, there exists for n large enough a constant C with ||Bn−1 Pn || ≤ C. Thus we have sk (An ) ≤ C||An Pn Pker A Pn || → 0 as n → ∞ . Now consider sk+1 (An ). By using the well-known inequality sk+1 (A∗n An ) ≤ ≤ sk+1 (An )||A∗n || and that ||A∗n || is bounded (recall that A∗n Pn converges strongly to A∗ = 0) it has to be shown that sk+1 (A∗ An ) is bounded away from zero (n large enough). We have sk+1 (A∗n An ) = = ≥ =

min

|| (A∗n An − B) Pn ||

min

|| ((A∗n An + Pn Pker A Pn ) − B − Pn Pker A Pn ) Pn ||

n B∈Flln −k−1

n B∈Flln −k−1

min

n B∈Flln −1

|| ((A∗n An + Pn Pker A Pn ) − B) Pn ||

s1 (A∗n An + Pn Pker A Pn ) ≥ δ > 0

for n large enough since (A∗n An + Pn Pker A Pn ) is stable, and we are done.

Corollary 3. If (An ) ∈ F T is Fredholm, then ind (An ) = 0 . Proof. One has only to use that the matrices A∗n An and An A∗n are unitarily equivalent. This shows that the splitting numbers of (An ) and (A∗n ) coincide. Example 4. The sequence (Pn V Pn ) belongs to both algebras F l and F W . This sequence is Fredholm in F W but not in F l . If it would be Fredholm in F l then ind V = 0; but ind V = −1. Theorem 13 has remarkable applications. Let us mention some simple results: • If T (a)(a ∈ C(T)) is Fredholm, then the Moore-Penrose inverses (Pn T (a)Pn )+ converge strongly to T (a)+ if and only if dim ker Pn T (a)Pn = α(Pn T (a)Pn ) for n large enough.

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A deeper study of this problem is presented in [3], Chapter 4. • Let T (a) (a ∈ C(T)) be Fredholm and K be compact. Since (Pn (T (a) + K)Pn ) is subject to the splitting property with splitting number α(Pn (T (a)+ a) and dim ker T (˜ a) is known by K)Pn ) = dim ker(T (a) + K) + dim ker T (˜ Coburns Theorem, dim ker(T (a)+K) can be found numerically (in principle). • If T (a) (a ∈ C(T)) is Fredholm and a is smooth then sα (Pn T (a)P n )βtends fast |k| |ak | < ∞ to zero. For instance, if the Fourier coeﬃcients ak of a fulﬁll k∈Z

for some β > 0 then sα (Pn T (a)Pn ) = O(1/nβ ). • It was mentioned before that multiplication operators Ma with continuous functions a are quasidiagonal in L2 (T). The corresponding sequence of ﬁnitedimensional projections can be taken as orthogonal projections on some spline spaces. To be more precise let T := {|z| = 1} be parametrized be ϕ : [0, 1] → T , ϕ(t) = e2πit . A sequence of partitions (∆k )k∈N , ∆k := {σ0k , . . . , σnk k } , 0 = k −σj ) → σ0k < σ1k < · · · < σnk k = 1, is said to be admissible if h∆k := max(σj+1 δ 0 as k tends to inﬁnity. We denote by S˜ (∆k ) the space of all ψ ∈ C(T) such that ψ ◦ ϕ is (δ − 1) times continuously diﬀerentiable and the restriction of k ) is a polynomial of degree ≤ δ (smoothest ψ ◦ ϕ to each interval (σjk , σj+1 splines). Let P∆k denote the orthogonal projections of L2 (T) onto S˜δ (∆k ). Then (see [8], Section 2.14) (I − P∆k )f P∆k → 0 ,

P∆k f (I − P∆k ) → 0

as k → ∞, where – · stands for the operator norm in L2 (T), – f is continuous, – (∆k ) is admissible. Consider the singular integral operator A with continuous coeﬃcients: 1 g(τ ) dτ Ag = ag + bSg , where (Sg)(t) := πi τ −t T

(Recall that this integral exists for g ∈ L (T) almost everywhere in the sense of Cauchy’s principal value). A singular integral operator A is called strongly (locally) elliptic, if there is a continuous function c on T, a linear operator T with T < 1 and a compact operator K such that 2

A = c(I + T ) + K , c(t) = 0 for all t ∈ T . It follows that A is Fredholm with index 0 (even invertible). It is well known that A is strongly elliptic if and only if a(t) + λb(t) = 0 ∀t ∈ T und ∀λ ∈ [−1, 1] . If A is strongly elliptic and (∆k ) admissible then (P∆k AP∆k ) is stable

50

B. Silbermann (use P∆k AP∆k = P∆k cIP∆k (I + T )P∆k + P∆k KP∆k + C∆k , C∆k → 0). If a and b are merely continuous N × N -matrix functions, then A is strongly elliptic if and only if det(a(t) + λb(t)) = 0 for ∀t ∈ T and ∀λ ∈ [−1, 1]

(see [8], Section 13.31). In this case A is Fredholm of index 0, but might be not invertible. In any case, (P∆k AP∆k ) is Fredholm and α(P∆k AP∆k ) = dim ker A where (∆k ) is admissible. Now we turn to general Fredholm sequences. Deﬁnition 8. Let B be a unital C ∗ -algebra. An element k ∈ B is of central rank one if, for every b ∈ B, there is an element µ(b) belonging to the center of B such that kbk = µ(b)k. An element of B is of ﬁnite central rank if it is the sum of a ﬁnite number of elements of central rank one, and it is centrally compact if it lies in the closure of the set of all elements of ﬁnite central rank. We denote the set of all centrally compact elements in B by J (B). It is easy to check that J (B) forms a closed two-sided ideal in B. Proposition 9. ([7], Proposition 6.33) A sequence (An ) ∈ F is centrally compact if and only if, for every ε > 0, there is a sequence (Kn ) ∈ F such that sup An − Kn < ε and sup dim im Kn < ∞ . n

Deﬁnition 9. A sequence (An ) ∈ F is a Fredholm sequence if it is invertible modulo the ideal J(F ) of the centrally compact sequences. Theorem 14. ([7], Theorem 6.35) A sequence (An ) ∈ F is Fredholm if and only if there is an l ∈ Z+ such that lim inf sl+1 (An ) > 0 .

n→∞

Conclusion. If (An ) ∈ F T is Fredholm then it is also Fredholm in the sense of Deﬁnition 9.

6. Applications continued: Around ﬁnite sections of operators with almost periodic diagonals This material is based on [10]. 6.1. Example. Let ˜l2 := {(xn )n∈Z : |xn |2 < ∞}. n∈Z

The Almost Mathieu Operator is the operator Hα,λ,θ : ˜l2 → ˜l2 , which acts on a sequence x = (xn )n∈Z ∈ ˜l2 by (Hα,λ,θ x)n := xn+1 + xn−1 + λxn cos 2π(nα + θ).

C ∗ -algebras and Asymptotic Spectral Theory

51

Only recently the long-standing Ten Martini problem was solved, see [1], [9], and for a introduction to the topic [2]. The problem consists in describing the spectrum of Hα,λ,θ . The result says (in a somewhat incomplete form) that • If α is rational, α = pq and p, q relatively prime with q > 0, then the spectrum of Hα,λ,θ is the union of exactly q closed and pairwise disjoint intervals, θ πp ∈ / Z. • If α ∈ [0, 1) is irrational, then the spectrum is a Cantor type set (it means: nowhere dense, closed, and does not contain isolated points). This result is a qualitative one! It does not allow to say that a given number µ belongs to the spectrum (or not). There is (at least in present time) only one way to tackle this problem, namely the use of approximation methods. For, introduce the projection operators Pn and P˜n : P˜n x = {. . . , 0, x−n , . . . , xn , 0, . . . } Pn x = {. . . , 0, x0 , . . . , xn , 0, . . . } . First idea: consider the operators (matrices) (P˜n Hα,λ,θ P˜n ) (restricted to im P˜n and with respect to the standard basis) and compute the eigenvalues using Matlab or something else. Then the question arises, is this spectrum some how related to the spectrum of the Almost Mathieu Operator Hα,λ,θ ? The following is devoted to some theory around this problem. However, we will merely make use of the projections Pn . The operator Hα,λ,θ is an example of a band operator with almost periodic diagonals. 6.2. Band-dominated operators with almost periodic diagonals and related Toeplitz-like operators Recall that a general band-dominated operator A : ˜l2 → ˜l2 is the norm limit of band operators, that is of operators of the kind k

al U l ,

−k

where Um : ˜l2 → ˜l2 is the shift operator given by (Um x)(n) = x(n − m) and al I : l → l are multiplication operators with al ∈ l∞ (Z). If we replace the elements al ∈ l∞ (Z) by almost periodic ones, that is the set {Um a}m∈Z is relatively compact in the norm of l∞ (Z), then we obtain the class of band and band-dominated operators with almost periodic diagonals. It is easy to see that the collection AP (Z) ⊂ l∞ (Z), of all almost periodic sequences as well as the collection AAP of all band-dominated operators with almost periodic diagonals form C ∗ -algebras. Simple example: The Laurent operator L(a) with continuous generating function a ∈ C(T) is obviously an element of AAP (Z). ˜2

˜2

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B. Silbermann

Let us introduce the following class of Toeplitz-like operators. Clearly, l2 can be thought of as a subspace of ˜l2 . Let P denote the orthogonal projection onto l2 and Q := I − P . Consider T (A) : l2 → l2 , A ∈ AAP (Z), T (A) := P AP |imP . If A = L(a), a ∈ C(T), then T (L(a)) is denoted simply by T (a), and this is a familiar Toeplitz operator. Introduce J : ˜ l2 → ˜l2 (ﬂip operator) by (xn ) → ˜ (x−n−1 )) and H(A) := P AQJ , A := JAJ. Then one has ˜ (A, B ∈ AAP (Z)) , T (AB) = T (A)T (B) + H(A)H(B) which reminds the basic identity relating Toeplitz and Hankel operators and it is this identity for A = L(a), B = L(b), a, b ∈ C(T). ˜ are compact operators! Notice: H(A), H(B) Let AAP (Z+ ) denote the smallest C ∗ -subalgebra of B(l2 ) containing all operators T (A), A ∈ AAP (Z). Then • T (A) = A (6.1) 2 ˙ ) • AAP (Z+ ) = {T (A) : A ∈ AAP (Z)}+K(l The ﬁrst identity is based on a remarkable fact which plays an important role in what follows. Let us have a closer look. Let H refer to the set of all sequences h : Z+ → Z which tend to +∞ or −∞. Deﬁnition 10. An operator Ah ∈ B(˜l2 ) is called a norm limit operator of the operator A ∈ B(˜l2 ) with respect to the sequence h ∈ H if U−h(k) AUh(k) → Ah as k → ∞ in norm. The set of all norm limit operators is called the norm operator spectrum σop (A). Theorem 15. A ∈ AAP (Z+ ) is Fredholm if and only if each Ah ∈ σop (A) is invertible. Proposition 10. If A ∈ AAP (Z) then A ∈ σop (A). This result is in force since so-called distinguished sequences exist for A ∈ AAP (Z). Deﬁnition 11. A monotonically increasing sequence h : Z+ → Z+ is called distinguished if Ah exists and equals A. The ﬁrst assertion in (6.1) is now easy to prove: Consider U−h(k) P AP Uh(k) = U−h(k) P Uh(k) U−h(k) AUh(k) U−h(k) P Uh(k) ! " ! " ! " ↓ strongly ↓ in norm ↓ strongly I A I and apply the Banach-Steinhaus Theorem.

C ∗ -algebras and Asymptotic Spectral Theory

53

Further conclusion: ess sp A = spA for A ∈ AAP (Z) and T (A) is Fredholm if and only if A is invertible (ess sp A = sp(A + K(˜l2 )). Example 5. Almost Mathieu operators. We have ak (n) = =

U−k Hα,λ,θ Uk = U−1 + U1 + ak I , a(n + k) = λ cos 2π((n + k)α + θ) λ(cos 2π(nα + θ) cos 2π(kα) − sin 2π(nα + θ) sin 2π(kα)) .

Let α ∈ (0, 1) be irrational. We write α as a continued fraction with nth approximant pqnn such that 1

α = lim

n→∞

1

b1 + b2 +

1 ..

. bn−1 +

1 bn

with uniquely determined positive integers. Write this continued fraction as pn /qn with positive and relatively prime integers pn , qn . These integers satisfy the recursions pn = bn pn−1 + pn−2 , qn = bn qn−1 + qn−2 with p0 = 0, p1 = 1, q0 = 1 and q1 = b1 , and one has for all n ≥ 1 # # # # #α − pqnn # < q12 , n ⇒ |αqn − pn | ≤ q1n → 0 . Now it is not hard to see that (qn ) is a distinguished sequence for Hα,λ,θ (note: (qn ) is independent on λ and θ). 6.3. Distinguished sequences and ﬁnite sections In what follows we ﬁx a strongly monotonically increasing sequence h : Z+ → Z+ and deﬁne AAP,h (Z) := {A ∈ AAP (Z) : Ah exists and Ah = A} . It is easy to check that AAP,h (Z) is a C ∗ -subalgebra of B(˜l2) which is moreover shift invariant, i.e., U−k AUk again lies in AAP,h (Z+ ) whenever A does. Let AAP,h (Z+ ) refer to the smallest closed subalgebra of B(l2 ) which contains all operators T (A) with A ∈ AAP,h (Z). For instance, all Toeplitz operators with continuous generating functions lie in this algebra. Thus • K(l2 ) ⊂ AAP,h (Z+ ) and 2 ˙ • AAP,h (Z+ ) = {T (A) : A ∈ AAP,h (Z)}+K(l ).

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Let us turn over to ﬁnite sections. For, let Fh denote the set of all bounded sequences (An ) of matrices An ∈ Ch(n)×h(n) . Provided with pointwise deﬁned operations and the supremum norm, Fh becomes a C ∗ -algebra (An – norm of the operator deﬁned by An on im Ph(n) ). Finally, we let SAP,h (Z+ ) denote the smallest closed subalgebra of Fh which contains all sequences (Ph(n) T (A)Ph(n) ) with operators A ∈ AAP,h (Z). Deﬁne Rn : l2 → l2 , (xn )n≥0 → (xn , xn−1 , . . . , x0 , 0, 0, . . . ). Theorem 16. The C ∗ -algebra SAP,h (Z+ ) consists exactly of all sequences of the form Ph(n) T (A)Ph(n) + Ph(n) KPh(n) + Rh(n) LRh(n) + Ch(n) (6.2) with A ∈ AAP,h (Z) , K, L ∈ K(l2 ) , Ch(n) → 0 as n → ∞, and each sequence in SAP,h (Z+ ) can be written in the form (6.2) in a unique way. ˜ : SAP,h (Z+ ) → AAP,h (Z+ ) by Deﬁne mappings W, W W (An ) = ˜ (An ) = W

s- lim Ph(n) An Ph(n) , s- lim Rh(n) An Rh(n) .

These mappings are ∗-homomorphisms. Their importance is given by the following stability theorem. Theorem 17. A sequence (An ) ∈ SAP,h (Z+ ) is stable if and only if the operators ˜ (An ) are invertible, that is, if W (An ), W An = Ph(n) T (A)Ph(n) + Ph(n) KPh(n) + Rh(n) LRh(n) + Ch(n) ˜ +L with K, L, Ch(n) as above, then (An ) is stable if and only if T (A) + K , T (A) are invertible. Moreover, SAP,h (Z+ ) is fractal. The proof is basically the same as in the Toeplitz case. 6.4. Spectral approximations The last theorem in Section 6.3 is one of the keys to study spectral approximations. Theorem 18. Let A := (An ) ∈ SAP,h (Z+ ) be a self-adjoint sequence. Then the ˜ (A). spectra sp An converges in the Hausdorﬀ metric to sp W (A) ∪ sp W Theorem 19. Let A := (An ) ∈ SAP,h (Z+ ). Then the set of the singular values ˜ (A)). (An ) converges in the Hausdorﬀ metric to (W (A)) ∪ (W Theorem 20. A sequence A = (An ) ∈ SAP,h (Z+ ) is Fredholm if and only if its ˜ (A) is a Fredholm operstrong limit W (A) is a Fredholm operator. In this case W ator too, and ˜ α(A) = dim ker W (A) + dim ker W(A) ; moreover, lim sα (An ) = 0. n→∞

These theorems can be completed by results concerning ε-pseudospectra and the so-called Arveson’s dichotomy (the last for self-adjoint sequences).

C ∗ -algebras and Asymptotic Spectral Theory

55

Arveson’s dichotomy: Given a self-adjoint sequence A := (An ) ∈ SAP,h (Z+ ) and an open interval U ⊂ R, let Nn (U ) refer to the number of eigenvalues of An in U , counted with respect to their multiplicity. A point λ ∈ R is called essential for A, if for every open interval U containing λ, lim Nn (U ) = ∞ ,

n→∞

and λ ∈ R is called a transient point for A is there is an open interval U containing λ such that sup Nn (U ) < ∞ . n

Theorem 21. Let A := (An ) ∈ SAP,h (Z+ ) be self adjoint, s-lim An = T (A) + K. (s-lim An is necessarily of this form because of Theorem 16). Then every point λ ∈ sp A is essential, and every point λ ∈ R\sp A is transient for A. Moreover, for every point λ ∈ R\sp A, the sequence A−λP where (P := (Ph(n) ) ∈ SAP,h (Z+ )) is Fredholm and there is an open interval U ⊂ R containing λ such that sup Nn (U ) = α(A − λP).

n

Theorem 21 is a consequence of Theorem 7.12 in [7] and of the fact that $ = ess sp A. $ ess sp A = sp A = sp A The ﬁrst assertion of Theorem 21 implies in particular that each real number is either essential or transient for A. This property is usually referred to as the Arveson’s dichotomy of that sequence. Remark. If A ∈ AAP (Z) is selfadjoint and h is a distinguished sequence, then Theorem 21 oﬀers the possibility to determine the spectrum of A numerically. Test calculations will be provided in the next section. Let us mention that we could also use the projection P$n for this aim. The related theory (see Section 7) is not simpler as that for the projections Pn , but gives essentially the same results. A deep study of the ﬁnite sections for general band-dominated operators in ˜l2 is carried out in [15]. Let us mention also the recent book [4], where spectral properties of banded Toeplitz matrices are studied. 6.5. Test calculations Here we shall demonstrate how the theory can be used to determine numerically the spectrum of the Almost Mathieu operator for some choices of the parameters α, λ and θ (using Matlab). For each of the triples √ √ 2 1 1 1 2 5−1 2 2 , 2, 0 , , 2, , , 2, 0 , , 2, , 2, , , 5 5 2 7 2 2 2 2

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B. Silbermann

in place of (α, λ, θ), we choose a distinguished sequence of the corresponding Almost Mathieu operator which depends only on & % √ √ 2 5−1 2 2 αj ∈ , , , , 5 7 2 2 namely α1 α2 α3

= = =

α4

=

2 5 : h1 (k) = 5k , 2 7 : h2 (k) = 7k , √ √ k √ k 2 1 2) , 5 : h3 (k) = 2 (1 + 2) +(1 − √ √ √ √ k √ k 5−1 5+ 5 1+ 5 5− 5 1− 5 : h (k) = + 4 2 10 2 10 2

.

