Kac Algebras and Duality of Locally Compact Groups

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Michel Enock Jean-Marie Schwartz

Kac Algebras and Duality of Locally Compact Groups Preface by Alain Connes

Postface by Adrian Ocneanu

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Michel Enock Jean-Marie Schwartz

CNRS, Laboratoire de Mathematiques Fondamentales Universite Pierre et Marie Curie 4 place Jussieu F-75252 Paris Cedex 05, France

The sculpture reproduced on cover and frontispiece is exhibited at the Museo de antropologia de la Universidad Veracruzana, Jalapa, E. U. de Mexico

Mathematics Subject Classification (1980): 22 D 25, 22 D 35, 43 A 30, 43 A 65

ISBN 3-540-54745-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54745-2 Springer-Verlag New York Berlin Heidelberg

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To Professor Jacques Dixmier "What's the matter?" Macbeth (11,2)

The question is the story itself, and whether or not it means something is not for the story to tell. Paul Auster (City of glass)

Preface

This book deals with the theory of Kac algebras and their duality, elaborated independently by M. Enock and J.-M. Schwartz, and by G.I. Kac and L.I. Vajnermann in the seventies. The subject has now reached a state of maturity which fully justifies the publication of this book. Also, in recent times, the topic of "quantum groups" has become very fashionable and attracted

the attention of more and more mathematicians and theoretical physicists. One is still missing a good characterization of quantum groups among Hopf algebras, similar to the characterization of Lie groups among locally compact groups. It is thus extremely valuable to develop the general theory, as this book does, with emphasis on the analytical aspects of the subject instead of the purely algebraic ones. The original motivation of M. Enock and J.-M. Schwartz can be formulated as follows: while in the Pontrjagin duality theory of locally compact abelian groups a perfect symmetry

exists between a group and its dual, this is no longer true in the various duality theorems of T. Tannaka, M.G. Krein,

W.F. Stinespring ... dealing with non abelian locally compact groups. The aim is then, in the line proposed by G.I. Kac in 1961 and M. Takesaki in 1972, to find a good category of Hopf algebras, containing the category of locally compact groups and fulfilling a perfect duality. It is natural to look for this category as a category of Hopf-von Neumann algebras since, first, by a known result of A. Weil, a locally compact group G is fully specified by the underlying abstract group with a measure class (the class of the Haax measure), and, second, by a result of M. Takesaki, locally compact abelian groups correspond exactly to co-involutive Hopf-von Neumann algebras which are both commutative and cocommutative. A co-involutive Hopf-von Neumann algebra is given by a morphism 1' : M --p M 0 M of a von Neumann algebra M

VIII

Preface

in its tensor square M ® M and a co-involution n which together turn the predual M* into an involutive Banach algebra. A Kac algebra is a co-involutive Hopf-von Neumann algebra with a Haar weight, i.e. a semi-finite faithful normal weight on

M which is left-invariant in a suitable way. In this book, the theory of Kac algebras and their duality is brought to a quite mature state, relying a lot on the modular theory of weights developped also in the seventies. The resulting category of Kac algebras fully answers the original duality problem, but is not yet sufficiently non-unimodular to include quantum groups. This of course opens a very interesting direction of research, undertaken recently by S. Baaj and G. Skandalis.

Paris

Alain Connes

Table of Contents

Introduction

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Chapter 1. Co-Involutive Hopf-Von Neumann Algebras 1.1

1.2 1.3 1.4

1.5 1.6

Von Neumann Algebras . . . . . . and Locally Compact Groups Co-Involutive Hopf - Von Neumann Algebras . Positive Definite Elements in a Co-Involutive . . Hopf-Von Neumann Algebra . . . . Kronecker Product of Representations . . . Representations with Generator . . . . . Fourier-Stieltjes Algebra . . . . . . . .

Chapter 2. Kac Algebras

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. . An Overview of Weight Theory . 2.2 Definitions . . . . . . . . . . . 2.3 Towards the Fourier Representation . 2.4 The Fundamental Operator W . . . 2.5 Haar Weights Are Left-Invariant . . . 2.6 The Fundamental Operator W Is Unitary 2.7 Unicity of the Haar Weight . . . . .

2.1

Chapter 3. Representations of a Kac Algebra; Dual Kac Algebra . . . . . . . . . . 3.1

3.2 3.3 3.4

3.5 3.6 3.7

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8 13

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19 23 30 36

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44

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45 55 58 60 66

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71

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76

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83

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84 89 92 97 101 104 111

The Generator of a Representation . . . . The Essential Property of the Representation A The Dual Co-Involutive Hopf- Von Neumann Algebra . . . . Eymard Algebra . . . . . . . . . Construction of the Dual Weight . . . Connection Relations and Consequences The Dual Kac Algebra . . . . . . .

1

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Table of Contents

X

Chapter 4. Duality Theorems for Kac Algebras and Locally Compact Groups . . . . . .

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Duality of Kac Algebras . . . . . . . . . 4.2 Takesaki's Theorem on Symmetric Kac Algebras 4.3 Eymard's Duality Theorem . . . . . . . for Locally Compact Groups 4.4 The Kac Algebra Ks (G) . . . . . . . . . 4.5 Characterisation of the Representations . . . . . . . . and Wendel's Theorem . 4.6 Heisenberg's Pairing Operator . . . . . . . 4.1

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5.1

5.2 5.3 5.4 5.5 5.6

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6.1

Unimodular Kac Algebras

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6.2 Compact Type Kac Algebras 6.3 Discrete Type Kac Algebras

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161

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162 166 172 174 176 184

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192

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Krein's Duality Theorem . . . . . . . . . 6.5 Characterisation of Compact Type Kac Algebras 6.6 Finite Dimensional Kac Algebras . . . . . .

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Bibliography

Index

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193 197 208 213 219 232

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6.4

Pos tf ace

136 140

144 152 158

Kac Algebra Morphisms . . . . . . . . H-Morphisms of Kac Algebras . . . . . . Strict H-Morphisms . . . . . . . . . . Preliminaries About Jordan Homomorphisms Isometries of the Preduals of Kac Algebras Isometries of Fourier-Stieltjes Algebras . .

Chapter 6. Special Cases: Unimodular, Compact, Discrete and Finite-Dimensional Kac Algebras

125 130

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4.7 A Tatsuuma Type Theorem for Kac Algebra Chapter 5. The Category of Kac Algebras

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243

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245

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255

Introduction

In the early nineteen thirties, L.S. Pontrjagin ([121]) established his famous duality theorem for abelian locally compact groups: he showed that the set of characters of an abelian locally compact group G is an abelian group,

called the dual group of G, and noted G, which is locally compact for a suitable topology. Moreover, the bidual G^^ is canonically isomorphic and homeomorphic to G; the Fourier transform carries the convolution algebra L1(G) into the multiplication algebra L' (G"'); conversely, the Fourier inverse transform carries the convolution algebra L1(G) into the multiplication algebra L°O (G) . These two transformations are transposed of each other in the following scheme: Fourier

L1 G (

)

LOO(G)

Fourier inverse

L°O( G )

L1 (G)

Since this fundamental result, which can be considered as one of the founding theorems of abstract harmonic analysis, a series of duality theorems for nonabelian locally compact groups has been gradually set up, but these dualities were not formulated within the category of locally compact groups. In 1938, T. Tannaka ([166]) proved a duality theorem for compact groups, involving the irreducible representations of a compact group G (one should bear in mind, that, in the case of an abelian locally compact group, the characters are the irreducible representations); although no group-like structure is to be put on that class (in particular, the irreducible representations being finite-dimensional, the tensor product of two irreducible representations is still finite-dimensional, but may no longer be irreducible), Tannaka showed

that it is possible to recover the compact group G from the set of its irreducible representations. In 1941, M.G. Krein ([82]) obtained an equivalent result, and, in 1949, ([83], [84] ), he took the system of matrix algebras generated by these repre-

sentations as the dual of the compact group G, and, without invoking the group G, gave an intrinsic description ("matrix block algebras") of these objects.

2

Introduction

The next step was done in 1959 by W.F. Stinespring's duality theorem for unimodular locally compact groups in [148]. There, Stinespring emphasizes the crucial role played by the unitary operator W on L2 (G x G) defined by:

(Wf)(s,t) = f (s, st)

(s,t E G; f E L2(G x G))

and by the left regular representation AG of G on L2 (G), which are linked by:

W*(AG(s) ®1)W = AG(S) ® AG(S)

(s E G)

The operator W allows us then to define a Hopf structure on the von Neumann algebra £(G) generated by the representation AG. Stinespring showed how it is possible to recover the group G from this Hopf structure. One must notice that this unitary W defines, too, a coproduct on L°O(G) by the formula:

(W(1® f )W*)(s, t) = f(st)

(s, t E G; f E L°O(G))

Then, in 1961, G.I. Kac ([66], [70]) introduced the notion of "ring-group", closely related to Hopf algebras; he gave a satisfactory abstract framework to Stinespring's results, and was again in a situation similar to Pontrjagin's theorem, where the initial objects and their duals are both of the same kind. These "ring-groups" generalize the algebra L' (G), for a unimodular locally compact group G, as well as the von Neumann algebra ,C(G), these two objects, for a given G, being dual to each other. In 1964, P. Eymard ([46] ), at last, gave a duality theorem valid for every locally compact group, even non-unimodular: taking an arbitrary pair f , g in the Hilbert space L2 (G), the convolution product f * gb (where gb (s) is equal to (_1) for all s in G) belongs to LOO (G), and the functions of that type form an involutive algebra A(G), called the Fourier algebra of G. With a suitable norm, it becomes an involutive Banach algebra, whose dual is the von Neumann algebra ,C(G). Moreover, the spectrum of A(G) is equal to G. In order to explain the importance of that theorem, let us assume again that G is abelian. Then, the Fourier-Plancherel transform carries respectively L2(G) onto L2(G), the left regular representation of L1(G) onto the Fourier tranform, and the von Neumann algebra £(G) onto the von Neumann algebra L°O(G), acting by multiplication on L2(G). So, the predual A(G) is isomorphic to the convolution algebra L1(G), whose spectrum is, by definition, the bidual G^^. By dealing with this particular case, Eymard's theorem allows us to recover Pontrjagin's duality theorem. In the general case, Eymard's theorem brings us to write down the following scheme, which is a generalization of the abelian case: AG