For irrational αk , this choice has been done via continued fractions. Notice that the sequences h3 and h4 are rapidly growing. For instance, h3 (13) = 47321 , h4 (23) = 46368. The results are plotted in Pictures 1–7. The results for α4 , λ = 2 , θ = 0, 5 and h5 (k) := 2k (non-distinguished!) are plotted in Picture 8. The computations clearly indicate the advantage of distinguished sequences over non-distinguished (compare Pictures 6 and 8). For irrational α the Cantorlike structure of the spectrum is also somehow reﬂected in the computations (see Pictures 4 and 5). The computations also show that the speed of converges is very high. There is only a guess why it should be, but not a proof.

Picture 1: Eigenvalues of Ph1 (k)Hα,λ,θ Ph1 (k) with α = 2/5, λ = 2, θ = 1/2.

C ∗ -algebras and Asymptotic Spectral Theory

Picture 2: Eigenvalues of Ph2 (k)Hα,λ,θ Ph2 (k) with α = 2/7, λ = 2, θ = 0.

Picture 3: Eigenvalues of Ph2 (k)Hα,λ,θ Ph2 (k) with α = 2/7, λ = 2, θ = 1/2.

57

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B. Silbermann

Picture 4: Eigenvalues of Ph3 (k)Hα,λ,θ Ph3 (k) with α =

√ 2/2, λ = 2, θ = 0.

Picture 5: The eigenvalues of Ph3 (k)Hα,λ,θ Ph3 (k) with α = which lie in the interval (−2.4, −2.8).

√ 2/2, λ = 2, θ = 0

C ∗ -algebras and Asymptotic Spectral Theory

59

√ Picture 6: Eigenvalues of Ph4 (k)Hα,λ,θ Ph4 (k) with α = ( 5 − 1)/2, λ = 2, θ = 0.5.

√ Picture 7: The eigenvalues of Ph4 (k)Hα,λ,θ Ph4 (k) with α = ( 5 − 1)/2, λ = 2, θ = 0.5, which lie in the interval (1.8, 2.6).

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B. Silbermann

√ Picture 8: Eigenvalues of Ph5 (k)Hα,λ,θ Ph5 (k) with α = ( 5 − 1)/2, λ = 2, θ = 0.5.

7. Applications continued: Band-dominated operators and their ﬁnite sections We reproduce in this section some recent results obtained in [12], [13], however with diﬀerent proofs. We give a complete study of the stability problem for ﬁnite sections of band-dominated operators and use these results to get an index formula for Fredholm operators belonging to that class. Recall that a general band-dominated operator A : ˜l2 → ˜l2 is the norm limit of band operators, that is of operators of the kind k

al U l ,

−k

where Ul are the earlier deﬁned shift operators, and al I : ˜l2 → ˜l2 are the multiplication operators (al x)m := al (m)xm with al = (al (m)) ∈ l∞ (Z). Let P, Q be the orthoprojections in ˜l2 , introduced in Section 6.2. Denote by H the set of all sequences h : N → Z which tend to −∞ or +∞. An operator Ah ∈ B(˜l2 ) is called the limit operator of A ∈ B(˜l2 ) with respect to the sequence h ∈ H if U−h(n) AUh(n) tends ∗-strongly to Ah as n → ∞. The set σ op (A) of all limit operators of a given operator A ∈ B(˜l2 ) is called the operator spectrum of A. It is not hard to see that every limit operator of a Fredholm operator is invertible. It is a remarkable fact that for band-dominated operators the reverse

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implication is true. More precisely, the following theorem is in force (see [11], Chapter 2): Theorem 23. Let A ∈ B(˜l2 ) be a band-dominated operator. Then (a) every sequence h ∈ H possesses a subsequence g such that the limit operator Ag exists, (b) the operator A is Fredholm if and only if each of its limit operators is invertible and if the norms of their inverses are uniformly bounded, (c) for each compact operator K one has σop (A + K) = σ op (A). Moreover, (A + K)g = Ag if one of the operators A (A + K)g , Ag with respect to the sequence g ∈ H exists. Notice that the collection of all band-dominated operators actually forms a C ∗ subalgebra S of B(˜l2 ) which contains all compact operators. Since P AQ, QAP are compact operators for all band-dominated operators, the study of the Fredholm properties of A ∈ S can be reduced to those of P AP + QBQ. ˜ n : ˜l2 → ˜l2 by Let P˜n be deﬁned as in Section 6.1 and deﬁne R (xn )n∈Z → (. . . , 0, 0, x−1 , x−2 , . . . , x−n , xn , xn−1 , . . . , x0 , 0, 0, . . . ) . ˜ n play an important role in the study of the stability of (P˜n AP˜n ), The operators R A ∈ S. It is not hard to see that the following version of Widom’s formula holds true (A, B ∈ S): ˜ n LR ˜ n + Cn (7.1) P˜n AB P˜n = P˜n AP˜n B P˜n + R with some compact operator L, and (Cn ) ∈ G (exercise; hint: consider expressions of the type P˜n P ABP P˜n − P˜n P AP P˜n P BP P˜n and those where P is replaced by Q). The ideas of 4.II are again not directly applicable. The point is that the strong ˜ n AR ˜n does not exist for A ∈ S in general. The situation changes when limit s-lim R we consider suitable subsequences. Theorem 24. (a) Let A ∈ S and h ∈ H be a sequence tending to +∞ for which the limit operator Ah of A with respect to the sequence h exists. Then there is a subsequence g of h such that the limit operator A−g of A exists. Moreover, ˜ g(n) AR ˜ g(n) = P JAh JP + QJA−g JQ . s∗ - lim R (b) An analogous result is true for sequences tending to −∞. ˜ g(n) ) tends weakly Proof. (a ) Since A− P AP − QAQ is a compact operator and (R to zero, it needs only to consider P AP + QBQ. Obviously, ˜ g(n) = P JU−g(n) P AP Ug(n) JP + QJUg(n) QAQU−g(n) JQ , ˜ g(n) (P AP + QAQ)R R where g is a subsequence of h for which the limit operator A−g of A exists (Theorem 23, (a)). A straightforward computation shows the claim.

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Let h ∈ H be a sequence tending to +∞ and introduce the algebra Fh with ˜ respect to the sequence (Ph(n) ) and let FhW stand for the C ∗ -subalgebra of Fh constituted by all sequences of Fh for which the limits ˜ h(n) Ah(n) R ˜ h(n) s∗ - lim P˜h(n) Ah(n) P˜h(n) and s∗ - lim R exist. Again, the set ( ' ˜ ˜ h(n) LR ˜ h(n) + Ch(n) : K, L − compact, (Ch(n) ) ∈ Gh JhW := P˜h(n) K P˜h(n) + R ˜

actually forms a closed two-sided ideal in FhW . (Gh stands here of the closed twosided ideal of Fh consisting of all sequences tending to zero in norm). The algebra ˜ FhW is of the type described in Section 5. Thus, the general theory mentioned in Section 5 applies. ˜

Corollary 4. A sequence (Ah(n) ) ∈ FhW is stable if and only if ˜ h(n) Ah(n) R ˜ h(n) are invertible, (a) the operators s-lim P˜h(n) Ah(n) Ph(n) and s-lim R ˜ ˜ ˜ W W W (b) the coset (Ah(n) ) + Jh is invertible in Fh /Jh . Corollary 5. Let h ∈ H be a sequence tending to +∞ and let A ∈ S be an operator for which ˜ h(n) AR ˜ h(n) s∗ - lim R ˜ exists. Then (P˜h(n) AP˜h(n) ) belongs to FhW .

Proposition 11. Let h ∈ H be a sequence tending +∞ and let A ∈ S be Fredholm. Then there exists a subsequence g of h such that (Pg(n) APg(n) ) is a Fredholm ˜ sequence in FgW . Proof. It is easy to see that A has a regularizer B in S, that is AB = I +M1 , BA = I + M2 , where M1 , M2 are compact operators. Using (7.1) we get ˜ n LR ˜ n + Cn , K, L-compact and (Cn ) ∈ G . P˜n AP˜n B P˜n = P˜n + P˜n K P˜n + R By Theorem 24 and Corollary 5 there is a subsequence h1 of h such that ˜ h (n) AR ˜h (n) exists. By the same reasoning there is a subsequence g1 of h1 s∗ -lim R 1 1 ˜ g (n) B R ˜g (n) exists. Hence, (Pg (n) APg (n) ), (Pg (n) BPg (n) ) ∈ such that s∗ -lim R 1 1 1 1 1 1 ˜

)

˜

˜

FgW1 , and (pg1 (n) APg1 (n) ) + JgW1 is invertible from the right in FgW1 /JgW1 . Again choosing a suitable subsequence g of g1 one obtains in the same fashion as above ˜ that (Pg(n) APg(n) ), (Pg(n) BPg(n) ) belong to FgW and ˜

˜

˜

(Pg(n) APg(n) ) + JgW is already invertible in FgW /JgW .

op op (A), σ− (A) denote the sets of all limit operators of A ∈ S Theorem 25. Let σ+ with respect to sequences tending to +∞ and −∞, respectively. Then the sequence (P˜n AP$n ) is stable if and only if the operator A, and all operators op (A) , QAh Q + P with Ah ∈ σ+

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and all operators op P Ah P + Q with Ah ∈ σ− (A)

are invertible on ˜l2 . Proof. Suﬃciency: Suppose that (P˜n AP˜n ) is not stable. Then there is a sequence h tending to +∞ such that lim (P˜h(n) AP˜h(n) )−1 P˜h(n) equals +∞ (if P˜h(n) AP˜h(n) is not invertible on im P˜h(n) , then we set (P˜h(n) AP˜h(n) )−1 Ph(n) = ∞). Then there is by Proposition 10 a subsequence g of h such that (Pg(n) APg(n) ) is a Fredholm ˜ sequence in FgW . By the computations in the proof of Theorem 24 we may assume that ˜ g(n) AR ˜ g(n) s∗ - lim R

= P JAg JP + QJA−g JQ = (P JAg JP + Q)(P + QJA−g JQ) .

Since JP J = Q and JQJ = P , we have J(P JAg JP + Q)J = QAg Q + P . Hence, P JAg JP + Q is invertible. By the same reasoning it follows that also P + QJA−g JQ is invertible. Using Corollary 4 we get that (Pg(n) APg(n) ) is stable. But this is a contradiction. The proof of the necessity is much more easier and is left to the reader. Our next topic is the index of a Fredholm band-dominated operator A ∈ S. For A ∈ S we put A+ = P AP + Q and A− = P + QAQ . It is easy to see that op op (A) ∪ {I} and σ op (A− ) = σ− (A) ∪ {I} . σ op (A+ ) = σ+

Further, the operator A ∈ S is Fredholm if and only if A+ and A− are Fredholm operators. Moreover, the equality P AP + QAQ = A+ A− = A− A+ shows that, for a Fredholm operator A ∈ S, ind A = ind A+ + ind A− . In addition, we shall use the notation ind+ A := ind A+ and ind− A := ind A− and call ind+ A and ind− A the plus- and the minus-index, respectively. Theorem 26. op (A) have the same plus1. Let A ∈ S be Fredholm. Then all operators in σ+ index, and this number coincides with the plus-index of A. Analogously, all op (A) have the same minus-index, and this number coincides operators in σ− with the minus-index of A.

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2. If A ∈ S is Fredholm, then ind A = ind+ B + ind− C , op op where B ∈ σ+ (A) and C ∈ σ− (A) are arbitrary taken. op (A) choose a sequence h tending Proof. It needs only to prove 1: Given B ∈ σ+ to + inﬁnity and for which B is the limit operator for A. Then B ∈ σop (A+ ). Notice that the limit operator of A+ with respect to −h exists and equals I. Then ˜ (Ph(n) A+ Ph(n) ) ∈ FhW . The Fredholmness of A+ implies that (Ph(n) A+ Ph(n) ) is a Fredholm sequence in Fh(n) and, by Corollary 3, ind A+ + ind (QBQ + P ) = 0. Since B is invertible it follows that ind (QBQ + P ) = −ind (P BP + Q) (use that ind (P BP + QBQ) = 0). Hence, ind A+ = ind+ B. Analogously, ind A− = ind− C, op where C ∈ σ− (A) is arbitrarily chosen.

Finally, let us mention an application of Theorems 25 and 26 to ﬁnite sections of band-dominated operators with slowly oscillating coeﬃcients. A function a ∈ l∞ (Z) is slowly oscillating if lim (a(x + k) − a(x)) = 0 for all k ∈ Z .

x→±∞

It is one of the remarkable features of band-dominated operators with slowly oscillating coeﬃcients that their limit operators are Laurent operators with continuous generating functions deﬁned on the unit circle in the complex plane (see [11], Chapter 2). Theorem 27. Let A ∈ S be a band-dominated operator with slowly oscillating coeﬃcients. Then the sequence (P˜n AP˜n ) is stable if and only if A is invertible and ind A+ = 0. Proof. Let A be invertible and ind A+ = 0. Then we have also ind A− = 0. By op op (A) (σ− (A)) have the same plus-index (minusTheorem 26 all operators in σ+ index) and this number coincides with the plus-index (minus-index) of A. Thus, all these indices are equal to zero. By a theorem of Coburg (usually formulated op for Toeplitz operators) the operators P BP + Q(P + QCQ) with B ∈ B+ (A) op (C ∈ σ− (A)) are invertible (see also part 4.II). Now it remains to apply Theorem 26. The reverse statement is obvious. Theorem 25 can be found in [11] under the additional assumption that all op op operators QAh Q + P, Ah ∈ σ+ (A) and all operators P Ah P + Q, Ah ∈ σ− (A), are uniformly invertible. Without this assumption it occurs in [12]. Theorem 26 was originally proved in [14] by help of K-theory and reproved recently in [13] using asymptotic spectral theory. Theorem 27 is Theorem 6.2.4 in [11], however for band operators with slowly oscillating coeﬃcients. In the general setting Theorem 27 was proved by S. Roch in [16] and is now simply a corollary to Theorems 25 and 26.

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The approach pointed out in Section 7 can be also used to get relatively simple proofs of some further results obtained by S. Roch in [15]. Let me mention only two: • computation of the α-number of a Fredholm sequence (P˜n AP˜n ), A ∈ S. • Any sequence (P˜n AP˜n ), A ∈ S and A = A∗ , has the Arveson dichotomy.

References [1] A. Avila and S. Jitomirskaja: The Ten Martini problem. -arXiv:math. DS/ 0503363. [2] F.-P. Boca: Rotation C ∗ -algebras and Almost Mathieu Operators. Theta Series in Advanced Mathematics 1, The Theta Foundation, Bucharest 2001. ¨ ttcher and B. Silbermann: Introduction to Large Truncated Toeplitz Ma[3] A. Bo trices. Springer-Verlag, New York 1999. ¨ ttcher and S.M. Grudsky: Spectral Properties of Bounded Toeplitz Ma[4] A. Bo trices. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA 2005. [5] N.P. Brown: Quasidiagonality and the ﬁnite section method. Math. Comp. 76 (2007), no. 257, 339–360 (electronic). [6] N.P. Brown: AF Embeddings and the Numerical Computation of Spectra in Irrational Rotation Algebras. Num. Funct. Anal. Optim. 27 (2006), no. 5-6, 517–528. [7] R. Hagen, S. Roch, and B. Silbermann: C ∗ -Algebras and Numerical Analysis. Marcel Dekker, Inc., New York, Basel 2001. ¨ ssdorf and B. Silbermann: Numerical Analysis for Integral and related [8] S. Pro Operator Equations. Akademie Verlag, Berlin 1991. [9] J. Puig: Cantor spectrum for the almost Mathieu Operator. Comm. Math. Phys. 244 (2004), 2, 297–309. [10] V. Rabinovich, S. Roch, and B. Silbermann: Finite sections of band-dominated operators with almost periodic coeﬃcients. Operator Theory: Advances and Applications 170, Birkh¨ auser Verlag, Basel, Boston, Berlin 2006. [11] V. Rabinovich, S. Roch, and B. Silbermann: Limit Operators and Their Applications. Operator Theory: Advances and Applications 150, Birkh¨ auser Verlag, Basel, Boston, Berlin 2004. [12] V. Rabinovich, S. Roch, and B. Silbermann: On ﬁnite sections of banddominated operators. Preprint 2486 (2006), Fachbereich Mathematik, TU Darmstadt. [13] V. Rabinovich, S. Roch, and B. Silbermann: The ﬁnite section approach to the index formula for band-dominated operators. Preprint 2488 (2006), Fachbereich Mathematik, TU Darmstadt. [14] V. Rabinovich, S. Roch, and J. Roe: Fredholm indices of band-dominated operators. Integral Equations and Operator Theory 49 (2004), 2, 221–238. [15] S. Roch: Finite sections of band-dominated operators. Preprint Nr. 2355 (2004), Fachbereich Mathematik, TU Darmstadt.

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[16] R. Roch: Band-dominated operators on lp -spaces: Fredholm indices and ﬁnite sections. Acta Sci. Math (Szeged) 70 (2004), 783–797. [17] A. Rogozhin and B. Silbermann: Banach algebras of operator sequences: Approximation Numbers. J. Operator Theory, in print. Bernd Silbermann Technische Universit¨ at Chemnitz Fakult¨ at f¨ ur Mathematik D-09107 Chemnitz, Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 67–118 c 2008 Birkh¨ auser Verlag Basel/Switzerland

Toeplitz Operator Algebras and Complex Analysis Harald Upmeier

1. Introduction The aim of this survey article is to present the recent work concerning Hilbert spaces of holomorphic functions on hermitian symmetric domains of arbitrary rank and dimension, in relation to operator theory (Toeplitz C ∗ -algebras and their representations), harmonic analysis (discrete series of semi-simple Lie groups) and quantization (covariant functional calculi and Berezin transformation). Acknowledgment A substantial part of the results presented here is joint work with J. Arazy (University of Haifa). The ﬁnancial support of the German-Israeli foundation (GIF No. 696-17.6/2001) is gratefully acknowledged. The author would like to thank the referee for valuable comments.