Gelfand transform

Introduction

3

In 1967, N. Tatsuuma ([167]. [168]) gave another duality theorem for arbitrary locally compact groups, recovering G, not as Eymard did, only from the left regular representation of G, but, in the spirit of Tannaka, from all the unitary representations of G. In [44], J. Ernest showed how Tatsuuma's theorem was closely related to the Hopf structure of the algebra ,C(G) introduced by Stinespring, and to the Hopf structure of the enveloping algebra W * (G) that he introduced in [43] and [44].

In 1968, M. Takesaki, using both Kac's work and these recent duality theorems, gave a characterization of group algebras: every co-involutive com-

mutative von Neumann algebra with a left invariant measure corresponds to a locally compact group ([157] ). This crucial result, making the link between the abstract constructions of Kac and the duality theorems for locally compact groups, is actually the keystone of the theory. Afterwards, Takesaki endeavoured, in 1972 ([159]), to complete Kac's programme for non-unimodular groups, i.e. to establish a general duality theory which would work inside a wider category than the one of locally compact groups, and would restrict itself to the Eymard-Tatsuuma-Ernest theorem for locally compact groups. Due to the incompleteness of non-commutative integration theory at that moment, he did not reach a perfect symmetry. This was done in 1973, independently by L.I. Vainermann and G.I. Kac ([170], [179], [180] ), and the authors ([34], [35], [36]) who named "Kac algebras" this wider category, because of Kac's 1961 basic work in that direction. The duality obtained may be sketched in the following scheme:

M* - M M4-M* where M and M are von Neumann algebras, their preduals M* and M* being involutive Banach algebras, and A and A non-degenerate faithful representations. Many improvements were made, by E. Kirchberg ([78], [79] ), J. De Canniere ([18] ), the authors ([135], [136], [38], [40] ), J. De Canniere and the authors ([21], [22]). In particular, was done in [22] the generalization of M.E. Walter's work, who had noticed ([193], [194]) in 1970, that the Banach algebra A(G) characterizes the group G (although A(G) is always abelian, even when G has not the same property), exactly like L1(G) does ([198] ), and had strongly indicated these two results should be just two particular cases of a stronger one (see also Akemann-Walter [2]). As of now, the theory of duality of Kac algebras, and the links with duality of locally compact groups, seem to have reached a state of maturity, which

is described in the present book. Surveys of Kac algebras theory may be found in [45], [20], [139] and [175]. For a historical view of abstract harmonic analysis, we refer to J.-P. Pier ([117] ).

4

Introduction

In the theory of Kac algebras, the crucial idea is the role played by the coproduct on L°O(G); it is well known that, for a locally compact semi-group,

the product (s, t) -- st is completely given by all the functions on G x G defined by 1'(f) : (s, t) -> f (st), for all f in a suitable space of functions on G (which separates the points of G, as, for example, C, ,(G) or L°O (G) for a suitable measure on G); if G is a group, the inverse s -+ s-1 will be given by all the functions ic(f) : s -+ f (s-1), for all f in C0(G) or L°O (G). Moreover,

it is known, thanks to a A. Weil's result ([197]), that the topology of the locally compact group G is completely given by its Haar measure. These simple remarks show how, on L°O(G), the coproduct, the co-involution, and the Haar measure give back the group G, both algebraically and topologically. Many important results have been obtained about actions of Kac algebras on von Neumann algebras and crossed-products (i.e. non-commutative dynamical systems) ([33], [37], [39], [103], [61], [106]) and are to be linked to the numerous works which have been made about duality and crossed products ([161], [156], [151], [102], [58], [88], [89], [188], [132], [190], [104], [153], [187], [77], [134], [139], [90], [60] ). In [37], the authors have proved a dual-

ity theorem for Kac algebras crossed-products, which generalized Takesaki's theorem ([161]) about crossed-products by group actions. In [107], [108], [60], [61], [106] is developped, by A. Ocneanu and others, a Galois theory for inclusions of von Neumann algebras, in which Kac algebras play a crucial role; are obtained intrinsic characterizations of crossed-products by a Kac algebra in term of position of the initial algebra in the crossed-product. More precisely ([106] ), if Mp C M1 is an inclusion of factors, with a faithful semi-finite normal operator valued weight from M1 to M0, then M1 is the crossed product of Mp by an outer action of some Kac algebra, eventually twisted by a cocycle, if and only if the relative commutant Mo n M1 is C and Mo n M3 is a factor (where Mp C M1 C M2 C M3 C ... is the canonical tower of von Neumann algebras constructed from the initial inclusion); these results open a wide range of opportunities for further study of Kac algebras. Other directions have been studied: amenability of Kac algebras ([112], [192], [41], [115], [116])); G*-Kac algebras ([111], [96], [187], [4], [42]), and actions of G*-Kac algebras ([187], [3], [4]). Precise examples of Kac algebras which are neither abelian, nor symmetric (i.e. which are neither groups, nor group duals) have been found and studied in [73], [74], [75], [19], [97], [4]. To exhibit and classify these examples, even in the finite-dimensional case, is a difficult problem, which is far from being solved. We refer to [4] for recent researches on that question. This theory, which remained in a kind of shade for some years, is nowadays

valuable to more and more mathematicians, first of all those who want to give an even more general framework for duality theory, for instance Yu. M. Berezanskii ([6], [7], [8], [9]), L.I. Vainerman ([182], [171], [172], [173], [174], [175], [181], [176], [178]), S.L. Woronowicz ([200], [201], [202], [203], [120], [204], [205], [206]), A. Ocneanu ([107], [108], [109]), S. Baaj and G. Skandalis

Introduction

5

([3], [4]), M.E. Walter ([195], [196]) or S. Doplicher and J. Roberts ([26], [27]) (we apologize to the others when we may have ommitted them). In the early eighties, many theoretical physicists from the Leningrad school

introduced the notion of quantum group, as it appeared in the quantum inverse scatterring transform method ([144], [140], [141], [143], [165], [85], [87], [137] ). A quantum group is generally considered to be mean a Hopf algebra obtained from a Lie group by deforming the envelopping algebra of its Lie algebra, in a way to obtain a non-commutative algebra. Historically, the first example of such a construction is to be found in [73], where Kac and Paljutkin gave a non-trivial infinite dimensional example of a Kac algebra. These ideas were developped in ([86], [142], [30], [63], [64] ), and the link was made with Lie bialgebras ([10], [5], [28], [29], [51]), mostly by V.G. Drinfel'd. This theory has become very fashionable, thanks to V.G. Drinfel'd's lecture at the ICM-86 ([31] ), and his Fields medal at the ICM-90, and many mathematicians are nowadays involved in quantum groups, from Yang-Baxter equation ([52], [53], [54], [55] ), Poisson-Lie groups ([91], [138] ), studying representations of quantum groups ([92], [93], [100], [101], [128], [80], [81]), or

making the link with a geometrical point of view ([99], [129], [169], [164], [123], [17] ). The non-commutative function algebras on quantum groups are studied in ([183], [184], [185], [145], [186], [146]). Other recent works are [32], [98], [122], [147], [119]. Constructing such objects in a C*-algebra setting has been done, mostly by S.L. Woronowicz ([200], [201], [202], [203], [204], [205] ); related works has been made by M.A. Rieffel ([125], [126] ), A. Van Daele ([189] ), P. Podles ([118] ), P. Podles and S.L. Woronowicz ([120]), S.L. Woronowicz and S. Zakrzewski ([206] ), I. Szymczak and S. Zakrzewski ([155] ). M. Rosso has shown ([127], [130]) the link between Drinfel'd's work and Woronowicz's "pseudo-groups" (see also J. Grabowski [50] ). For a survey of quantum group theory, we refer to [31], [191], [13], [131]. Let us now describe briefly the link between Kac algebras and Woronow-

icz's point of view. Roughly speaking, in Woronowicz's "compact matrix pseudo-groups" ([201], [205] ), the co-involution ic satisfies weaker axioms:

in Kac algebras, it is involutive (i.e. rc(x*) = rc(x )* for all x in M) and is an involution (i.e. rc2(x) = x for all x in M); in Woronowicz's algebras, ic is defined only on a dense sub-involutive algebra A and satisfies r.(,C(x )* )* = x

for all x in A. Woronowicz then obtains a left-invariant state, i.e. satisfying (i ®Sp)-P = co, which is not a trace. On the contrary, in Kac algebra theory, whenever the Haar weight is finite (we shall say then that the Kac algebra is "of compact type"), it is a trace, invariant by the co-involution (we shall say then that the Kac algebra is "unimodular" ). For the links (and the differences) between compact type Kac algebras and Woronowicz's compact matrix pseudo-groups, see [177].