2. Complex geometry of bounded symmetric domains In the following consider a complex vector space Z = Cd , endowed with a norm · , and let D = {z ∈ Z : z < 1} be its unit ball. In general, the boundary ∂D = {z ∈ Z : z = 1} will not be smooth. Consider the set S = ∂ex D ⊂ ∂D of all extreme points and let K = U (Z) = {g ∈ GL(Z) : gz = z}

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be the isometry group. Then K ⊂ GL(Z) is a closed subgroup, therefore K is a compact Lie group. Clearly, the natural K-action satisﬁes K · D = D, K · ∂D = ∂D, K · S = S. Example. For Z = Cd , deﬁne the scalar product zi w i (z|w) = zw∗ = i

and let z = (z|z)

1/2

be the Hilbert norm. Then D = {z ∈ Cd : (z|z) < 1}

is the Hilbert unit ball. In this case K = U (d) is the unitary group and S = ∂D = S 2d−1 is the odd sphere. Example. More generally, consider now the matrix space Z = Cr×(r+b) , endowed with the operator norm z = sup σ(zz ∗ )1/2 , and let D = {z ∈ Cr×(r+b) : z < 1} = {z ∈ Cr×(r+b) : I − zz ∗ > 0} be the corresponding matrix unit ball, with closure D = {z ∈ Cr×(r+b) : I − zz ∗ ≥ 0}. In this case we have K = U (r) × U (r + b) (u, v), via the action z → uzv ∗ , and S = {z ∈ Cr×(r+b) : zz ∗ = I} consists of all isometries. In the special case b = 0 we obtain S = U (r). In this case the domain D is said to be of tube type. We now introduce holomorphic functions. By deﬁnition, a (possibly vectorvalued) function f : D → Cm is holomorphic, if it has a power series expansion f (z) = cα z1α1 · · · znαn α∈Nn

(compact convergence) with coeﬃcients cα ∈ Cm . By collecting monomials of equal (total) degree, we obtain the expansion cα z1α1 · · · znαn = pk (z) f (z) = k≥0 |α|=k

k≥0

into a series of k-homogeneous polynomials. Let Pk (Z, Cm ) be the space of all k-homogeneous polynomials pk : Z → Cm satisfying pk (λz) = λk pk (z) ∀ λ ∈ C. The space Pk (Z, Cm ) P(Z, Cm ) = k≥0

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of all polynomials is dense in the space of all holomorphic functions O(D, Cm ) under compact convergence. Deﬁnition 2.1. Let G = Aut (D) = {g : D → D biholomorphic} denote the holomorphic automorphism group. It is a real Lie group (by a deep theorem of H. Cartan). Moreover, K = {g ∈ G : g(0) = 0} G is a compact subgroup. Deﬁnition 2.2. The domain D is called symmetric iﬀ G acts transitively on D, i.e., ∀ z ∈ D ∃ g ∈ G, z = g(0)). Clearly, this is equivalent to D = G/K. For arbitrary domains, the condition of symmetry requires that at each point there exists a globally deﬁned geodesic reﬂection. In general, this is a stronger condition than homogeneity, but in our case the unit ball D is circular and has the obvious reﬂection s0 (z) = −z at the origin. Example. If Z = Cr×(r+b) and D is the matrix unit ball, we obtain the pseudounitary group ∗ ∗ ab ab 1 0 a c 1 0 G = SU (r, r+b) = ∈ GL (2r + b) : = cd b d 0 − 1 b∗ d∗ 0 −1 acting on D via Moebius transformations ab (z) = (az + b)(cz + d)−1 . cd Its maximal compact subgroup is a0 K= : a ∈ U (r), d ∈ U (r + b) 0d Example. If Z = Cd = C1×d and D is the Hilbert unit ball, we obtain as a special case G = SU (1, d). Example. For dimension d = 1, we have Z = C and D is the unit disk. In this case, we may identify G =

SU (1, 1) ≈ SL(2, R),

K

U (1)

=

It is a fundamental fact [K], [K1], [LO2], [U6] that hermitian symmetric domains have an algebraic description in terms of the so-called Jordan algebras and Jordan triples. In order to explain this connection, consider the Lie algebra exp g −→ G of the real Lie group G = Aut (D), which can be realized as follows:

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Consider a 1-parameter group t → gt ∈ G, and deﬁne, for f ∈ O(D, Cm ), the inﬁnitesimal generator ∂ ∂f (gt (z)) ## f (z). = (Xf )(z) = h(z) f (z) = h(z) t=0 ∂t ∂z Here h : D → Z = Cd is a holomorphic vector ﬁeld, with commutator bracket ∂ ∂k ∂h ∂ ∂ , k = h −k . h ∂z ∂z ∂z ∂z ∂z Let s0 (z) = −z be the symmetry at the origin 0 ∈ D. For the adjoint action of G on g, deﬁned via ∂ ∂ = h(g(z)) g (z) , Ad (g) h ∂z ∂z we obtain the Cartan decomposition g = k ⊕ ℘, where k = {X ∈ g : Ad (s0 ) X = X} is the 1-eigenspace and ℘ = {X ∈ g : Ad (s0 ) X = −X} is the (−1)-eigenspace. By deﬁnition, we obtain a Lie triple system [k, k] ⊂ k ⊃ [℘, ℘], [k, ℘] ⊂ ℘ ⊃ [℘, k]. The crucial step towards Jordan triples is the following Theorem 2.3. Let D ⊂ Z be a bounded symmetric domain. Then there exists a Jordan triple product Z × Z × Z → Z, denoted by u, v, w → {uv ∗ w} = {wv ∗ u}, such that ∂ ℘ = (v − {zv ∗ z}) : v∈Z ∂z ∂ k = {h(z) : h ∈ gl(Z) linear , h{zv ∗ z } = {h(z ) h(v )∗ h(z )}}. ∂z In order to deﬁne the Jordan triple property, let Z × Z → End (Z) be the mapping u, v → u v ∗ , given by (u v ∗ ) z = {uv ∗ z}. Since [[℘, ℘] ℘] ⊂ [k, ℘] ⊂ ℘, we obtain the Jordan triple identity [u v ∗ , x y ∗ ] = {uv ∗ x} y ∗ − x {yu∗v}∗ .

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Example. In the matrix case Z = Cr×(r+b) , let at b t (z) = (at z + bt )(ct z + dt )−1 ∈ SU (r, r + b) ct dt be a 1-parameter group of Moebius transformations. Its diﬀerential is computed as d ## ˙ + b˙ − z cz ˙ − z d˙ # (at z + bt )(ct z + dt )−1 = az dt t=0 ˙ with ac˙˙ db˙ ∈ su (r, r + b). The eigenspace decomposition is given by ∂ ˙ ˙ : a˙ ∈ u(r), d ∈ u(r + b) k = (az ˙ − z d) ∂z and

∂ ∗ r×(r+b) ˙ ˙ ˙ : b∈C ℘ = (b − z b z) . ∂z Therefore, according to Theorem 2.3 we obtain {zv ∗ z} {uv ∗ w} and u v∗ =

= =

zv ∗ z, 1 ∗ ∗ 2 (uv w + wv u)

1 (Luv∗ + Rv∗ u ). 2

Any bounded symmetric domain D ⊂ Z has an important K¨ ahler geometric structure called the Bergman metric. To deﬁne this metric, consider the Bergman operators Z × Z → End (Z), denoted by u, v → B(u, v), and deﬁned as B(u, v)z = z − 2{uv ∗ z} + {u{vz ∗ v}∗ u}. Example. For the matrix case Z = Cr×(r+b) , we have B(u, v)z

=

z − uv ∗ z − zv ∗ u + u(vz ∗ v)∗ u

= =

z − uv ∗ z − zv ∗ u + uv ∗ zv ∗ u (1 − uv ∗ ) z(1 − v ∗ u).

Therefore B(u, v) = L1−uv∗ R1−v∗ u . Deﬁnition 2.4. Deﬁne a G-invariant Hermitian metric on D as follows: ∀ z ∈ D ∀ u, v ∈ Tz D ≈ Z we put hz (u, v) = (u|v)z = (B(z, z)−1 u|v), using the normalized K-invariant inner product (u|v) on T0 D = Z given by (u|v) = where p is the “genus” of D.

2 trZ u v ∗ , p

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Example. For matrices Z = Cr×(r+b) we have 2r + b Luv∗ + Rv∗ u = tr uv ∗ 2 2 and hence the Bergman metric is given by trZ u v ∗ = trZ

hz (u, v) = tr (1 − zz ∗ )−1 u(1 − z ∗ z)−1 v ∗ . In terms of the Bergman operator, the Bergman kernel function K : D × D → C of D can be expressed as K(z, w) = det B(z, w)−1 . It is a fundamental fact that there exists a sesqui-polynomial called the Jordan triple determinant N : Z × Z → C such that the Bergman kernel function has the form K(z, w) = det B(z, w)−1 = N (z, w)−p for all z, w ∈ D. Example. In the matrix case Z = Cr×(r+b) we have det B(z, w) = detZ [L1−zw∗ R1−w∗ z ] = detZ L1−zw∗ detZ R1−w∗ z = detCr (1r − zw∗ )r detCr+b (1r+b − w∗ z)r+b = det (1r − zw∗ )2r+b . It follows that N (z, w) = det (1 − zw∗ ). Jordan triples are closely related to Jordan algebras which were originally introduced in quantum mechanics. Deﬁnition 2.5. A real vector space X ≈ Rd is called a real Jordan algebra iﬀ X has ◦ a commutative, non-associative product X × X −→ X, denoted by x, y → x ◦ y = y ◦ x, which satisﬁes the Jordan algebra identity x2 ◦ (x ◦ y) = x ◦ (x2 ◦ y). If, in addition, we require x2 + y 2 = 0 =⇒ x = y = 0, the Jordan algebra X is called formally-real or Euclidean since, in this case it has a strictly positive inner product which agree with (u|v) on the complexiﬁcation Z = X C . We have the following classiﬁcation [JNW]: Every (irreducible) Euclidean Jordan algebra has a realization X ≈ Hr (K) = {self-adjoint r × r-matrices x = (xij ) ∈ Kr×r , x∗ij = xji } for the anti-commutator product 1 (xy + yx). 2 More precisely, we have the following possibilities, depending on the “rank” r x◦y =

if r ≥ 4, then K = R, C, H = quaternions if r = 3, then K = R, C, H, O = octonions.

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If r = 2, then K = (0) is a real Hilbert space, i.e., we obtain “formal” 2 × 2 matrices α b H2 (K) ∗ b δ with α, δ ∈ R, b ∈ K. It follows from the classiﬁcation that X is uniquely characterized by only two invariants: r = rank X and the characteristic multiplicity a = dimR K. Every Euclidean Jordan algebra X has a positive cone Ω = {x2 : x ∈ X \ {0}}, corresponding to the positive deﬁnite matrices Hr+ (K) and a Jordan algebra determinant N : X → R, which is a degree r polynomial deﬁned via Cramer’s rule ∇x N . x−1 = N (x) Example. For the rank 2 case, N

α b = αδ − (b|b) b∗ δ

is the Lorentz metric on R1, 1+a . If (X, ◦) is a real Jordan algebra, the complexiﬁcation Z := X C becomes a complex Jordan ∗-algebra, and the associated Jordan triple (of tube type) on Z is deﬁned as follows: {uv ∗ w} = (u ◦ v ∗ ) ◦ w + (w ◦ v ∗ ) ◦ u − v ∗ ◦ (u ◦ w). For the ﬁne structure of Jordan triples and the corresponding symmetric domains, we need the notion of tripotent (= triple idempotent) and associated Peirce decomposition. This is analogous to the well-known root decomposition of (semi-simple) Lie algebras which underlies the Lie theoretic approach to symmetric spaces. In any Jordan triple Z, an element c ∈ Z is called a tripotent iﬀ {cc∗ c} = c. In this case we have the Peirce decomposition Z = Z1 (c) ⊕ Z1/2 (c) ⊕ Z0 (c), where Zα (c) = {z ∈ Z : {cc∗ z} = αz} is the α-eigenspace for α = 0, 1/2, 1. Example. For matrices Z = Cr×(r+b) a tripotent c = cc∗ c is a partial isometry. In case k r−k b 0 0 r 1k , c= r−k 0 0 0

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the Peirce decomposition is given by Z1/2 (c) Z1 (c) Cr×(r+b) = Z1/2 (c) Z0 (c)

Z1/2 (c) Z0 (c)

.

The Peirce decomposition can be constructed more generally for any maximal frame e1 , . . . , er of orthogonal tripotents, where r = rank Z. Here orthogonality means ei e∗j = 0 for i = j, a stronger condition than orthogonality for the scalar product of Z. In this case ⊕ Zij , Z= 0≤i≤j≤r

where

1 i (δ + δkj ) z}. 2 k Proposition 2.6. If D is an irreducible symmetric domain, the joint Peirce decomposition yields the characteristic multiplicities ⎫ Z00 = (0) ⎪ ⎪ ⎬ Zii = C ei 1 ≤ i ≤ r a 1≤i<j≤r ⎪ dimC Zij = ⎪ ⎭ dimC Z0j = b 1≤j≤r Zij = {z ∈ Z : {ek e∗k z} =

In particular, b = 0 iﬀ D is of tube type. Example. In the matrix case Z = Cr×(r+b) let e1 , . . . , er be the diagonal matrix units. The associated Peirce decomposition looks as follows ⎛ ⎞ Z11 Z12 Z1r Z01 ⎜ Z12 Z22 Z02 ⎟ ⎜ ⎟ ⎜ ⎟ .. ⎝ ⎠ . Z1r

Zrr

Z0r

and we have a = 2 (“complex case”).

3. Weighted Bergman spaces of holomorphic functions We will now introduce an important class of Hilbert spaces belonging to the holomorphic discrete series. Let Z be an irreducible Jordan triple of rank r, with characteristic multiplicities a, b. The dimension is then given by d = r + a r(r−1) + rb 2 or, equivalently, a d = 1 + (r − 1) + b. r 2 The precise meaning of the “second” multiplicity b will be given in Proposition 2.6 below. Deﬁne the genus p = 2 + a(r − 1) + b. The tube type case corresponds to b = 0 or, equivalently, to p2 = dr . Fix a parameter λ > p − 1 = 1 + a(r − 1) + b

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and consider the probability measure on D given by dµλ (z) =

ΓΩ (λ) N (z, z)λ−p dz, π d ΓΩ (λ − d/r)

where ΓΩ denotes the Koecher-Gindikin Γ-function of the symmetric cone Ω discussed in more detail below. The weighted Bergman space of holomorphic functions on D is deﬁned as 2 Hλ (D) = {ψ ∈ O(D, C) : dµλ (z) |ψ(z)|2 < ∞}. D

For λ = p, dµp (z) is a multiple of the Lebesgue measure dz and Hp2 (D) = {ψ ∈ O(D, C) : dz |ψ(z)|2 < ∞} D

is the standard (unweighted) Bergman space. The inner product (conjugate-linear in the ﬁrst variable) is given by ΓΩ (λ) dz N (z, z)λ−p ψ1 (z) ψ2 (z). (ψ1 |ψ2 )λ = dµλ (z) ψ1 (z) ψ2 (z) = d π ΓΩ (λ − dr ) D

D

Here we follow the convention of mathematical physics, where Hilbert state spaces, such as Hλ2 (D), have a scalar-product which is conjugate-linear in the ﬁrst variable, whereas the underlying phase space Cn (in its K¨ahler polarization) has the K¨ahler metric conjugate-linear in the second variable. Proposition 3.1. Hλ2 (D) has the reproducing kernel Kλ (z, w) = N (z, w)−λ = det B(z, w)−λ/p . It follows that the orthogonal projection Pλ : L2 (D, dµλ ) → Hλ2 (D) has the form (Pλ ψ)(z) = dµλ (z) N (z, w)−λ ψ(w). D

Deﬁne an irreducible projective unitary representation Uλ : G → U (Hλ2 (D)) of G by putting (Uλ (g −1 ) ψ)(z) = det ψ (z)λ/p ψ(g(z)). Viewed as a quantum deformation, λp = h1 corresponds to the inverse Planck constant. The representations Uλ constitute the scalar holomorphic discrete series of G. Using the Jordan triple determinant, we may describe the analytic continuation of the scale of weighted Bergman spaces [L3], [FK1], [VR], [WA]: Deﬁne theWallach set W (D) = {λ ∈ C : (N (zi , zj )−λ )1≤i, j≤n

0 ∀ z1 , . . . , zn ∈ D}.

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Theorem 3.2. The Wallach set is a disjoint union ' a ( ' ( a W (D) = : 0 ≤ < r ∪ λ > (r − 1) 2 2 consisting of r discrete points and a continuous part. More explicitly, the Wallach parameters are given as follows:

Figure 1. The scale of Wallach parameters z

a (r − 1) 2 }|

{z a (r − 1) 2

0 a 2

1+b }|

{z d r Hardy

a (r − 1) 2 }|

{z p−1

d a + r 2

weighted Bergman }|

{

p Bergman

Example. In the matrix case Z = Cr×(r+b) we have a = 2, and hence dr = r+b, p = 2r + b. The Shilov boundary S is a homogeneous space under the group K, therefore S = K/L where L := { ∈ K : e = e} for some e ∈ S. In case D ⊂ X C is of tube type and e is the unit element of the associated Euclidean Jordan algebra X, L = Aut (X) agrees with the automorphism group of X. The Shilov boundary S = K/L, endowed with the K-invariant probability measure, corresponds to the Wallach parameter d a λ = = 1 + (r − 1) + b r 2 giving rise to the Hardy space H 2 (S) = {ψ ∈ L2 (S) : ψ holomorphic on B}. The associated Szeg˝ o projection P : L2 (S) → H 2 (S) has the form (P ψ)(z) = ds N (z, s)−d/r ψ(s). S

The proof of the fundamental Theorem 3.2 depends on harmonic analysis (PeterWeyl decomposition) for the compact Lie group K. Consider the algebra P(Z) of all polynomials p : Z → C and the natural algebra action K × P(Z) → P(Z) deﬁned by (Uk−1 p)(z) := p(kz). A deep result [S] asserts that Pm1 ≥...≥mr (Z) P(Z) = m1 ≥...≥mr ≥0

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is a direct sum of irreducible K-modules, labelled by all partitions m = (m1 , . . . , mr ) ∈ Nr+ . Moreover, the highest weight vector has the form Nm (z) = N1 (z)m1 −m2 N2 (z)m2 −m3 · · · Nr (z)mr , where Nk is the Jordan determinant of

Zij

1≤i≤j≤k

(generalized minor). This is the main result of [U3]. Example. For the simplest partition m = (1, 0, . . . , 0), P10···0 (Z) = Z is the dual space of linear forms on Z. In this case N10···0 (z) = (e1 |z), where we ﬁx a frame (e1 , . . . , er ). As a Hilbert space completion, we obtain the Segal-Bargmann-Fock space 1 dz e−(z|z) |ψ(z)|2 < ∞} H 2 (Z) = {ψ ∈ O(Z) : d π Z

of entire functions, with reproducing kernel expanded in a series e(z|w) = Km (z, w), m1 ≥···≥mr ≥0

where Km (z, w) denotes the reproducing kernel of Pm (Z) ⊂ H 2 (Z). The FarautKoranyi binomial formula [FK1] is N (z, w)−λ = (λ)m Km (z, w), m

where (λ)m =

r . Γ(λ + mj − (j − 1) a2 ) ΓΩ (λ + m) = ΓΩ (λ) Γ(λ − (j − 1) a2 ) j=1

denotes the Pochhammer symbol. By checking positivity of the coeﬃcients, one obtains the desired conclusion a a W (D) = {λ ∈ C : (λ)m ≥ 0 ∀ m ∈ Nr+ } = { : 0 ≤ < r} ∪ {λ > (r − 1)}. 2 2

4. Toeplitz operators on symmetric domains A deep relationship between analysis on symmetric domains and operator theory concerns C ∗ -algebras generated by Toeplitz operators in the multi-variable setting. In general, for any complex Hilbert space H, let L(H) be the C ∗ -algebra of all bounded linear operators on H. For f ∈ L∞ (D), deﬁne the Bergman-Toeplitz operators Tfλ ∈ L(Hλ2 (D)) for parameter λ > p − 1 by Tfλ ψ = Pλ (f ψ) ∀ ψ ∈ Hλ2 (D).