6

Introduction

More generally, if a coproduct r on a von Neumann algebra M is given by a unitary W by the formula:

F(X) - w(l 0 X)yy* as it is the case for L' (G), then, the co-associativity of r leads to a pentagonal relation for W, which occurs both in Kac algebras and in pseudo-groups. This relation is the starting point of S. Baaj and G. Skandalis ([4]), where such unitaries are called "multiplicative". They define supplementary conditions to get "Kac systems", where duality theorems are proved. Both Kac algebras and Woronowicz's pseudo-groups are Kac systems, and, within Kac systems, pseudo-groups are characterized. So, the close relationship between Kac algebras, Woronowicz's work and quantum groups appears more and more often. We do hope our work will be useful for researches in that direction. We would like to express our profound gratitude to Jacques Dixmier, who oriented us to this subject and who always provided us with true help and

support. Special thanks are due to Alain Connes, who kindly accepted to write the preface of this book. We are indebted to Masamichi Takesaki, whose

work is the founding stone of the present one, and who gave us permanent encouragement. Thanks also to Adrian Ocneanu, who contributed most to bring back Kac algebras into bright light, and wrote a very encouraging postface. We are thankful to many other colleagues for fruitful exchanges, specially to Francois Combes for so many substantial conversations, and to the C.N.R.S. to which we both belong and which made this research possible. Last but not least, we are obliged to Ms. C. Auchet, who processed the typescript.

Chapter 1 Co-Involutive Hopf-Von Neumann Algebras

This chapter is devoted to the structure of co-involutive Hopf-von Neumann algebras, which has been introduced by Ernest ([44] ), and mostly studied by Kirchberg ([79] ), and de Canniere and the authors ([21] ). The paradigm, from which the whole theory comes, is the algebra L°°(G) of all the (classes

of) essentially bounded measurable (with respect to a left Haar measure) complex valued functions on a locally compact group G, equipped with a coproduct and a co-involution, which are nothing but the duals of the usual product and involution of the involutive Banach algebra L1(G) of all (classes of) integrable (with respect to a left Haar measure) complex valued functions on G (let us recall that L1(G) is the predual of L°°(G)). Other examples will be found later: the von Neumann algebra of G (1.6.8 ),

the von Neumann algebra generated by the left regular representation of G (3.3.6 ), the dual of the involutive Banach algebra M1(G) of bounded measures on G (4.4.1). A co-involutive Hopf-von Neumann algebra (1.2.5) is a von Neumann algebra M, equipped with a coproduct, i.e. a normal injective unital morphism r from M to the von Neumann tensor product M 0 M, which is co-associative, i.e. which satisfies:

(r®i)r=(i®r)r

and a co-involution x satisfying:

Fx=s(K0K)r where c(a 0 b) = b ® a for any elements a, b of M. The predual M* is then, in a natural way, an involutive Banach algebra, and so we may study the theory of continuous positive forms on it (called positive definite elements in M (1.3.1)) and the theory of representations of M*. In the case of L' (G), we recover the usual positive definite functions on G (1.3.11) and the unitary representations of G (1.1.6 ). Convenient morphisms (H-morphisms) for co-involutive Hopf-von Neumann algebras are defined and allow us to define a category (1.2.6).

8

1. Co-Involutive Hopf-Von Neumann Algebras

A crucial result is the construction of the Kronecker product of two representations, due to Kirchberg (1.4.2 ), which, in the case of L°O (G), is the tensor product of unitary representations of G (1.4.7) and may be considered as an important and useful tool for the general theory as well. The Kronecker product enables us to put, on the enveloping von Neumann algebra of M*, almost a structure of co-involutive Hopf-von Neumann algebra (eventually, we only miss the coproduct being non-degenerate; (1.6.5) and (1.6.6)). So, its predual (which is, too, the dual of the enveloping C*-algebra of M* ), is therefore an involutive Banach algebra, called the Fourier-Stieltjes algebra (1.6.9 ); it has a faithful representation in M, called its FourierStieltjes representation. In the case of L°O(G), we get B(G), the FourierStieltjes algebra of G, studied by Eymard ([46] ), and the canonical imbedding of B(G) into L°O(G). It is well known that non-degenerate representations of L' (G) are given by unitary representations of G. A similar situation, for a general co-involutive Hopf-von Neumann algebra, is given by the "representations with gener-

ator", which, after Kirchberg, are studied in 1.5. The particular case of finite-dimensional representations is especially studied, and will be useful in Chap. 6.

1.1 Von Neumann Algebras and Locally Compact Groups 1.1.1 Notations and Definitions. We refer to the usual text books about operator algebras ([105], [12], [24], [25], [133], [124], [113], [152], [162]); here follow some basic facts about C*-algebras and W*-algebras:

(i) A C*-algebra A is a complex Banach algebra, equipped with an antilinear involution x -+ x* such that (xy)* = y*x* and llx*xil = Iix112 for all x, y in A. Let H be a Hilbert space; we shall denote by NC(H) the involutive Banach algebra of all bounded linear operators on H equipped with the norm:

lxii =

6 E H, 11611 ijk 0 Xk,i7

®we,y,)

kEI

by 1.2.8 kEI

_ E (ic(xj),

(Xk,j,

by 1.2.5

kEI

_ E I (x,

I2

by the result above

kEI

The same calculation proves that, for every finite subset J if I, we have:

_ 1: I(xk,i7 kEJ

kEJ

1: 1 (X k, i)

Lv "q) 12

kE7

(F(Xili)(J 0 0 1 J710 77) and, by linearity and continuity, we have, for every 1, ," 2 in the closed convex

cone c(J) introduced in 1.3.2:

xz,k ®xk,i 1 I =2 kEJ

< (r(,)s1 u2

26

1. Co-Involutive Hopf-Von Neumann Algebras

As every element S in H ® H may be written as S = L i - u 2 + i. 3 - i=4, with E in c(j) (n = 1.... , 4), and En=1 IISn 112 = Iiu II27 we then easily get that II >.kEJ xi,k 0 xk,i II < IIxi,iII.