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More precisely, (Tfλ ψ)(z)

=

dµλ (z) Kλ (z, w) f (w) ψ(w) D

=

ΓΩ (λ) d π ΓΩ (λ − d/r)

dz N (z, z)λ−p N (z, w)−λ f (w) ψ(w).

D

∞

Similarly, for f ∈ L (S), deﬁne Hardy-Toeplitz operators Tf ∈ L(H 2 (S)) for parameter λ = dr by Tf ψ = P (f ψ) ∀ ψ ∈ H 2 (S). More precisely,

(Tfλ ψ)(z) =

ds N (z, s)−d/r f (s) ψ(s).

S

In both cases we consider the Toeplitz C ∗ -algebras Tλ (D) =

C ∗ (Tfλ : f ∈ C(D)) ⊂ L(Hλ2 (D))

T (S) =

C ∗ (Tf : f ∈ C(S)) ⊂ L(H 2 (S))

In principle this could be deﬁned for any Wallach parameter ·

λ ∈ W (D) = Wdisc (D) ∪ Wcont (D), since Hλ is a Hilbert space with reproducing kernel N (z, w)−λ (z, w ∈ D). However, there is a diﬀerence between the discrete and continuous part: Proposition 4.1. Hλ is a Hilbert-module iﬀ λ > a2 (r − 1). It follows that only in this case the Toeplitz operators Tz1 , . . . , Tzd form a d-tuple of bounded operators. For a deeper analysis of Toeplitz C ∗ -algebras, we need some geometric preparations concerning the boundary of symmetric domains. Proposition 4.2. The Shilov boundary has the Jordan theoretic characterization S = ∂ex D = {e ∈ Z : {ee∗ e} = e tripotent, e e∗ invertible, Z0 (e) = (0)}. In the “tube type” case, we obtain the simpler characterization S = {e ∈ Z tripotent : e e∗ = idZ , Z = Z1 (e)}. The boundary structure of a symmetric domain D can now be described in algebraic terms [LO2]. Let D ⊂ Z be the unit ball of a Jordan triple and consider a tripotent c = {cc∗ c}, with Peirce decomposition Z = Z1 (c) ⊕ Z1/2 (c) ⊕ Z0 (c). Then Z1 (c) is a Jordan ∗-algebra with unit element c such that k = rank Z1 (c). On the other hand, Z0 (c) is a Jordan subtriple and the norm satisﬁes z1 + z0 = max (z1 , z0 ) for all z1 ∈ Z1 (c), z0 ∈ Z0 (c). The unit ball D0 (c) = {z0 ∈ Z0 (c) : z0 < 1}

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is a symmetric domain of lower rank r − k and c + D0 (c) ⊂ ∂D, since c + z0 = max ( c , z0 ) = 1. !" !" =1

(r − 1) 2 in the continuous Wallach set (Hilbert modules), or the subclass λ > 1 + a(r − 1) + b corresponding to the weighted Bergman spaces (discrete series). Let c = {cc∗ c} be a tripotent of rank k. Then D0 (c) ⊂ Z0 (c) is an irreducible symmetric domain of rank r − k, with characteristic multiplicities a, b. Moreover, the shifted parameter a a λ − k > (r − k − 1) 2 2

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belongs to the “little” continuous Wallach set. Similarly, in the Bergman case a λ − k > 1 + a(r − k − 1) + b 2 belongs to the “little” discrete series. Theorem 4.6. Let Tλ (D) be the Toeplitz C ∗ -algebra acting on Hλ2 (D) (weighted Bergman space). Then (i) For any tripotent c = {cc∗ c}, there exists a unique irreducible ∗-representation σc : Tλ (D) → Tλ−k a/2 (D0 (c)) such that D (c)

0 σc (TλD (f )) = Tλ−k a/2 (fc )

for all f ∈ C(D), with restriction fc ∈ C(D0 (c)) deﬁned by fc (w) := f (c + w) for all w ∈ D0 (c). (ii) The representations σc are mutually inequivalent (iii) each irreducible ∗-representation of Tλ (D) has this form, i.e., similar to the Hardy case, we may identify Spec Tλ (D) with the set of all tripotents c = {cc∗ c} of Z. Note that the structure and representations of these Toeplitz C ∗ -algebras remain “rigid” along the continuous part of the Wallach set, since we are dealing with a “non-commutative” topological structure. The discrete points, however, show a diﬀerent behavior.

5. Convolution C ∗ -algebras on non-commutative Hardy spaces It turns out that the structure theorems for Toeplitz C ∗ -algebras belong to a much more general theory concerning convolution operators on compact symmetric spaces. Thus we may consider situations where only the Shilov boundary S, but not the domain D itself, is a symmetric space. In this section we use “left” quotient spaces which are more convenient for convolution algebras. (Accordingly the domain D, which in this section plays only a minor role, should be realized as D = K \ G.) Let S =L\K be a homogeneous compact manifold, with stabilizer group L = {k ∈ K : e · k = e}. Here e ∈ S is the “unit element”. L2 (K) carries the left-regular K-action (t f )(s) := f (t−1 s) ∀ s, t ∈ K, f ∈ L2 (K). Consider the group C ∗ -algebra C∗ (K) = C ∗ (u : u ∈ L1 (K))

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generated by all left-convolution operators (u f )(s) := du(t)(t f )(s). K

Its weak closure W ∗ (K) = W ∗ (t : t ∈ K) is the group W ∗ -algebra. Identify L2 (S) ≈ {f ∈ L2 (K) : f (k) = f (k) ∀ ∈ L} L2 (K) closed

with all left L-invariant functions, and consider the orthogonal projection π : L2 (K) → L2 (S) deﬁned via averaging over L:

dt f (k).

(πf )(ek) = L

Note that in this setting we write S as a left quotient space, endowed with a right K-action. Proposition 5.1. For symmetric domains, the Szeg˝ o distribution dE(s) = N (s, e)−d/r on S deﬁnes via convolution the orthogonal projector E : L2 (K) → H 2 (S). π

P

Proof. Consider the orthogonal projections L2 (K) −→ L2 (S) −→ H 2 (S). The substitutions t = e · τ and k = στ −1 yield (P ◦ π) f (e · σ) = dt N (e · σ, t)−d/r (πf )(t) = dτ N (e · σ, e · τ )−d/r d f (τ ) S

K

−d/r

dτ N (e · σ, e · τ )

= =

dτ N (e · στ

f (τ ) =

K

L

−1

−d/r

, e)

f (τ )

K

dk N (e · k, e)−d/r f (k −1 σ) = (N (−,e)−d/r f )(σ).

K

In this special case, the binomial formula yields the expansion d Km (s, e) N (s, e)−d/r = r m m1 ≥···≥mr ≥0

into K-spherical functions.

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We will now consider “non-commutative” Hardy spaces more generally. If K is a compact Lie group with involution σ : K → K and K σ = {k ∈ K : σ(k) = k} denotes the ﬁxed point subgroup, then S = K σ \ K = Kσ is a compact symmetric space, and we have the Peter-Weyl decompositions Kα ⊗ Kα , L2 (K) = ˆ α∈K

ˆ is the unitary dual of K (irreducible representations), and where K L2 (S) = Kα ⊗ φα , ˆσ α∈K

where ˆ σ = {α ∈ K ˆ : Kα contains a non-zero K σ -invariant vector} Sˆ = K is the subset of all spherical representations. Here Kα denotes the (ﬁnite-dimenˆ In terms of the Lie algebra decomposition sional) representation space of α ∈ K. k = kσ ⊕ kσ , consider a maximal torus t = tσ ⊕ tσ , where kσ ⊃ tσ = i a is a maximal abelian “Cartan subspace”. Deﬁne r := dim tσ to be the rank of S. Then there exist natural embeddings as discrete subsets ˆ → i t = L(t, i R) (highest weight) K and Sˆ → a (restricted highest weight). Now let V ⊂ a be an arbitrary open polyhedral cone. The basic result for Fourier analysis on H 2 (S) is the following [L1], [L2]. Theorem 5.2. Let V ⊂ a denote the (closed) dual cone. Then the Hardy space corresponding to the (open) cone V has a Peter-Weyl decomposition HV2 (S) = Kα ⊗ φα . ˆ α∈S∩V

Moreover, the Cauchy-Szeg˝ o distribution can be expanded into a series EV (s) = φα (s) ˆ α∈S∩V

of spherical functions, and HV2 (S) can be realized via boundary values of holomorphic functions on the K-invariant domain DV = K exp (V )(e) ⊂ S C .

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Here the complexiﬁcation S C = (K σ )C \ K C exp

is a complex manifold and, choosing a “Weyl chamber” t+ σ ⊂ tσ ⊂ k −→ K, we obtain a polar decomposition S C = K exp (i tσ ) and may deﬁne the complex phase space (“non-commutative tube domain”) by DV := K exp (V ) where V ⊂ t+ σ is an open polyhedral convex cone. Letting Λ = V ⊂ i tσ be the polar cone, the “non-commutative” Hardy space HV2 (S) := Kα ⊗ φα ˆ σ ∩Λ α∈K

plays the role of “Hilbert state space” in this setting. Theorem 5.3. If DV is pseudoconvex (i.e., a domain of holomorphy), then one may identify HV2 (S) = {h ∈ L2 (S) : h holomorphic on DV }. Using Proposition 5.1, we show next that the Hardy space of hermitian symmetric domains (at least of tube type) ﬁts into the more general framework. In fact, let K be the compact Lie group of Jordan (triple) automorphisms, with involution subgroup K σ consisting of all Jordan algebra automorphisms. In this case, on the Lie algebra level, kσ corresponds to the Jordan multiplication operators and the Cartan subspace i tσ is obtained by diagonal multiplication operators, isomorphic to Rr via spectral theory. The associated cone V = Rr> is precisely the positive octant, a very special case of a polyhedral cone. The Hardy space decomposition ⊕ Pm1 ,...,mr H 2 (S) = m1 ≥···≥mr ≥0

into irreducible K-modules labelled by integer partitions, with Pm1 ,...,mr , generated by the highest weight vector N1 (z)m1 −m2 N2 (z)m2 −m3 · · · Nr (z)mr , where N1 , . . . , Nr are the Jordan theoretic minors ⎞ ⎛ N1 ⎟ N2 ⎜ ⎟ ⎜ ⎟, ⎜ N3 ⎝ ⎠ .. .. . .

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is now understood by realizing the partitions within the dual cone of V . Moreover, the Cauchy-Szeg˝ o kernel N (z, w)−d/r = det (I − zw∗ )−d/r will now be interpreted as a convolution kernel on the Shilov boundary. In principle, the same could be done for the Bergman kernel N (z, w)λ = det (I − zw∗ )−λ , for parameter λ > p − 1. Thus also the Berezin quantization could be realized on the Shilov boundary. Returning to the general case of a compact symmetric space S = L \ K, let P : L2 (S) → HV2 (S) be the orthogonal projection, realized as a convolution operator (PV h)(s) := EV (st−1 ) h(t) dt, K

where EV (s) =

φα (s)

ˆ σ ∩V α∈K

is the Cauchy-Szeg˝o distribution. We may regard KV (s, t) := EV (st−1 ) as a generalized “reproducing kernel”. If f ∈ C(S) is a continuous symbol function, let TS (f ) = PV f PV be the associated Toeplitz operator (depending on V ), given by TS (f ) h := PV (f h). These operators are in general non-commuting, but satisfy TS (f )∗ = TS (f ). The Toeplitz C ∗ -algebra TV (S) := C ∗ (TS (f ) : f ∈ C(S)) is the uniform closure of the algebra generated by all operators TS (f ) with continuous symbol f . In order to describe the generalization of Theorem 4.6 to the general case of compact Lie groups K, let again V be a polyhedral cone, with polar cone V realized in the (dual) Cartan subspace. The basic geometric idea, generalizing the boundary stratiﬁcation of symmetric domains, is as follows: For any face A of V there exists a subgroup KA (not necessarily closed) with Cartan subspace A spanned by the face. Consider the associated foliation K/KA of Kronecker type, which is Hausdorﬀ iﬀ KA is closed. + Theorem 5.4. There exist C ∗ -ideals IKA IK TV (S) in the Toeplitz C ∗ -algebra, A + with subquotient IKA /IKA ≈ C ∗ (K/KA ) isomorphic to the foliation C ∗ -algebra.

The proof is based on C ∗ -duality theory [LPRS], [NT]. As a ﬁrst step, TV (S) is identiﬁed with a corner of the co-crossed product Cˆ ∗ (K) ⊗ C 0 (K) δ

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for a suitable co-action δ on a “completed” C ∗ -algebra Cˆ ∗ (K). On the other hand, the foliation C ∗ -algebra C ∗ (K/KA ) = C0 (K) ⊗ C ∗ (KA ) can be realized as a crossed product [CO]. In view of the well-known Katayama duality K(L2 (K)) = C ∗ (K) ⊗ C 0 (K) δ

one obtains C ∗ -representations of TV (S) on L2 (KA ). This is of course the crucial step, and we give a sketch of the construction. Let T (A) be the tangent space of ˜ such that T (A) ⊂ m ˜ is maximal A. Then there exists a Lie subalgebra ˜k = ˜l ⊕ i m, abelian. Now consider the (non-closed) analytic subgroup KA := exp ˜k = Kc × Ke , which can be factored into a compact times Euclidean part. o distribution relative to the face A. Then there Theorem 5.5. Let EA be the Szeg˝ exists a C ∗ -representation σ

A TV (S) −→ L(L2 (KA )),

which maps the Toeplitz operator T (f ) to TA (fA ), where fA := f |KA is the restriction and TA is the Toeplitz operator deﬁned relative to EA . Proof. Let Cˆ ∗ (K) be the C ∗ -algebra generated by all left convolution operators λgE with g ∈ A(K), the Fourier algebra of K. Then there exist representations A(K) f

−→ L(L2 (KA )) → fA

and

Cˆ ∗ (K) −→ L(L2 (KA )) → λgA EA . λgE The latter representation is best described via the Fourier decomposition W (K) ∼ = ∗

(∞)

L∞ (Kα ),

ˆ α∈K

with λu → u ˆα =

u(s)∗ sα ds

K

denoting the Fourier coeﬃcient. Here sα denotes the action of s ∈ K for the representation α. Now let g(s) = (φ|sγ ψ) be a matrix coeﬃcient function, with φ, ψ ∈ Hγ belonging to a not necessarily irreducible K-representation. Then an / ˆ ˆ (γ ⊗ α) ψ. Now let β ∈ K ˆ A , and let α ∈ K easy computation shows gE(α) = φ∗ E ˜ β ⊂ Kα as a KA -invariant be a sequence such that α|KA β. Realizing K / subspace, one shows that gE(α| ˜ β ) g A EA (β). In this way the representation K is constructed via a limiting argument. For more details, cf. [U5], [W].

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Once the representations σA are constructed, Theorem 5.4 can be expressed as follows: Deﬁning 0 + IA = Ker σA IA = Ker σB B A

it follows that

+ IA /IA ∼ = C ∗ (FA ),

where FA = K/KA is the Kronecker foliation associated with the non-closed subgroup KA . This C ∗ algebra is of type I iﬀ KA is closed. The “local version” can be reformulated in the following global version: There exists a C ∗ -ﬁltration K = I1 I2 · · · Ir TV (S), such that for every k ⊕ ∼ Ik+1 /Ik = C ∗ (FA ). codim A=k

In the special case of symmetric domains, the composition factors Ik+1 /Ik ∼ = C(Sk ) ⊗ K of the ﬁltration are even realized as ﬁbre bundles over the compact manifold Sk of all tripotents (partial isometries) of rank k. Hence in the case of symmetric domains we obtain a ﬁbration of boundary faces instead of a foliation in the more general setting. A few remarks about the associated index theory. In full generality the analytic index is given by a mapping ⊕ Ind K1 C ∗ (FB ) −→A K0 (C ∗ (FA )), BA

and should be related to a longitudinally elliptic operator DA along the foliation FA . The details of this construction have only been worked out for a basic class of non-symmetric domains, the so-called Reinhardt domains. In this case K = S = Tn is the n-torus, whereas L = {1}. The complexiﬁcation S C = (C \ {0})n ⊂ Cn is open and dense. A domain D ⊂ Cn is called Reinhardt iﬀ (z1 , . . . , zn ) ∈ D =⇒ (ei t1 z1 , . . . , ei tn zn ) ∈ D. Let |D| = {(|z1 |, . . . , |zn |) : z ∈ D} ⊂ Rn+ be the associated “absolute” domain and deﬁne V = log |D| ⊂ Rn− ,

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assuming that D is contained in the unit polydisk. Important examples, for n = 2, are the L-shaped domains and the so-called Hartogs’ wedge, shown in Figure 2. Figure 2. L-shaped domains and the Hartogs’ wedge |z2 |

|z2 |

T2

2 (1, 1) T

T2 log |z2 |

|D| log |z1 |

V = log |D|

|D| log |z2 |

|z1 |

|z1 |

slope − θ1

V = log |D|

log |z1 |

Via groupoid methods, Toeplitz operators on Reinhardt domains have been studied in [CM], [SSU]. In the special case of Hartogs’ wedge, one obtains the following Theorem 5.6. [SSU]: Let V = VΘ1 , Θ2 be the convex cone with slopes Θ1 and Θ2 , coming from Hartogs’ wedge. Then the Toeplitz C ∗ -algebra has a 2-step ﬁltration K I2 T (T2 ). Here T (T2 )/I2 = C(T2 ) is the commutator ideal, and I2 /K is stably isomorphic to AΘ1 ⊕ AΘ2 , where AΘ = C ∗ (u, v : uv = e2πiΘ vu)

unitary:

denotes the irrational rotation algebra (non-commutative torus) induced by the Kronecker foliation T2 /RΘ (cf. Figure 3). Figure 3. The Kronecker foliation T2 /RΘ

Θ

Based on this C ∗ -algebra ﬁltration, one obtains a real-valued index theorem: tr

Z2 ≈ K 1 (T2 ) −→ K0 (AΘ ) −→ R, ≈

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89

corresponding to the dense embedding (m, n) → m + n Θ. These results have also applications to complex analysis, for example the existence of smooth (Reinhardt) domains with non-compact ∂-Neumann operator, and a characterization of proper holomorphic maps between Reinhardt domains in terms of their associated Toeplitz C ∗ -algebras. It is interesting to note that Reinhardt domains are closely related to multi-variable Wiener-Hopf operators [MR], [U12], by embedding higher-dimensional cones in compact spaces via the Cayley transform.