The algebraic tensor product M* O M* being dense in (M 0 M)*, we can conclude that the sum EkEI xi,k 0 xk,i is ultra-weakly convergent, and is equal to P(xi,i ), which, by polarization, implies (ii). Let us assume (ii). Let S be an element of CM, which can be considered as a dense subset of £2(I). Therefore there exists a finite subset Io of I, and complex numbers ai (i E Io) such that:

iEIo

Thanks to the calculation of lemma 1.4.1, we know that: 2

ai(xk,i)w) E kEI iEIo i

is finite and equal to: 2

aiaj xi,j, w° * w

la,

aici

iii° * w II

Ilwll2 =11=11211w11

Since: 2

kEI iEIo

ai (xk,i, w)

2

kEI iEIo

ai (xk,i, w)ek

we have, by this way, defined a linear mapping y (w) from CM to t2 (I) such

that:

(w)u =

ai(xk,i7 w)Ck kEI iEIo

This mapping is continuous and such that II,i(w)hI < 11w II , and we have:

E(Xk,i,W)Ck kEI

which implies: (z(w)ei I

= (Xj,j,W)

Thank to its continuity, it is possible to extend µ(w) to 12(I). For all W1, W2 in M*, we have:

1.4 Kronecker Product of Representations (Y(W1 * W2)6i I Cj)

27

= (Xj,j,Wl *W2) = (1'(x,),wi (3 w2) by hypothesis

= (1: x j,k ® xk,i, w1 0 w2

Therefore: µw1 * w2) = µ(Wl)M(w2)

We have: (/1(w°)ei I kj) = (Xj,j,W°) = (K(xj,i)* I w)

= (x,,w)by hypothesis _ (ft(w)Ej I i ) _ (µ(w)*Ei I ej )

therefore y(w°) = µ(w)* and it is an involutive representation of M. on t2(I). The remaining of (i) follows immediately, which completes the proof. 1.4.3 Theorem ([79]). Let H = (M, I', i) be a co-involutive Hopf-von Neumann algebra. Let µ1 and µ2 be two representations of M*. There exists a unique

representation v of M*, such that A C Aµ1 0 Aµ2 and that, for every 01 in (Aµ1)* and S22 in (Aµ2)*, we have: v*((Q1 0

Y1*(fl1)µ2*(f22)

We shall write v = p1 X µ2 and we shall call it the Kronecker product of Y1 by 92. This operation is obviously associative. If M is abelian, this operation is commutative.

Proof. Let I1 and 12 such that fl, and fµ2 be respectively isomorphic to and £2(12). Let {e; i E I1 } and {e; m E I2} be respectively the

R2 (I1)

canonical basis of £2 (I1) and £2 (12). Let us put, for i, j in I1, and m, n in 12:

xi j = 111*(f1 i El) and xm,n =

262

28

1. Co-Involutive Hopf-Von Neumann Algebras

The Hilbert space £2 (I1 X I2) can be identified with £2 (Il) ® £2(12), its canonical basis being then {e' ® 62=}. Thus if we put x(i,m),(j,n) = x=,jxm,n) we have: IC(x*i,m ,

,n) = Iqxi,j)IC(xm*,n) 1 2 = xj,ixn,m

by 1.4.2

= x(j,n),(i,m) and:

r(x(i,m),(j,n)) = r(xz j)r(xm,n )

x ®j 1

1

xm)I ®xl)n

kEIi 1

IEI2 2

1

2

xi,kxm,l ®xk,jxl,n k,l

x(i,m),(k,l) ® x(k,l),(j,n) (k,l)EI1 X12

We are then faced with the conditions 1.4.2 (ii) applied to £2 (I1 X J2); therefore

it exists a representation v of M* on £2 (I1 x 12) such that: X(i,m),(j,n)

v*(fle,1®En,Ei®E,21)

Using the natural identification of Q

1®E2 )Ei ®E2

get : (Jnn

I2 1 *

nn

nn

with DCI)ej (9 f262

)E2

we

n

`Ej1,Ei )p2* (QEri ,Em) = v* (' `E ,Ei ® DEr2i,E n

By linearity and continuity, this implies that for all 1, 771 in 'Hy, and 2, 772 in 'we have: 12 1171)Y2*(` 2,772) = v*(Qel)

0 Q 2,772)

Let a1 be in Al and a2 in A'2. For all e1,'71 in Hy1 and 2, 712 in ?-lµ2, and w in M*, we have: (u(.,)(a1 0 a2)(6 (9

2) 1111 0 772)

(u'(w),i2011,1 0

a26,172

(z'*(Qaiei,,,i 0 Qa 2 e2 ,,72) , w ) = (121*(SIaiei,,71)/22*((1 22,172 )lw)

= (/2l*(Qi,a17i)/22*(Q2,a;172),w)

by 1.3.9

(V*(i2l,a171 0 2,a;172 )' w) = (zi(w)(e' 0'11) I (a1 0 a2)*(e2 0 112))

from what follows that v(w) belongs to Aµ1®Aµ2 'which completes the proof.

1.4 Kronecker Product of Representations

29

1.4.4 Corollary. Let H be a co-involutive Hopf-von Neumann algebra; the set PR(H) is stable by multiplication.

Proof. With the notations of 1.3.6, let x = p 1 * (Qi) and x2 = µ2* (Sl2) be in PR(H); Q1 and S?2 are therefore positive. We have: XlX2

/I1*(fl1)P2*(J?2)

by 1.4.3

_ (lei x 92)*(J?i 0 ,fta) As f21 ® S?2 is positive, it completes the proof.

1.4.5 Proposition. Let H = (M, F, rt.) be a co-involutive Hopf-von Neumann algebra. Let µ, /t1,µ2 be three representations of M*. (i) Let w be in Hom(p1, 112). Then 1®w belongs to Hom(p X 91, /i X µ2). (ii) Let 4$ be a normal morphism from Aµ1 to Aµ2 such that 4$(1) = 1 and OP o /11 = 92. Then, we have: (i (9 Ifi) 0 (Y X YO = Y X P2

-

Proof. Let w in M*, , 17 in ?,Cµ, 1 in 'Hµl , 712 in mµ2 . We have: w)(Y X Y1)(wM 0 61) I ii 0 772)

= ((µ X 91)(w)(6 0 61) 17l 0 w*772) = ((Y X POMI fk,77 (9 f26)W*q2)

= ((µ X µ2)(w), 0 '?wfl IriO = ((µ X µ2)(w)(l 0 w)(6 0 60177 (9 712)

by 1.4.3 by 1.3.9 by 1.4.3

which by linearity, density and continuity completes the proof of (i). We have also, for all 5? in (Aµ)* and Q2 in (Aµ2 )*: 49 0 (Y X YOM, fl 0 fl2) _ (Ut X Y1)Gw), ,fl & ,f12 0 4i)

(9*(fl)Y1*(fl2 0 by hypothesis

_ ((µ X µ2)(w)

0 S22)

which by linearity, density and continuity completes the proof of (ii).

30

1. Co-Involutive Hopf-Von Neumann Algebras

1.4.6 Remark. Let H = (M, I', n) be a co-involutive Hopf-von Neumann algebra. It is straightforward from what is above that the operation which associates p1 x 112 to the couple (µl, P2) is a functor from Rep M* x Rep M* to Rep M*. It is easy to check that the representation 1 (1.3.7) is a unit for that product. Since the Kronecker product is also associative, it is clear that (Rep M*, x, 1) is a strict monoidal category in the sense of [95].

1.4.7 Example. Let G be a locally compact group and p, v non-degenerate representations of L1(G). Let PG, vG be the corresponding continuous unitary representations of G. We have seen, in 1.3.10, that, for 12 in (Aµ)*, p* (Q) is a.e. equal to the function s -+ (PG(s), d2).

So, using 1.4.3, we get that, for 12 in (A,,)*, 121 in (A), the element (p x v)* (Q ®12') is a.e. equal to the function: s -' (PG(s), Sl) (VG (S), 12r) = (PG(s) 0 ZG(s),12 (9 '2r)

It is easy to conclude that p x v is the non-degenerate representation of L1(G) associated to the tensor product µG 0 VG.

1.5 Representations with Generator 1.5.1 Proposition. Let (M, I') be a Hopf-von Neumann algebra, A be a von Neumann algebra on a Hilbert space 7, V be in A 0 M. We define a bounded linear mapping p : M. -+ A by, for all w in M*: Y(W) = (i 0 WXV)

Then we have: (i) The two following assertions are equivalent: (a) p is multiplicative

(b) V satisfies the formula:

(i ®I')(V) = (V 01)(10 )(V ®1)(10 a) where o is the flip operator from H ®l to 7-1®H, or from H ® H

to HOX

(ii) If re is a co-involution on M such that (M, F,,%) is a co-involutive Hopfvon Neumann algebra, then the three following assertions are equivalent: (c) p is involutive (d) for all w in M*, we have:

(t

W 0 K)(v) = (t 0 w)(V*)