6. Quantization of Hermitian symmetric domains A diﬀerent area of interaction between analysis on symmetric domains and Hilbert space operator theory concerns quantization methods in the sense of mathematical physics. Let D = G/K be a hermitian symmetric space of non-compact type, realized as the open unit ball of a Jordan triple Z = Cd . The group G = {biholomorphic automorphisms of D} = Aut(D) is a semi-simple Lie group, with maximal compact subgroup K = {Jordan triple automorphisms} = Aut(Z) ⊂ GL(Z). Example. In case D = {z ∈ Cp×q : z < 1} is the matrix ball for the operator norm, Z = Cp×q has the Jordan triple product uv ∗ w + wv ∗ u , 2 and we obtain the Moebius transformations ab G = SU (p, q) (z) = (az + b)(cz + d)−1 , cd whereas K = S(U (p) × U (q)) consists of the linear isometries. Here r = min (p, q) is the rank of D. {uv ∗ w} =

Example. In case D = {z ∈ Cr×r : z < 1, z t = z} consists of symmetric matrices, the tangent space Z = {z ∈ Cr×r : z t = z} becomes a Jordan sub-triple of Cr×r , and G = Sp (2r, R) ⊃ K = U (r). In the following we will also consider the ﬂat case D = Cn ≈ T ∗ (Rn ), identiﬁed with the cotangent bundle. Here G = U (n) Cn is a semi-direct product.

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The quantization Hilbert spaces are the weighted Bergman spaces described in more detail in Section 3. Consider the G-invariant measure dµ0 (z) = N (z, z)−p dz, where N (z, w) denotes the “Jordan triple determinant” and p is the genus. Example. For the matrix ball, N (z, w) = Det (I − zw∗ ) and the genus is p + q. Deﬁne the scalar holomorphic discrete series as follows. For ν > p − 1, ΓΩ (ν) N (z, z)ν−p dz π d ΓΩ (ν − d/r) is a probability measure and the associated weighted Bergman space dµν (z) =

Hν2 (D) := {ψ ∈ L2 (D, µν ) : ψ holomorphic} has the projective unitary representation (Uν (g −1 )ψ)(z) = (Det g (z))ν/p ψ(g(z)) and the reproducing kernel ν Kν (z, w) = N (z, w)−ν = Kw (z).

Accordingly, we have

ψ(z) =

dµν (w) Kν (z, w) ψ(w)

∀ ψ ∈ Hν2 (D).

D

For the ﬂat case,

ν d

e−ν(z|z) dz π denotes the Gauss measure, with corresponding Segal-Bargmann-Fock space dµ (z) =

H2 (Cn ) := {ψ ∈ L2 (Cn , dµ ) : ψ holomorphic} and its unitary representation (π (tb ) ψ)(z) = eν(z|b)−ν(b|b)/2 ψ(z − b) for the translations tb induced by b ∈ Cn . Here = ν1 plays the role of Planck’s constant. By deﬁnition [AU4] a covariant functional calculus (or “quantization”) is determined by a densely deﬁned ∗-linear mapping L2 (D, dµ0 )

A

−→

∗−linear

L2 (Hν2 (D)) (Hilbert-Schmidt operators),

associating to f the operator Af with active symbol f . The covariance condition can be expressed as Af ◦g−1 = Uν (g) Af Uν (g)−1 .

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The adjoint mapping A∗ : L2 (Hν2 (D)) −→ L2 (D, dµ0 ), associating to an operator T the passive symbol A∗T of T , is formally deﬁned by duality dµ0 (w) A∗T (w) f (w) = tr T ∗ Af . D

Composing both mappings, we obtain the link transform (or generalized Berezin transform) B = A∗ A : L2 (D, dµ0 ) → L2 (D, dµ0 ), which is a G-invariant pseudo-diﬀerential operator. It is often convenient to express the quantization by an integral representation Af = dµ0 (w) f (w) Aw , D

where w → Aw is a covariant ﬁeld of (bounded) operators, satisfying the covariance condition Ag(w) = Uν (g) Aw Uν (g)−1 . For any (non-compact) Hermitian symmetric space, we have the Plancherel decomposition dλ 2 Gλ L (G/K) = |c(λ)|2 Rr

into a direct integral of principal series representations πλ, where c(λ) is HarishChandra’s c-function [HE]. Here, by deﬁnition, the representation space Gλ contains a spherical vector φλ, explicitly given by φλ = dk πλ(k) eλ. K

For symmetric domains of tube type, eλ(z) = N (z + z ∗ )λ+ρ denotes the conical functions for λ = (λ1 , . . . , λr ), where as before we put N (x)λ = N1 (x)λ1 −λ2 N2 (x)λ2 −λ3 · · · Nr (x)λr and N1 , N2 , . . . , Nr = N are the Jordan algebraic minors. The half-sum ρ of positive roots has the components 1+b a (j − 1) + . 2 2 Now let A be a G-covariant calculus. The Berezin transform ρj =

σ ◦ A = A∗ ◦ A

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is G-invariant and hence, by diagonalization, ∗ A(λ) · id σ ◦ A|Gλ = A Gλ , ∗ A(λ) = σ(A where A eλ )(0) is the λ-eigenvalue. For the spectral resolution of Berezin-type transforms it suﬃces to determine these eigenvalues or, equivalently, to express σ ◦ A in terms of some set of generators ∆1 , ∆2 , . . . , ∆r of the algebra of G-invariant diﬀerential operators on D, where ∆1 is the Laplace-Beltrami operator. The primary example of a covariant quantization is the Toeplitz-Berezin calculus

T ν : L∞ (D) −→ L(Hν2 (D))

(bounded operators),

associating to f the Toeplitz operator Tfν deﬁned via Tfν ψ = Pν (f ψ), where Pν : L2 (D, dµν ) → Hν2 (D) is the orthogonal projection [B2], [B3]. Its integral representation has the form ν (Tf ψ)(z) = dµν (w) Kν (z, w) f (w) ψ(w). D

Therefore the adjoint (TT∗ )(w) =

ν ν (Kw |T Kw ) Kν (w, w)

coincides with the Berezin symbol and the deﬁning operator ﬁeld Twν is given by ν the rank 1 projection onto C · Kw . It follows that the Berezin transform f → Bf = ∗ T T f has the integral representation Kν (z, w) Kν (w, z) f (w), (Bf )(z) = cν dµ0 (w) Kν (z, z) Kν (w, w) D

where cν is chosen so that B1 = 1. In the ﬂat case of the Segal-Bargmann-Fock space H2 (Cn ), it has been shown in [G] that Bf = e−∆ f coincides with the heat semi-group. In the curved setting, we have instead [BLU], [E]: Theorem 6.1. For any bounded symmetric domain, the Toeplitz quantization satisﬁes the correspondence principle in the following sense (i) Tfν Tgν − Tfνg → 0 ν → 0. (ii) ν[Tfν , Tgν ] − i T{f,g} Here {f, g} denotes the G-invariant Poisson bracket.

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For the explicit calculation of the eigenvalues of the Toeplitz-Berezin transform, we pass to the Siegel domain realization Z = U ⊕ V , via the biholomorphic Cayley transform D −→ {(u, v) ∈ Z : u + u∗ − {ev ∗ v} ∈ Ω} C

≈

onto a symmetric Siegel domain. Here U = X C is the complexiﬁcation of a Euclidean Jordan algebra, x ◦ y denotes the Jordan product and Ω = {x2 : x ∈ X invertible} is the positive cone of X. In general, V is a complex representation space of X, with V = {0} corresponding to the tube type. The Cayley transform is given by √ C(u, v) = ((e + u) ◦ (e − u)−1 , 2 v ◦ (e − u)−1 ), where e is the unit element of X. By the classiﬁcation, we may identify X with the algebra Hr (K) of all self-adjoint (r×r)-matrices over K, under the anti-commutator . The associated matrix entries belong to product x ◦ y = xy+yx 2 ⎧ r≥4 ⎨ R, C, H R, C, H, O r = 3 exceptional K= ⎩ Euclidean space r = 2 spin factors, and we call a = dimR K the characteristic multiplicity. Using the conical functions N s (x) = N1 (x)s1 −s2 N2 (x)s2 −s3 · · · Nr (x)sr ,

the Euler integral r d1 −r . a ΓΩ (s) = dx N (x)−d1 /r N s (x) e−(x|e) = (2π) 2 Γ (sj − (j − 1)) 2 j=1 Ω

is called the Koecher-Gindikin Γ-function and d1 = d − rb is the dimension of X. Using this fundamental concept, the Toeplitz-Berezin eigenvalues have been computed in [UU], yielding the beautiful formula T ∗ T (λ) =

ΓΩ (ρ + ν −

d r

+ λ) ΓΩ (ρ + ν −

ΓΩ (ν −

d r ) ΓΩ

(ν)

d r

− λ)

,

where ρ denotes the half-sum of positive roots. A second example of a covariant calculus is the Weyl quantization W : L2 (D, dµ0 ) −→ L2 (Hν2 (D)), associating to f the Weyl operator Wf deﬁned via its integral representation Wf = dµν (w) f (w) Uν (sw ) D

[UU1], [UU2]. Here sw ∈ G denotes the geodesic symmetry, uniquely determined by sw (w) = w, sw (w) = −Id.

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H. Upmeier Figure 4. The geodesic reﬂection

sw (z) w z

By duality, we obtain the Weyl symbol WT∗ (w) = tr T Uν (sw ). In the ﬂat case, we have the formulas sw (z) = 2w − z, s0 = −id, sw = tw ◦ s0 ◦ t−w which induce for the complex wave Hilbert space H2 (Cn ) the representations = e(z|b)−(b|b)/2 ψ(z − b) and

(π(tb )ψ)(z)

= ψ(2w − z) e2(z−w|w).

(π(sw )ψ)(z) This yields the explicit form (WfC ψ)(z) =

dw f ( Cn

z + w (z−w|z+w)/2 )e ψ(w). 2

Remark 6.2. Both in the setting of bounded symmetric domains and for the ﬂat case, the quantization Hilbert spaces have also a real representation via a socalled Segal-Bargmann transform. For the real wave Hilbert space L2 (Rn ), the representations (π(ty,η ) φ)(x)

=

ei(y−2x)η φ(x − y),

π(sy,η φ)(x)

=

e4i(y−x)η φ(2y − x)

lead to (WfR

ψ)(x) =

dy Rn

dη f

Rn

x+y ,η 2

e2i(y−x)η φ(y).

It should be noted that the well-known Moyal ∗-product arises this way. In order to determine the eigenvalues of the link transform for the Weyl calculus, let G = KAN be the Iwasawa decomposition, and for λ ∈ a let eλ denote the conical function of G/K, with φλ = dk k eλ K

the associated spherical function. Realized as a Siegel domain G/K = {(u, v) : u + u∗ − 2{ev ∗ v} ∈ Ω},

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the conical functions have the form eλ(u, v) = N λ+ρ (u + u∗ − 2{ev ∗ v}). Taking o = (e, 0) as a base point, we ﬁrst apply the so called product formula [AU4], [AU6], which asserts that whenever A, B are two covariant operator calculi on Hν2 (D), the eigenvalues of the link transform can be expressed as ∗ B (λ) = A

(Aeλ Ko |Ko )(Ko |Beλ Ko ) ∗ T (λ) T

.

Hence it suﬃces to compute (Aeλ Ko |Ko ) = (Aφλ Ko |Ko ). After this simpliﬁcation, the Weyl transform eigenvalues (at least for domains of rank 1) have been determined in [AU6]: Theorem 6.3. Let D = {z ∈ Cd : (z|z) < 1} be the unit ball of rank 1. Then the Weyl calculus satisﬁes Γ(ν − d2 + λ)Γ(ν − d2 − λ) d−ρ+λ d−ρ−λ α , (Ko |Weλ Ko ) = 2 F1 Γ(ν)Γ(ν − d) α−1 ν where 2 F1 is the Gauss hypergeometric function. Note that the product of Γ-factors, analogous to the Toeplitz-Berezin case, is still present, but has to be supplemented by the hypergeometric function. It is expected that a similar phenomenon holds for higher rank and also for more general functional calculi. A common generalization of both the Toeplitz-Berezin calculus and the Weyl calculus is the so-called interpolating calculus [AU7] which depends on an additional parameter α. For the Fock space we assume α ∈ C, whereas for bounded symmetric domains we assume that α ∈ B, the closed unit disk. Consider the map z → sα (z) := αz on the domain D. By conjugation, we deﬁne for any w ∈ D, α −1 sα w := g s g where g ∈ G satisﬁes g(0) = w. The interpolating calculus for parameter α is deﬁned by the local operator ﬁeld α α (Aα w ψ)(z) := ψ(sw (z)) j(sw (z))

where j denotes the Jacobian cocycle. In the ﬂat case D = Cd we have sα w (z) = αz + (1 − α) w and therefore ν(1−α)(z−w|w) . (Aα w ψ)(z) = ψ(αz + (1 − α) w) e

If α = 0, (A0w ψ)(z) = ψ(w) eν (z−w|w)

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and one obtains the Toeplitz calculus ν d (A0f ψ)(z) = dw f (w) ψ(w) e−ν(w|w) eν(z|w) . π Cd

If α = −1,

(z) = 2w − z s(−1) w is the geodesic symmetry and one obtains the Weyl calculus ν (−1) (Af ψ)(z) = ( )d dw f (w) ψ(2w − z) e−2ν(w|w) e2ν(z|w) . π Cd

It is shown in [AU4] that for α → ∞, the Wick calculus, formally given by hk → Th Tk∗ , is obtained via a suitable limit procedure. Theorem 6.4. For the interpolating α-calculus, in the ﬂat case, the link transform is given by 1 − |α|2 ∆ , A∗α Aα f = exp |1 − α|2 ν where ∆ is the Laplacian. The general set-up for Moyal products of quantized operators (also called ∗-products) is as follows: Let A, B, C be covariant calculi and f, g ∈ C ∞ (D) be symbol functions. Then one has the quantized operators Af , Bg and deﬁnes the “weak” Moyal product by f • g := C ∗ (Af Bg ). This is in general not associative. In contrast, the “strong” Moyal product f g ∈ C ∞ (D), uniquely deﬁned by Cf g = Af Bg , is associative on the formal level, but in general there is no such function f g which yields the required operator product. In the Toeplitz operator calculus, one has instead an asymptotic expansion [E] 1 Tf Tg = TCk (f,g) , νk k≥0

where Ck (f, g) are uniquely determined G-covariant bidiﬀerential operators. Therefore, in this case, 1 Ck (f, g). f g = νk k≥0

The Correspondence Principle [BLU] gives the following lower-order terms (i) Tfν Tgν − Tfνg → 0 implies that C0 (f, g) = f g, ν → 0 implies that C1 (f, g) is related to the Poisson (ii) ν [Tfν , Tgν ] − i T{f,g} bracket.

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In the ﬂat case on the Segal-Bargmann-Fock space one has a more or less complete theory of Moyal products, even in the general setting of the interpolating calculus. Let f, g, f g ∈ C ∞ (Cd , C) be symbol functions such that γ β Aα f Ag = Af g .

Then the Moyal product f g can be expressed as an integral (f g)(ζ) = dξ dη f (ξ) g(η) M(ξ, η; ζ). Cd

Cd

Equivalently,

Aα ξ

Aβη

dζ Aγζ M(ξ, η; ζ)

= Cd

for the corresponding operator ﬁelds. The following result appears in [AU7]. Theorem 6.5. For α, β, γ in the unit disk, subject to the restrictions αβ = γ and Re

(1 − γ)2 (1 + αβ) > 0, (αβ − γ)(1 + γ)

the 3 × 3 matrix ⎡

(1 − α)(β − γ) (1 − α)(1 − β) γ Φ = ⎣ (1 − α)(1 − β) (1 − β)(α − γ) (γ − 1)(1 − α) (γ − 1) α (1 − β)

⎤ (γ − 1)(1 − α) β (γ − 1)(1 − β) ⎦ (1 − γ)(1 − αβ)

has all rows and columns summing up to 0, and the integral kernel for the Moyal product is given as follows ⎛ ⎞ ξ ν (ξ, η, ζ) Φ ⎝ η ⎠ . M(ξ, η; ζ) = exp αβ − γ ζ Note that M is invariant under U (d) × Cd , acting jointly on ξ, η, ζ.

7. Deformation of real symmetric domains It turns out that similar results hold for the more general class of real symmetric domains, which are deﬁned as follows. Let D = G/K be a hermitian symmetric domain, with G a semi-simple Lie group of hermitian type, and let z → z be an antiholomorphic involution of D. Deﬁne the real form DR = {z ∈ D : z = z} = GR /KR , where GR = {g ∈ G : g(z) = g(z)} is a reductive Lie group, with maximal compact subgroup KR = K ∩ GR .

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Example. If D = T (Ω) = Ω + iX is a tube domain, the involution (x + iy)∗ = x − iy is the hermitian adjoint, and DR = Ω considered as a real symmetric domain. In this case, GR is only reductive with a non-trivial center. Example. If D = {z ∈ Cp×q |z ∗ z < I} = U (p, q)/U (p) × U (q) is the complex matrix ball, and z → z = (z ij ) is the conjugation, then DR = {x ∈ Rp×q |x∗ x < I} = O(p, q)/O(p) × O(q) coincides with the real matrix ball. Figure 5. Complex and real symmetric domains DC DC = T (Ω)

DR

DR = Ω

− d −c , one Example. For D = {z ∈ C2×2 : z ∗ z < I} and the involution ac db := −b a can show that DR ⊂ H is realized as the quaternion unit ball. Example. (product case) In case DR = GR /KR is itself hermitian symmetric, the complexiﬁcation D = DR × D R is endowed with the ﬂip involution (z1 , z 2 )− := (z2 , z 1 ) and G = GR × GR contains GR as the “diagonal” subgroup. We now introduce the “real” version of a covariant calculus. Let D ⊃ DR be a real symmetric domain, irreducible of rank r, with complexiﬁcation D not necessarily irreducible. Let Hν2 (D) denote the νth Bergman space over D, in case D is irreducible, whereas in the product case we consider Hν2 (DR × DR ) = Op Hν2 (DR ). Here one could also take pairs (ν1 , ν2 ) not necessarily equal. In both cases we obtain the restricted representation Uν |GR which is not irreducible but has a Plancherel decomposition which is multiplicity free. By deﬁnition [AU3], a covariant calculus in the “real” setting is a densely deﬁned linear map A : C ∞ (DR ) → Hν2 (D)

Toeplitz Operator Algebras and Complex Analysis

99

associating to f a holomorphic function Af , subject to the covariance condition Af ◦g−1 = Uν (g) Af ∀ g ∈ GR . The inﬁnitesimal version corresponds to an integral representation Af = dµR 0 (ζ) f (ζ) Aζ , DR

where Aζ ∈ O(D) is a ﬁeld of “states” satisfying the covariance condition Ag(ζ) = Uν (g) Aζ , g ∈ GR . µR 0

Here is a (suitably normalized) GR -invariant measure on DR . Similar to the complex case, the adjoint transformation A∗ : Hν2 (D) → L2 (DR , dµR 0) is deﬁned by duality ∗ ∗ dµR 0 (ζ) f (ζ) AT (ζ) = (f |AT ) = (Af |T )Hν2 (D) . DR

By composition we obtain the link transform B = A∗ A acting on C ∞ (DR ) which commutes with GR -translations. As in the complex setting, the primary example of a real deformation is the Toeplitz calculus [AU3], [Z] T : C ∞ (DR ) → Hν2 (D), which associates to f the holomorphic function Tf = dµR 0 (ζ) f (ζ) Tζ DR

explicitly given by Tζ (z) =

Kν (z, ζ) ∈ Hν2 (D), ζ ∈ D. Kν (ζ, ζ)1/2

The normalization of µR 0 is chosen so that T1 = I is the identity operator, and hence depends on ν and also on the type of covariant calculus. The adjoint transformation T ∗ : Hν2 (D) → L2 (DR , µR 0) coincides with the so-called weighted restriction (T ∗ h)(ζ) = Kν (ζ, ζ)1/2 h(ζ). In the product case D = DR × DR , one computes easily that Tζ,ζ (z, w) =

Kν (z, w; ζ, ζ) Kν (z, ζ) Kν (ζ, w) = 1/2 Kν (ζ, ζ) Kν (ζ, ζ; ζ, ζ)

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H. Upmeier

is just the integral kernel of kζ ⊗ kζ∗ . Hence we recover the “complex” Toeplitz calculus. In order to determine the eigenvalues of the Toeplitz-Berezin transform in the real setting consider, as before, the exponentials λ+ρ

eλ (x + y, v) = NX

(x − {ev ∗ v})

and the corresponding Euler type integral Kν (e, ξ) ∗ T (λ) = T (e) = eλ (ξ) dµR T eλ 0 (ξ) Kν (ξ, ξ)1/2 =

dx

dy

BR

λ+ρ (x − {ev ∗ v}) dv NX

−ν NX⊕Y (e + x − y) −ν/2

NX⊕Y (x − {ev ∗ v})

.

The following result [DP], [N], [Z], [AU3] is the real version of the UnterbergerUpmeier formula: Theorem 7.1. Let DR = {x + y + v ∈ X ⊕ Y ⊕ V : x − {ev ∗ v} ∈ Ω} be a real symmetric domain, realized as a real Siegel domain, where U =X ⊕Y is a semi-simple real Jordan ∗-algebra, the self-adjoint part X = {x ∈ U : x∗ = x} is a Euclidean irreducible Jordan algebra and Ω ⊂ X denotes the symmetric cone. Moreover, Y = {y ∈ U : y ∗ = −y}. In this context, the Toeplitz deformation eigenvalues have the form ∗ T (λ) = T

ΓΩ (κ + λ) ΓΩ (κ − λ) ΓΩ (κ + ρ) ΓΩ (κ − ρ)

where ρ is the half-sum of positive roots and κ =

ν 2r

rank (U ) + ρ +

d r

− p.

In order to treat more general deformations, we need the “product formula” for real symbolic calculi [AU3], [AU6]: Theorem 7.2. Let A, B : C ∞ (DR ) → Hν2 (D) be two covariant symbolic calculi, acting on Hν2 (D). The link transform 2 R A∗ B : L2 (DR , µR 0 ) → L (DR , µ0 )

has the Plancherel eigenvalues Aeλ (0) Beλ (0) ∗ B (λ) = 4 A ∗ T (λ) T ∗ T (λ) > 0 is independent of A, B. In particular for all λ ∈ a = Rr , where T ∗ A (λ) = A

1 |Aeλ (0)|2 . ∗ T T (λ)

Toeplitz Operator Algebras and Complex Analysis

Note that Aλ (0) =

101

∼ dµR 0 (ζ) φλ (ζ) σK0 (ζ) = σK0 (λ)

DR

coincides with the spherical Fourier transform of σK0 , K0 ∈ Hν2 (D) being the symbol of the kernel vector at the origin, which is a KR -invariant function. On a somewhat deeper level one can also deﬁne a “real” Weyl calculus W : C ∞ (DR ) → Hν2 (D), which associates to f the holomorphic function Wf = dµR 0 (ζ) f (ζ) Wζ DR

given by Kν (sζ (z), z)1/2 Kν (z, ζ)1/2 , Kν (sζ (z), ζ)1/2 where sζ ∈ GR is the symmetry about ζ ∈ DR ⊂ D. In the ﬂat case D = Cn , DR = Rn , the analogous mapping Wζ (z) =

Wν : L2 (Rn ) → Hν2 (Cn ) is the (unitary) Bargmann transform, given explicitly by ν n/2 ν (Wν f )(z) = dζ f (ζ) exp (2ν(z|ζ) − (z|z) − ν(ζ|ζ)). π 2 Rn

In order to compute the Weyl transform eigenvalues, let m = (m1 ≥ m2 ≥ · · · ≥ mr ≥ 0) be an integer partition, and deﬁne the Pochhammer symbol ΓΩ (α + m) ΓΩ (α) and the multivariable hypergeometric function (α)m (β)m αβ (x1 , . . . , xr ) = Jm (x1 , . . . , xr ). 2 F1 (γ)m (1)m γ m (α)m =

Here Jm are the well-known Jack polynomials [ST], which are proportional to the spherical functions on the symmetric cone Ω regarded as a real symmetric domain of root type (A), restricted to the “diagonal” C e1 + · · · + C er for a Jordan algebra frame e1 , · · · , er of X. The reason why only “discrete” parameters m occur lies in the fact that one really considers the “compact dual” S of Ω. For r = 1, the symmetric domains in K = R, C, H, O (m = 2, Cayley plane) can be expressed as a hyperbolic space m D = {x ∈ Km | xi x∗i < 1}. i=1

102

H. Upmeier Figure 6. Rank 1 real symmetric domains ZR ZC rC ρ

Rm Cm 1

Cm m Cm × C 2

m−1 4

m 2

Hm C2×2m 2 m + 12

O2 C16 V 2 11 2

In this case 2ρ + 1 = a+d 2 , where a = dim K. It follows that ZR is uniquely determined by d = dim ZR and ρ. Generalizing both the Toeplitz and the Weyl calculus, we may also consider two covariant calculi of interpolating type Aα , Aβ : C ∞ (DR ) → Hν2 (DC ) and the associated link transform A∗ α Aβ : C ∞ (DR ) → C ∞ (DR ) ∗ α Aβ (λ) for λ ∈ a . The which is uniquely determined by the eigenvalues A R product formula gives ∗ α Aβ (λ) = A

where

Aeλ (0) = DR

β Aα eλ (0) Aeλ (0) A0eλ (0)

dµR 0 (ζ) eλ(ζ) Aζ (0) =

dµR 0 (ζ) φλ (ζ) Aζ (0)

DR

and φλ is the spherical function. In view of the product formula, the following result [AU7], [AU9] solves the eigenvalue problem for rank 1 domains. Theorem 7.3. For real rank 1 domains DR we have d/2−ρ+λ d/2−ρ−λ α Γ (ν − ρ + λ) Γ (ν − ρ − λ) 2 F1 α−1 d/2+ν−2ρ R α Aeλ (0) = . d/2 d/2−2ρ α Γ (ν) Γ(ν − 2ρ) 2 F1 α−1 d/2+ν−2ρ For the special case of the Weyl calculus (α = −1) considered in [AU6] we have Theorem 7.4. For real symmetric domains of rank 1, the Weyl transform eigenvalues are determined by the formula d/2−ρ+λ d/2−ρ−λ 1 (2) Γ (ν − ρ + λ) Γ (ν − ρ − λ) 2 F1 d/2+ν−2ρ Weλ (0) = . d/2 d/2−2ρ 1 Γ (ν) Γ(ν − 2ρ) (2) 2 F1 d/2+ν−2ρ

Toeplitz Operator Algebras and Complex Analysis

103

Remark 7.5. In the complex setting ZR = Cm we have d/2 = 2ρ = m and obtain the simpler formula m Γ (ν − m m−ρ+λ m−ρ−λ α R α 2 + λ) Γ (ν − 2 − λ) Aeλ (0) = 2 F1 Γ (ν) Γ(ν − m) α−1 ν which generalizes Theorem 6.3 (for α = −1). Analogous to the complex setting, the real symmetric domains give rise to a so-called Moyal restriction [AU7]: For the Fock space and F ∈ C ∞ (Cd , C), the Moyal restriction F ∈ C ∞ (Rd , C) is deﬁned by C

R γ 2 d Aα F I = AF ∈ Hν (C ).

Writing

dw M(w, ζ) F (w),

(F )(ζ) = Cd

we have C

dζ R Aγζ M(w, ζ)

Aα wI = Rd

where I(ζ) = Kν (ζ, ζ)1/2 . Theorem 7.6. For α, γ in the unit disk, subject to the restrictions α2 = γ and Re the symmetric 3 × 3 matrix ⎡ ⎢ Φ=⎣

(1 − γ)2 (1 + α2 ) > 0, (α2 − γ)(1 + γ)

(1−α) γ 2 α−γ 2 α(γ−1) 2

α−γ 2 1−α 2 γ−1 2

⎤

α(γ−1) 2 γ−1 2

⎥ ⎦

(1 − γ)(1 +

α 2)

has all rows and columns summing up to 0, and ⎞ w ν(1 − α) M(w, ζ) = exp (w w ζ) Φ ⎝ w ⎠ α2 − γ ζ ⎛

for all w ∈ Cd , ζ ∈ Rd .

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8. Vector-valued Bergman spaces and intertwining operators A current area of research in harmonic and complex analysis on symmetric domains concerns vector-valued holomorphic functions. Let D = G/K be a hermitian symmetric domain, and let G = Aut (D) be its holomorphic automorphism group. G is a semi-simple Lie group of non-compact type, and K = {g ∈ G : g(0) = 0} is its maximal compact subgroup. As always, we realize D ⊂ Z as the unit ball of a Jordan triple with triple product {uv ∗ w}. Then K = Aut (Z) = {k ∈ GL (Z) : k {uv ∗ w} = {ku(kv)∗ kw}}. Example. If Z = Cp×q is the matrix triple, uv ∗ w + wv ∗ u {uv ∗ w} = 2 and D = {z ∈ Z : zz ∗ < I} is the matrix ball. Here G = SU (p, q) consists of all Moebius transformations ab (z) = (az + b)(cz + d)−1 cd 6 5 and K = a0 d0 : a ∈ U (p), d ∈ U (q) consists of all linear isometries z → a z d−1 . Example. In case Z = {z ∈ Cp×p : z t = z} is the Jordan subtriple of all symmetric matrices, G = Sp (2p, R) is the symplectic group and K = U (p). One frequently 0} called Siegel’s uses the unbounded realization D = {z = z t ∈ Cp×p : z + z ∗ half-space. Not let E be a unitary representation space for K, with dimC E < ∞. There π is a homomorphism K −→ U (E) into the corresponding unitary group. Example. For K = SU (2), putting E = z1m z2n−m : 0 ≤ m ≤ n = SU (2)n yields all irreducible representations. Example. If K = U (k) and E = Ck is the deﬁning representation, the exterior powers E = Λj Ck , 0 ≤ j ≤ k, are called the fundamental representations. For the group SU (k), the well-known Schur functor construction Em for integer partitions m = m1 ≥ m2 ≥ · · · ≥ m k ≥ 0 of length ≤ k yields all irreducible representations. Example. If K = U (p) × U (q) is the direct product, we may consider the tensor product E = Ep ⊗ Eq∗ = L(Eq , Ep ) of all Hilbert-Schmidt operators, with its canonical U (p) × U (q)-representation, given by a0 (T ) = a T d∗ = πp (a) T πq (d)∗ 0d for T ∈ L(Eq , Ep ).

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In order to make contact with symmetric domains, let N : Z × Z → C be the Jordan triple determinant, which is given by N (z, w) = Det (I − zw∗ ) for matrices Z = Cp×q , and let B : Z × Z → End (Z) be the Bergman operator, denoted by z, w → B(z, w), and deﬁned as B(z, w) x = x − 2{zw∗ x} + {z{wx∗ w}∗ z}. Example. In the matrix case Z = Cp×q , it has been shown above that B(z, w) x = (1 − zw∗ ) x (1 − w∗ z) and hence B(z, w) = L1−zw∗ R1−w∗ z . Proposition 8.1. If z, w ∈ D, then B(z, w) ∈ K C ⊂ GL (Z). In particular, we obtain for matrices: 0 1 − zw∗ ∈ K C, B(z, w) = 0 (1 − w∗ z)−1 whereas in case of the unit disk, B(z, w) x = (1 − zw)2 x, and for the unit ball, B(z, w) x = (1 − (z|w))(x − (x|w) z). Recall that N (z, w) = det B(z, w)1/p . Now let π : K → U (E) be a unitary ﬁnite-dimensional representation, which is not necessarily irreducible. Then, for z ∈ D, π(B(z, z)) =: B(z, z)π is positive deﬁnite and we deﬁne as an induced representation the vector-valued Bergman space Hν2 (D, E) = {ψ : D → E holomorphic: dµ0 (z)N (z, z)ν (ψ(z)|B(z, z)π ψ(z)) < ∞}, D

where dµ0 (z) = N (z, z)−p dz = det B(z, z)−1 dz. If E = C, we recover the scalar-valued Bergman spaces. In case of the unit ball D ⊂ Cd , we have K = U (d) and the inner product has the form dz (1 − (z|z))ν−d−1 ((1 − (z|z))ψ(z)| (I − z ∗ z)π ψ(z)). ! " D

∈GL (d)=K C

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If ψ ∈ Hν2 (D, E) and φ ∈ E (corresponding to the constants), one has the reproducing property (φ|ψ(z))E

= (Bz−π φ|ψ)H 2 (D,E) = dµ0 (w) N (z, z)ν (B(w, z)−π φ|B(w, w)π ψ(w))E . D

This shows that the mapping D × D → GL (E), given by z, w → B(z, w)−π , plays the role of a matrix-valued reproducing kernel. The scalar-valued reproducing kernel N (z, w)−ν = det B(z, w)−ν/p arises as a special case, either by keeping ν and letting π be the trivial representation or, alternatively, by setting ν = 0 and considering the 1-dimensional representation K → U (1) given by k → (det k)ν/p . The full holomorphic discrete series, not necessarily scalar-valued, is related to the following construction. Lemma 8.2. For g ∈ G = Aut (D), z ∈ D the holomorphic derivative satisﬁes (i) g (z) ∈ K C ⊂ GL (Z) and the covariance property (ii) g (z) B(z, w) g (w)∗ = B(g(z), g(w)). Thus we may deﬁne the holomorphically induced representation πν,E : G → U (Hν2 (D, E)) via the formula (πν,E (g −1 ) ψ)(z) = det g (z)ν/p g (z)π [ψ(g(z))] ∈ E or, equivalently, as (πν,E (g −1 ) ψ)(z) = j(g, z)π ψ (g, z), where j : G × D → K C is the group-valued holomorphic cocycle j(g, z) = g (z). The cocycle property means j(g1 g2 , z) = j(g1 , g2 (z)) j(g2 , z). It is well known that for symmetric domains, the group K has a rich and interesting representation theory related to the Hua-Schmid decomposition introduced in Section 3. Let Z be an irreducible hermitian Jordan 7 triple, and consider the algebra P(Z) of all polynomials on Z, identiﬁed with sym Z , where Z is the dual space. Then K = Aut (Z) acts on P(Z) via (π(k) p)(z) = p(k −1 z). As shown in [S] there is a multiplicity-free K-decomposition P(Z) = Pm (Z) m

labelled by integer partitions m = (m1 ≥ m2 ≥ · · · ≥ mr ≥ 0)

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of length ≤ r = rank (Z). Moreover, in [U3] it has been shown that Pm (Z) has the highest weight vector Nm (z) = N1 (P1 z)m1 −m2 N2 (P2 z)m2 −m3 · · · Nr (Pr z)mr , where Nk is the kth Jordan algebra minor, and Pk is the kth Peirce projection. Example. We have Nm,0,...0 (z) = (z|e1 )m and Nm,m,...m (z) = N (z)m (for tube type domains). Combining the previous concepts, for the compact group U = K, and the holomorphic cocycle j : G × D → K C given by the holomorphic derivative j(g, z) = g (z) ∈ K C , we obtain a unitary irreducible representation πm : K → U (Pm (Z)) via translation (πm (k) p)(ζ) = p(k −1 ζ). The induced K-homogeneous multiplier representations are realized on the vectorvalued Bergman space Hλ2 (D, Pm (Z)) = {ψ : D → Pm (Z) holomorphic: dz N (z, z)λ−p (ψ(z, ζ)|ψ(z, B(z, z) ζ))Pm (Z) < ∞} D

where, as above, B(z, w) ζ = (1 − zw∗ ) ζ(1 − w∗ z) is the Bergman operator acting on Z. This space carries a projective unitary G-action, which is irreducible and belongs to the holomorphic discrete series λ (g −1 ) ψ)(z, ζ) = det g (z)λ/p ψ(g(z), g (z) ζ). (Um

More generally, let U be a compact Lie group such that K ⊂ U and let J : G × D → UC be a holomorphic multiplier (cocycle), denoted by g, z → J (g, z), which is holomorphic in z ∈ D and satisﬁes J (g1 g2 , z) = J (g1 , g2 (z)) · J (g2 , z). Let π : U → U (E) be a unitary irreducible representation with dimC E < ∞, and denote by π C : U C → GL (E) its holomorphic extension. Let o be the base point of D and choose a real analytic cross-section γz ∈ G, γz (o) = z. Then K(z, z) = J (γz∗ , o)π J (γz , o)π belongs to GL (E) via π C . We let K(z, w) ∈ GL (E) be its sesqui-holomorphic extension. The induced U -homogeneous multiplier representation is realized on

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the vector-valued Bergman spaces

H (D, E) = {Ψ : D → E holomorphic: 2

dµ0 (z)(Ψ(z)|K(z, z) Ψ(z))E D

dµ0 (z)(J (γz , o)π Ψ(z)|J (γz , o)π Ψ(z))E < +∞}

= D

endowed with the G-action (Uπ (g −1 ) Ψ)(z) = J (g, z)π Ψ(g(z)). An important special case is given by the symmetric Grassmann manifolds. It is well known [LO2] that GC = Aut (M ), where D ⊂ Z ⊂ M is the compact symmetric dual space. Consider the ﬁnite-dimensional space P m (Z), P n (Z) = m1 ≤n

with its reproducing kernel K n (ω, ζ) = Kωn (ζ) = N (ζ, −ω)n =

(−n)m (−1)|m | Km (ζ, ω).

m1 ≤n

Figure 7. The partitions occurring in P n (Z) n

r

Deﬁne a GC -action on P n (Z) via the kernel vectors γ −1 Kωn = det γ (ω)−n/p Kγn∗ (ω) . Here GC × M → M is the Moebius type action extending the action of G on D. Let tz ∈ GC be the translation, deﬁned by tz (ζ) = z + ζ and consider the “adjoint” (relative to the compact form U ⊂ GC ) t∗−z ζ = ζ z = B(ζ, z)−1 (ζ − {ζz ∗ ζ}), which is called the quasi-inverse. Example. For matrices Z = Cp×q , we have ζ z = (1 − ζ z ∗ )−1 ζ.

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A holomorphic cocycle G × D → GC , denoted by g, z → J (g, z), is deﬁned via C J (g, z) = t−1 g(z) g tz ∈ G .

The cocycle property J (g1 g2 , z) = J (g1 , g2 (z)) J (g2 , z) is easily checked. In this situation, we obtain the “Grassmann homogeneous” vector-valued Bergman space Ψ

Hν2 (D, P n (Z)) = {D −→ P n (Z) : D × Z z, ζ → Ψ(z, ζ), hol ν−p dz N (z, z) Ψ(z, B(z, z)1/2 ζ z )2 < +∞} D

since, according to [LO2], γz (ζ) = z + B(z, z)1/2 ζ z ∈ G deﬁnes a real-analytic cross-section, i.e., γz (0) = z, with the property J (γz , 0) ζ = B(z, z)1/2 ζ z . The unitary G-action has the form (Unν (g −1 ) Ψ)(z, ζ) = det g (z)ν/p det g (z + ζ)−n/p Ψ(g(z), g(z + ζ) − g(z)). Theorem 8.3. There is a multiplicity-free G-decomposition ⊕ Hν2 (D, P n (Z)) = Hν2 (D, Pm (Z)), n≥m1 ≥···≥mr ≥0

showing that the representation is not irreducible. Nevertheless, the Toeplitz C ∗ -algebra T generated by coordinate multiplications acts irreducibly on Hν2 (D, P n (Z)). Conversely, one may ask whether every irreducible T -module is of this kind? This is closely related to the problem of classiﬁcation of homogeneous holomorphic vector bundles [KM]. The explicit intertwining operators use the Faraut-Koranyi binomial formula N (z, w)−λ = det (1 − zw∗ )−λ = (λ)m K m (z, w) m

where e(z|w) =

K m (z, w)

m

is expanded into the series of reproducing kernels for Pm (Z). Here r .

a (i − 1))mi 2 i=1 partitions having the is the multi-Pochhammer symbol, and we consider the n+r r form n ≥ m1 ≥ m2 ≥ · · · ≥ mr ≥ 0. (λ)m =

(λ −

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In the simplest (scalar-valued) case the intertwiner is given by (−n)m (I f )(z, ζ) = Km (ζ, ∂z ) f (z). (λ)m m

9. Analytic continuation of Wallach parameters Another recent generalization of Bergman spaces concerns the analytic continuation of the Wallach set. Let Pm (Z) be the space of all polynomials of type m = m1 ≥ · · · mr ≥ 0, and let K C Pm (Z) be the irreducible action deﬁned by (h p)(z) = p(h−1 z). Consider the Bergman operator B : Z × Z → End (Z) deﬁned by B(z, w) = id − 2 z w∗ + Qz Qw where Qz ζ := {zζ ∗ z}. Then B(z, w) ∈ K C if z, w ∈ D. Example. For matrices Z = Cp×q the Bergman operator is given by B(z, w) x = (1 − zw∗ ) x (1 − w∗ z). As discussed already, the vector-valued Bergman spaces of type m Hν2 (D, Pm (Z))

F

= {D −→ Pm (Z) : D × Z z, ζ → F (z, ζ) hol dz N (z, z)ν−p (F (z, ζ)|F (z, B(z, z) ζ)ζ < ∞} D

carry the irreducible projective G-representation ν (Um (g −1 ) F )(z, ζ) = det g (z)ν/p F (g(z), g (z) ζ)

induced by the holomorphic cocycle G × D → K C coming from the complex derivative g (z) and satisfying the chain rule (g1 g2 ) (z) = g1 (g2 (z)) g2 (z). In Jordan theoretic terms, a real analytic cross-section gz ∈ G, satisfying gz (0) = z, can be realized in exponential form gz = exp ((v − zv ∗ z)

∂ ). ∂z

Then gz (0) = B(z, z)1/2 is a positive operator. The classical concept of Grassmann manifolds has the following generalization. Let Z be a hermitian Jordan triple, which is irreducible of rank r, and consider the compact manifold S of all tripotents of rank ≤ r. As a special case, Sr is the

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Shilov boundary. Two tripotents u ∼ v are called Peirce-equivalent iﬀ they have the same Peirce spaces 1 Zk (u) = Zk (v), k = 0, , 1. 2 The quotient space M = S / ∼ is called the th Jordan Grassmann manifold [LO2]. In general, it is a compact hermitian symmetric space under the semi-simple Lie group K := K/center. Example. If Z = Cr×s is the matrix triple for r ≤ s, S consists of all partial isometries, satisfying uu∗ u = u, of rank u = ≤ r. One can show that M ≈ Gr (Cr ) × Grs− (Cs ) is the product of “classical” Grassmannians. In general, let U = Z1 (u) ∈ M be the Peirce 1-space. Then the tangent space TU (M ) = Z1/2 (u) coincides with the Peirce

1 2 -space.

Therefore M carries the tautological bundle Z1 ↓ M

U ↓ U

and the tangent bundle Z1/2 ↓ M

Z1/2 (u) ↓ U = Z1 (u).

On the analytic side, let 1 (p|q) = (∂p q)(0) = d π

dz e−(z|z) p(z) q(z)

Z

be the Fischer-Fock inner product on P(Z), with reproducing kernel e(z|w) = K m (z, w) m

expanded into the kernels for Pm (Z). Example. We have P10··0 (Z) = Z (dual space) and K10··0 (z, w) = (z|w) coincides with the normalized K-invariant scalar product. On the other hand, if Z = X C is of tube type, then Pm···m (Z) = C · N m and hence m 1 Km···m (z, w) = N (z)m N (w) . (d/r)m···m

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The Faraut-Kor´ anyi binomial series expansion N (z, w)−ν = (ν)m Km (z, w), m

expressed via the Pochhammer symbol (ν)m =

r . ΓΩ (ν + m ) a = (ν − (k − 1))mk , ΓΩ (ν) 2 k=1

can be used to deﬁne the discrete part of the Wallach set, the so-called discrete Wallach spaces H2a/2 (D) = Pm1 ,...,m , 0,...,0 (Z) m+1 =0

as the space of “-harmonic” functions, with inner product (φm |ψm ) (φ|ψ) a/2 = . ( a/2)m m =0 +1

2 Figure 8. The partitions occurring in Ha/2 (D) n

r It is an important problem to obtain an integral expression for the inner product [AU1], [AU2], [AU5]. Deﬁne GL (Ω)-invariant diﬀerential operators on the symmetric cone Ω ⊂ X by ΓΩ (α) n N (x)d/r−α ∂N Tαn = N (x)α+n−d/r ΓΩ (α + n) and consider the unique extension to a holomorphic diﬀerential operator T˜αn on X C . By construction [AU5, Section 3.1], we have the eigenvalues (α + n)m T˜αn pm = pm . (α)m Theorem 9.1. For φ, ψ ∈ H2a/2 (D) the inner product can be expressed as an integral (φ|ψ) a/2 = dU (φ|S(U ) |T˜U ψ|S(U ) )Hardy M

over the th Jordan Grassmann manifold M . Here S(U ) denotes the Shilov boundary of U ∩ D, so that · S = U × S(U ) U

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as a disjoint union. Moreover, (T˜U )U ∈M is a K-covariant family of holomorphic (pseudo) diﬀerential operators on U , which can be constructed (in the sense of spectral theory) from the building blocks T˜αn (restricted to U ) by requiring that the eigenvalues are given by (ra/2)m (d/r)m , ( a/2)m where m runs over all partitions such that m+1 = 0. Note that in contrast to the situation in Proposition 4.3, the union of the respective Shilov boundaries is in fact disjoint since everything is “centered at the origin”, whereas in Proposition 4.3 one considers the translated Shilov boundaries as subsets of S ⊂ ∂D. Another part of the analytic continuation are the generalized Dirichlet spaces [A], deﬁned for 1 ≤ n ∈ N. Let d1 /r = 1 + a(r − 1)/2 correspond to the dimension d1 of the Peirce 1-space of Z. Taking all partitions m1 ≥ m2 ≥ · · · ≥ mr ≥ n, one obtains as a completion ˜ d /r−n = Pm (Z), H 1 mr ≥n

which is a unitarizable G-space called the Dirichlet space, endowed with the inner product ∼ (fm |gm )F (f |g)n = . (1 + a2 (r − 1) − n)m mr ≥n

Figure 9. The partitions occurring in the Dirichlet space n

r

It is proved in [FK1] that, in case Z = X C is of tube type, there is a unitary isomorphism ∂n

N 2 ˜ d/r−n ≈ Hd/r+n (D) H

sending the parameter sense that

d r

− n = λ to p − λ =

d r

+ n, which is G-equivariant in the

j ∂N (Ud/r−n (g) ψ) = Ud/r+n (g) ψ. This is false for non-tube type domains, as pointed out by G. Zhang, who has found a vector-valued generalization using the so-called transvectants and Shimura operators [PZ], [HLZ]. Using the Jordan Grassmann manifolds the intertwiner can

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also be realized in more geometric terms [AU8]. For 1 ≤ ≤ r consider the homogeneous line bundle Ln = S × C = {[u, ξ] = [v, η] : u u∗ = v v ∗ , η = Nu (v)n ξ} ∼

over M , i.e., the nth power of the determinant bundle, which is the basic holomorphic line bundle over M . Here Nu is the Jordan determinant of Z1 (u). Theorem 9.2. There exists a unitary G-intertwiner into the vector-valued Bergman space ˜ d /r−n ∼ H (D, Pn···n (Z)). = H2 a 1+ 2 (r−1)+n

1

In case Z if of tube type, Pn···n (Z) = C · N n is 1-dimensional, but if Z is not of tube type, e.g., if D is the unit ball in Cd , d > 1, then dimC Pn···n (Z) > 1. The geometric framework for the intertwiner, using the Jordan-Grassmann manifolds, can be described as follows [AU8]: Proposition 9.3. As a special case of the Borel-Weil-Bott theorem, one may realize K-equi Pn··n (Z) −→ Γhol (Mr , Lnr ) ≈

via holomorphic sections over Mr , using the restriction p → p|Sr . More generally, we have an identiﬁcation K-equi Pn·· n 0···0 (Z) −→ Γhol (M , Ln ) !" ≈

via holomorphic sections over M . Theorem 9.4. Using the Jordan-Grassmann manifold, we obtain the G-equivariant intertwiner I ˜ d /r−n −→ H Hd21 /r+n (D, Pn···n (Z)) ≈ Hd21 /r+n (D, Γhol (Mr , Lnr )) 1 ≈

in the form n (I f )(z, U ) = (∂N f |U )(PU z), u

where U = Z1 (u) for some tripotent u. More generally, for 1 ≤ ≤ r, there are similar intertwiners ˜ 2a H (−1)+1−n (D) 2

I

−→ H 2a (−1)+1+n (D, Γhol (Mk , Ln )) ≈

≈

2

˜ 2a H (−1)+1+n (D, Pn··n 0··0 (Z)). 2

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[AU4]

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[LO1] O. Loos, Jordan Pairs, Springer Lect. Notes Math. 460 (1975). [LO2] O. Loos, Bounded Symmetric Domains and Jordan Pairs, University of California, Irvine 1977. [MR] P. Muhly, J. Renault, C ∗ -algebras of multivariable Wiener-Hopf operators, Trans. Amer. Math. Soc. 274 (1982), 1–44. [N]

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[NT] Y. Nakagami, M. Takesaki, Duality for Crossed Products of von Neumann Algebras, Springer Lect. Notes Math. 731 (1979). [PZ]

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[SSU] N. Salinas, A. Sheu, A. Upmeier, Toeplitz operators on pseudoconvex domains and foliation algebras, Ann. Math. 130 (1989), 531–565. [ST]

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[UU1] A. Unterberger, J. Unterberger, La s´erie discr`ete de SL(2, R) et les op´erateurs ´ Norm. Sup. 17 (1984), 83– pseudo-diﬀ´ erentiels sur une demi-droite, Ann. Sc. Ec. 116. [UU2] A. Unterberger, J. Unterberger, Quantiﬁcation et analyse pseudo-diﬀ´erentielle, ´ Norm. Sup. 21 (1988), 133–158. Ann. Sc. Ec. [UU] A. Unterberger, H. Upmeier, The Berezin transform and invariant diﬀerential operators, Comm. Math. Phys. 164 (1994), 563–597. [U1]

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H. Upmeier H. Upmeier, Jordan Algebras in Analysis, Operator Theory and Quantum Mechanics, CBMS Ser. Math. 67, Amer. Math. Soc., Providence RI, 1987. H. Upmeier, Toeplitz C ∗ -algebras and non-commutative duality, J. Oper. Th. 26 (1991), 407–432. H. Upmeier, Symmetric Banach Manifolds and Jordan C ∗ -Algebras, NorthHolland 1985. H. Upmeier, Multivariable Toeplitz Operators and Index Theory, Birkh¨ auser 1996. H. Upmeier, Toeplitz operators and solvable C ∗ -algebras on hermitian symmetric spaces, Bull. Amer. Math. Soc. 11 (1984), 329–332. H. Upmeier, Toeplitz operators on symmetric Siegel domains, Math. Ann. 271 (1985), 401–414. H. Upmeier, Index theory for Toeplitz operators on bounded symmetric domains, Bull. Amer. Math. Soc. 16 (1987), 109–112. H. Upmeier, Fredholm indices for Toeplitz operators on bounded symmetric domains, Amer. J. Math. 110 (1988), 811–832. H. Upmeier, Index theory for multivariable Wiener-Hopf operators, J. reine angew. Math. 384 (1988), 57–79. M. Vergne, H. Rossi, Analytic continuation of holomorphic discrete series of a semi-simple Lie group, Acta Math. 136 (1975), 1–59. N. Wallach, The analytic continuation of the discrete series I, II, Trans. Amer. Math. Soc. 251 (1979), 1–17, 19–37. A. Wassermann, Alg`ebres d’op´erateurs de Toeplitz sur les groupes unitaires, C.R. Acad. Sci. 299 (1984), 871–874. G. Zhang, Berezin transform on real bounded symmetric domains, Trans. Amer. Math. Soc. 353 (2001), 3769–3787.

Harald Upmeier Fachbereich Mathematik University of Marburg 35032 Marburg Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 121–142 c 2008 Birkh¨ auser Verlag Basel/Switzerland

Rotation Algebras and Continued Fractions Florin P. Boca Abstract. This paper discusses two problems related with the approximation of rotation algebras: (i) estimating the norm of almost Mathieu operators and (ii) studying a certain AF algebra associated with the continued fraction algorithm. The Eﬀros-Shen AF algebras naturally arise as primitive quotients of this algebra. Mathematics Subject Classiﬁcation (2000). Primary 46L05; Secondary 11A55, 47A30, 47B36. Keywords. Rotation algebras, almost Mathieu operators, continued fractions, AF algebras.

1. Introduction Let Γ be a ﬁnitely generated discrete group. Let λ denote the left regular repre2 sentation of Γ on the Hilbert space (Γ). For 2any ﬁnite set S ⊂ Γ, consider the 1 Markov operator µS = |S| s∈S λs acting on (Γ) as (µS f )(t) =

1 f (s−1 t). |S| s∈S

Suppose S is a symmetric generating set for Γ. A classical result of Kesten [23] shows that the group Γ is amenable if and only if the spectral radius σ(µS ) of µS is equal to 1. The natural example where Γ is the (non-amenable) free group Fn with generators g1 , . . . , gn and S = {g1±1 , . . . , gn±1√}, n 2, has also been considered by Kesten [22]. In this case σ(µS ) = µS = n1 2n − 1 and the spectrum of µS is the whole interval [−µS , µS ]. Things are diﬀerent when the representation λ is twisted by a cocycle. Here we are interested in the case where Γ is the abelian group Z2 . Given θ ∈ [0, 1), the mapping β(m, n) = eπiθm∧n deﬁnes a skew-symmetric bicharacter of Z2 × Z2 . The (left) regular representation of Z2 is twisted by β to (πs f )(t) = β(s, t)f (t − s),

s, t ∈ Z2 , f ∈ 2 (Z2 ).

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This formula deﬁnes a projective unitary representation π of Z2 on 2 (Z2 ) such that πs∗ = π−s and πs1 πs2 = β(s1 , s2 )πs1 +s2 = β(s1 , s2 )β(s2 , s1 )πs2 πs1 ,

s1 , s2 ∈ Z2 .

The unitaries Uθ = π(1,0) and Vθ = π(0,1) acting on the orthonormal basis {δ(m,n) } of 2 (Z2 ) consisting of Dirac functions as Uθ δ(m,n) = eπinθ δ(m+1,n) , Vθ δ(m,n) = e−πimθ δ(m,n+1) , commute as Uθ Vθ = e2πiθ Vθ Uθ . The rotation algebra Aθ is deﬁned as the universal C ∗ -algebra generated by two unitary operators uθ and vθ such that uθ vθ = e2πiθ vθ uθ . Using the simplicity of Aθ when θ is irrational and the canonical isomorphism between C ∗ (Uθq , Vθq ) and C(T2 ) when θ = pq in lowest terms, the mapping uθ → Uθ , vθ → Vθ , is seen to extend to an isomorphism ρθ : Aθ → C ∗ (Uθ , Vθ ) ⊂ B(2 (Z2 )). The C ∗ -algebra A θ is endowed with the tracial state τθ m n deﬁned on polynomials in uθ and vθ by τθ α u v m,n (m,n) θ θ = α(0,0) . It is easily seen that ρθ and the GNS representation of Aθ deﬁned by the state τθ are unitarily equivalent. Another representation, σθ : Aθ → B(2 (Z)), is obtained by mapping $θ , V$θ ), where the unitaries U $θ and V$θ act on an orthonormal basis (uθ , vθ ) → (U 2 2πinθ $ $ δn . Every irreducible representation {δn }n of (Z) by Uθ δn = δn−1 , Vθ δn = e of Ap/q has dimension q and is unitarily equivalent with one of the representations πz1 ,z2 deﬁned as πz1 ,z2 (uθ ) = z1 U0 , πz1 ,z2 (vθ ) = z2 V0 , with ⎤ ⎡ ⎡ ⎤ 0 1 0 ... 0 0 1 ⎥ ⎢0 0 1 . . . 0 0⎥ ⎢ ω ⎥ ⎢ ⎢ ⎥ 2 ⎥ ⎢ .. .. .. . . ⎢ ⎥ . . ω . . = U0 = ⎢ . . . , V ⎥ , ω = e2πip/q . ⎢ ⎥ 0 . . .⎥ ⎥ ⎢ ⎢ . .. ⎦ ⎣0 0 0 . . . 0 1⎦ ⎣ q−1 ω 1 0 0 ... 0 0 Rotation algebras are closely connected with the three-dimensional discrete Heisenberg group H3 (Z) generated by the set S = {x±1 , y ±1 , z ±1 }, with ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 1 0 1 0 0 1 0 1 x = ⎣0 1 0⎦ , y = ⎣0 1 1⎦ , z = ⎣0 1 0⎦ , 0 0 1 0 0 1 0 0 1 satisfying the commutation relations z = xyx−1 y −1 , zx = xz, and zy = yz. Since H3 (Z) is amenable, its reduced and full C ∗ -algebras coincide, being described as the universal C ∗ -algebra generated by three unitary elements u, v, w (corresponding to λx , λy , λz ) satisfying the commutation relations uv = vuw, wu = uw, and wv = vw. For every θ the mapping u → uθ , v → vθ , w → e2πiθ 1, extends to a representation βθ of C ∗ (H3 (Z)) onto Aθ . Conversely, for any irreducible unitary representation π of H3 (Z) there is some number θ ∈ [0, 1) such that πz = e2πiθ 1. Consequently πx and πy are unitaries such that πx πy = e2πiθ πy πx . When θ is irrational any two such representations are approximately unitarily equivalent as a consequence of Voiculescu’s noncommutative Weyl-von Neumann theorem. When θ = pq in lowest terms, πx and πy generate a copy of the C ∗ -algebra Mq of q × q matrices with complex entries, and there are z1 , z2 ∈ T such that πx = z1 U0

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and πy = z2 V0 (see, e.g., [14, Thm.VII.5.1]). These considerations show that the spectrum of any self-adjoint element h ∈ C ∗ (H3 (Z)) coincide with the closure of the union of the spectra of the elements βθ (h) ∈ Aθ , θ ∈ [0, 1). The C ∗ algebra of any ﬁnitely generated torsion-free nilpotent group contains no nontrivial projections (cf., e.g., [2, Prop.1]), hence spec(h) is always a compact interval. Using the equality spec(β√ θ (6µS )) = 2 cos(2πθ) + spec(Hθ ), it was proved in [2] that spec(µS ) = [− 13 (1 + 2), 1] for the set of generators S = {x±1 , y ±1 , z ±1 } of Γ = H3 (Z). The “twisted” Markov operator hθ = uθ + u∗θ + vθ + vθ∗ ∈ Aθ corresponding to the set S = {±(1, 0), ±(0, 1)} of generators of Z2 is called the Harper operator. More generally, one considers almost Mathieu operators hθ,λ = uθ +u∗θ + λ2 (vθ +vθ∗ ), λ ∈ R∗ . The norm of hp/q,λ is seen to be given by 8 8 8 8 λ ∗ ∗ 8 (V U + U + + V ) (1.1) hp/q,λ = π1,1 (hp/q,λ ) = 8 0 0 0 8. 8 0 2

1 0.8

Θ

0.6 0.4 0.2 0 -4

-2

0 spec hΘ

2

4

Figure 1. The Hofstadter butterﬂy (1 q 30) The spectrum of hp/q,λ is the union of q − 1, respectively q, disjoint intervals, according to whether q is even or odd. The union of the sets spec(hθ ), θ ∈ Q, forms the Hofstadter butterﬂy (see Figure 1). During the last three decades much eﬀort has been put in the spectral analysis of almost Mathieu operators. One of the crowning achievements is the recent result of Avila and Jitomirskaya [1] which shows that spec(hθ,λ ) is a Cantor set for every irrational θ whenever λ = 0, thus

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answering the ‘Ten Martini problem’ of Kac and Simon. In the next section we will discuss some results concerning the norm of almost Mathieu operators. Rotation algebras provide a noticeable role in noncommutative geometry [13]. A classical result of Pimsner and Voiculescu [26] shows that any irrational rotation algebra Aθ can be embedded into the Eﬀros-Shen AF algebra Fθ [15], which encodes the continued fraction expansion of θ. Combined with Rieﬀel’s construction [29] of projections of trace {nθ} in Aθ this led, via the (ordered) dimension group of Fθ , to the classiﬁcation of irrational rotation algebras: two irrational rotation algebras Aθ1 and Aθ2 are isomorphic if and only if θ2 = ±θ1 (mod Z). The order on the dimension group of Fθ also plays a role in the Elliott-Evans decomposition [16] of irrational rotation algebras as inductive limits of circle algebras. The results of Rieﬀel, Pimsner and Voiculescu yield {τθ (p) : p projection in Aθ , p = 1} = {{nθ} : n ∈ Z} for any irrational θ, thus gaps in the spectrum of almost Mathieu operators are canonically labelled by integers. The ‘dry’ Ten Martini problem asks / Q, is it whether all these canonical gap labels occur, i.e., given λ ∈ R∗ and θ ∈ true that for any n ∈ Z∗ there is p spectral projection of hθ,λ in Aθ such that τθ (p) = {nθ}? The answer is known to be positive for θ of Liouville type [12], but the problem is open in general (see also [1] and [28] for some recent thoughts). In Section 3 we will review some results from [6] concerning a certain AF algebra A associated with the continued fraction algorithm. This algebra is related with both the Eﬀros-Shen algebras and the GICAR AF algebra having the Pascal triangle as Bratteli diagram.

2. Norm estimates for almost Mathieu operators The problem of estimating the norm of Harper operators (i.e., to approximate the external boundary of the Hofstadter butterﬂy) has been ﬁrst considered in [2] in connection with the study of the spectrum of the Markov operator µS on the group C ∗ -algebra of the discrete Heisenberg group H3 (Z), generated by the set S = {x±1 , y ±1 , z ±1 }. Explicit estimates for the norm of almost Mathieu operators are provided by Theorem 2.1 ([7]). (i) For every λ ∈ R and θ ∈ [0, 12 ], hθ,λ Mλ (θ),

(2.1)

with

⎧9 2 ⎪ ⎨ 4 + λ + 4|λ|(cos πθ − sin πθ) cos πθ : ; Mλ (θ) = ⎪ 4 + λ2 − 1 − 1 1 − 1+cos2 4πθ min{4, λ2 } ⎩ tan πθ 2

In particular for every θ ∈ [ 14 , 34 ], hθ,λ

9 4 + λ2

if θ ∈ [0, 14 ], if θ ∈ [ 14 , 12 ].

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(ii) For every θ ∈ [0, 12 ], hθ m(θ) = max{f1 (θ), f2 (θ), f3 (θ)},

(2.2)

where f1 , f2 , f3 0 are the elementary functions given by f1 (θ)2 = 6 − 1+

1 sin πθ

4 ; + 1+

+ 1 sin πθ

8 cos2 πθ 2 + , 2 1 + 4 sin πθ cos πθ (1 + sin πθ)3/2

2 2 cos2 2πθ + f2 (θ)2 = 4 + 9 1 + | sin 4πθ| 1 + | sin 4πθ| < 2 = 2 = cos 2πθ 16 cos4 πθ sin2 2πθ > 1+ 9 + , +2 1 + | sin 4πθ| (2 + | tan 2πθ|)2 1 + | sin 4πθ| f3 (θ)2 = 4 + ? +

4(cos 2πθ + 2 cos2 2πθ + 2 cos4 2πθ) 5

4(cos 2πθ + 2 cos2 2πθ + 2 cos4 2πθ) 2− 5

2

√ 8 cos 2πθ + 10 + √ 10

2 .

Figure 2. The graphs of the functions M2 , h· and m on [0, 1/2] The map [0, 1] θ → hθ,λ was proved to be Lipschitz by Bellissard [3]. Figure 2 and the inequalities sup M2 (θ) − m(θ) 0.18962, M2 (θ) − m(θ) 0.16183, sup 0θ1/4

1/4θ1/2

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show that Theorem 2.1 provides a reasonably accurate numerical approximation for the boundary of the Hofstadter butterﬂy. It also shows that Kesten’s result does not extend to the case of twisted Markov operators, even in the case of the abelian group Γ = Z2 . Actually the norm of 14 (uθ + u∗θ + vθ + vθ∗ ) is much smaller √ than the norm 23 of 14 (λg1 + λ∗g1 + λg2 + λ∗g2 ) ∈ Cr∗ (F2 ). To prove hθ fj (θ), one simply uses the trivial inequalities hθ σθ (hθ )

σθ (hθ )ξ2 , ξ2

where ξ = {ξn }n ∈ 2 (Z) is given, according to j = 1, 2, 3, by ⎧ sin √ β ⎪ ⎪ ⎪ % √ 10 ⎪ ⎨ 2√sin β s|k| cos α if n = 2k, 5 ξ ξn = r|n| , ξn = = n ⎪ s|k| sin α if n = 2k + 1, cos β ⎪ ⎪ ⎪ ⎩ 0

if n = ±2, if n = ±1, if n = 0, else,

for appropriate choices of r = r(θ), s = s(θ) ∈ (0, 1), α = α(θ), β = β(θ) ∈ [0, 2π). The continuity of the map θ → hθ,λ and (1.1) reduce (2.1) to the situation where θ = pq is a rational number in lowest terms, when hθ,λ = Hθ,λ with Hθ,λ = π1,1 (hθ,λ ) = U0 + U0∗ + λ2 (V0 + V0∗ ). Let E be an eigenvalue of the selfadjoint q × q matrix Hθ,λ and ξ = {ξm }m∈Zq ∈ Cq = 2 (Zq ) be a unit eigenvector for E. The equality Hθ,λ ξ = Eξ can be written as a three-term recurrence relation ξm+1 = (E − λ cos 2πmθ)ξm − ξm−1 , or in matrix form as

E − λ cos 2πmθ ξm+1 = 1 ξm

m ∈ Zq ,

(2.3)

−1 ξm . 0 ξm−1

Set S = m∈Zq ξm ξm−1 cos π(2m − 1)θ. After some heavy trigonometry and intensive use of the recurrence relation (2.3), one can express E 2 in two ways, as (ξm+1 − ξm−1 + λξm sin 2πmθ)2 E 2 =4 + λ2 + 4λS(cos πθ − sin πθ) − m∈Zq (2.4) 2 4 + λ + 4λS(cos πθ − sin πθ), and respectively as λ2 2 (4 − sin2 2πθ) ξm sin2 2πmθ 4 m∈Zq 2 λ ξm sin 2πθ sin 2πmθ + ξm+1 − ξm−1 − 2

E 2 =4 + 4λ2 + 4λS cos3 πθ −

m∈Zq

4 + 4λ2 + 4λS cos3 πθ.

(2.5)

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127

Some more trigonometry and use of Cauchy-Schwarz inequality lead to |S| cos πθ for every θ ∈ [0, 14 ]. In conjunction with (2.4) this completes the proof of (2.1) in this range of θ. In the range θ ∈ [ 14 , 12 ] we have cos3 πθ 0 cos πθ − sin πθ. So regardless of the actual√sign of S, inequalities (2.4) and (2.5) yield E 2 4 + λ2 , and therefore hθ,λ 4 + λ2 . This upper bound can be improved to that given by Mλ (θ) by further enhancing this kind of arguments. Remark 2.2. (i) The lower bound estimate (2.2) gives in particular √ min hθ min f1 (θ) = 6.59303 . . . = 2.56769 . . . 1/4θ1/2

(2.6)

1/4θ1/2

Numerical computations suggest that the left-hand side in (2.6) is only fractionally larger than 2.59. It would be interesting to improve these estimates. (ii) Estimate (2.2) also gives min0θ1/4 hθ 2 7.82387 (which is pretty √ close to 8) and hθ 2 2 for all θ ∈ [0, √ 0.23441]. Numerical computations seem to suggest that the inequality hθ 2 2 may hold in the whole range θ ∈ [0, 14 ]. It would be interesting to prove/disprove this. (iii) The spectrum of the non-self-adjoint operator uθ + u∗θ + λvθ , λ ∈ C, was computed for irrational θ in [4, 30]. It would be interesting to ﬁnd accurate estimates for its norm.

3. An AF algebra associated with the continued fraction algorithm 3.1. Continued fractions and the Farey tessellation The regular continued fraction representation of real numbers establishes a oneto-one correspondence NN a = (a1 , a2 , . . .) ←→ θ ∈ (0, 1] \ Q.

(3.1)

In one direction deﬁne, for given a, the rational numbers [a1 , . . . , an ] =

1 a1 +

1 a2 +...+

= 1 an

pn . qn

(3.2)

The representation (3.2) is not unique because [a1 , . . . , an ] = [a1 , . . . , an − 1, 1]. However, the Euclidean algorithm shows that any rational number in (0, 1] can be uniquely represented as in (3.2), for some n and a1 , . . . , an ∈ N with an 2. Moreover, pn and qn are obtained by plain matrix multiplication as an 1 qn qn−1 a1 1 a2 1 ··· = . (3.3) 1 0 1 0 1 0 pn pn−1 Applying the determinant on both sides of (3.3) one gets the familiar relation pn−1 qn − pn qn−1 = (−1)n .

(3.4)

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It is easy to infer from (3.3) that qn is greater or equal than the nth Fibonacci number Fn . In conjunction with (3.4) this yields # # # pn+1 1 pn ## 1 # = − , # qn+1 qn # qn qn+1 Fn Fn+1 showing that { pqnn }n is a Cauchy sequence of rational numbers. Its limit is irrational and deﬁnes the number θ from the right-hand side of (3.1). In the opposite direction the digits an of θ are obtained by plain iterations of the Gauss map 1 1 1 = − , x = 0, G : [0, 1] → [0, 1], G(0) = 0, G(x) = x x x as 1 an = n−1 − Gn (θ), n 1. G (θ) The Gauss map acts as G[a1 , a2 , a3 , . . .] = [a2 , a3 , . . .], hence its periodic points are exactly the reduced (purely periodic) quadratic irrational numbers [a1 , . . . , an0 ].

F

B −1 F

BF

AF

A2 F 3

A F

A2 BF

1 4

0 1

ABF

1 3

AB 2 F

ABAF

2 5

1 2

3 5

3 4

2 3

1 1

Figure 3. The Farey tessellation Consider the matrices 1 0 1 1 A= , B= , 1 1 0 1 The equalities (3.3) and a 1 b M (a)M (b) = 1 0 1

J=

0 1 , 1 0

1 a 1 1 = 0 1 0 0

yield

a1

a2

B A

···B

a2m−1

a2m

A

B a1 Aa2 · · · Aa2m B a2m+1

M (a) =

a 1

b = B a Ab , 1

q = 2m p2m q = 2m p2m

q2m−1 , p2m−1 q2m+1 . p2m+1

1 ∈ GL2 (Z). 0

a, b ∈ Z,

(3.5)

(3.6)

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129

It is now plain to prove that the semigroup G generated by A and B in SL2 (Z) is canonically isomorphic to the free semigroup on two generators F+ 2 . This can be seen geometrically considering the Farey tessellation {gF : g ∈ G} of the upper half-plane H, where F = {0 Re z 1, |z − 12 | 12 } (see Figure 3). However, the group generated by A and B is not free, as one can see from the equality (BA−1 B)4 = I2 . Note also that conjugating by J in (3.6) one gets a b ∈ SL2 (Z) : 0 a c, 0 b d = AG. c d A ﬁrst estimate [21] on the number Ψ(N ) of elements C ∈ G such that 2 < Tr C N as N → ∞ was furthered in [5] to 3 ζ (2) N2 ln N + γ − − + Oε (N 7/4+ε ). (3.7) Ψ(N ) = ζ(2) 2 ζ(2) Interestingly, the contribution to the main term of even length words is which turns out to be much smaller than the contribution ζ (2) ζ(2) )

N2 ζ(2) (ln N

N 2 ln 2 ζ(2) ,

+ γ − ln 2 −

− of odd length words. The estimate on the number of even length words leads to good estimates of the number of reduced quadratic irrationals according to their natural length [18, 5]. 3 2

3.2. The mediant construction and AF algebras The mediant construction, pictured in Figure 4, associates to every two consecutive rational numbers pq < pq with p q − pq = 1 the new rational number p+p q+q . Clearly p q

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