1.5 Representations with Generator

31

(e) for all (1 in A*, we have:

i)(V)) = W 0 i)(V*)

-

Proof. Let wl, w2 in M* . We have, on the one hand: Y(wl * w2) = (i 0 (.'i * w2))(V )

= (i 0L01 0 w2)((i ®r)(V)) and, on the other hand: (i 0 wl)(V)(i 0 W2)(V) (i & wl)(V((i 0 W2)(V) 0 1)) (a 0 Wil(v((i 0 w2 & ,)(v 0 i)) _ (i (9 wl)(V(( i (9 W2)(1 (9 o)(V (9 1)(1 (9 a)) (i 0 wl)((i & i & W2)((V 0 1)(10 U)(V 0 1)(10 U)) (i 0 W1 0 W2)((V 0 1)(10 a)(V 0 1)(10 a))

For any (1 in A*, the equality: WW1 * W2), D) = WWOMP2), 12)

is therefore equivalent to: ((1 0 w, 0 W2)((= 0 fl(V)) = (D 0 w, 0 Ws)((v 0 1)(1 0 o)(v 0 1)(1 0 o))

which implies (i) by linearity and density. For every w in M*, (1 in A*, we have, on the one hand:

(QOwo,c,V*)_ (t2,(i ®w o K)(V * )}

On the other hand: fin, µ(w)) 021 (i 0 W) (V))

Therefore we get that (d) is equivalent to (c). Moreover, (d) can also be written: (S2 0 W 0 x)(V) = (12 (9 w)(V*)

(S2 E A*)

32

1. Co-Involutive Hopf-Von Neumann Algebras

which is trivially equivalent to:

w o,c((,fl 0 i)(V)) = w((Q ® i)(V*)) and therefore to (e), which completes the proof. 1.5.2 Definition. Let (M, F, x) be a co-involutive Hopf-von Neumann algebra;

let p be a representation of the Banach algebra M. on a Hilbert space fl . Let U be a partial isometry in ,C(1-ip) 0 M, whose initial and final supports are equal to P 0 1, where P is a projector in .C(fl, ). We shall say that U is a generator for p if it satisfies: Y(W) = (i 0 W)(U)

for w in M*

By linearity and density, U, if it exists, is unique; for any ,fl in ,C(f,2)*, we have p* (0) = (Q 0 i)(U); by the bicommutant theorem, it is easy to see that U belongs to Ap 0 M, and, thanks to 1.5.1 (i) and (ii), U satisfies: 0 -r)(U) = (U 0 1)(10 a)(U 0 1)(10 Cr) (i 0 W 0 K)(U) = (i 0 w)(U*)(W E M*)

1.5.3 Lemma. The projection P in 1.5.2 is the projection on the essential space of ,u. So U is unitary if and only if p is non-degenerate. Proof. Let us represent M on some Hilbert space H. Let a in ?-Cµ such that p(w)a = 0 for all w in M*. This is equivalent to (U(a ®/3) 7 0 6) = 0 for all

13,6in H,7in 'hp,or to U(a®/3)=0forall0inH,i.e. to Pa®,3=0for all P in H, and then to Pa = 0; this leads to the result. 1.5.4 Lemma. With the hypothesis of 1.5.2, let µ, (/lj)IEJ be representations of the Banach algebra M*, with generators Uµ, (Up,)jEI. Then: (i) If ji' is quasi-equivalent to p, then µ' has a generator. (ii) If p" is a sub-representation of /2, then p" has a generator. (iii) The representation ®iEl/2i has a generator. Proof. Let ofi be the morphism from Aµ to Aµ/ such that 4i o p = p'. It is easy to check that (4$ 0 i)(UU) is a generator for µ', which gives (i). There exists a projection P in (A/A)' such that p" = pp. It is then easy to check that (U)(p®1) is a generator for ,u", which gives (ii). Finally, ®zEIU, is a generator for ®iElpi.

1.5.5 Proposition. With the hypothesis of 1.5.2, let iz1 and 92 be two representations of M* having generators, respectively V1 and V2. Then the element

1.5 Representations with Generator

33

(cr 01)(1 0 V1)(cr 0 1)(1 0 V2) is the generator of µ1 X µ2, and the essential space of Ill x 92 is the tensor product of the essential spaces of Y1 and µ2. So, if µ1 and /2 are non-degenerate, 1U1 X /22 is non-degenerate too. Proof. For ,f11 in (Aµ1)* , (l2 in (Aµ2)* , we get: (/21 X µ2)*(J?1 (9 D2)

= µ1*(fl1)Y2*(,f12)

by 1.4.3

_ (Qi 0 z)(V1)92*(,(22)

by 1.5.2

_ (f21 0 0(V1(1 (9 µ2*(S?2)))

by 1.5.2 _ A 0 z)(V1(z 0 ,f12 0 i)(1 0 V2)) V2))) _ (21 0 z)((i 0 ,(l2 0 z)((U 0 1)(1 0 vl)(U because (i 0 ,f12 0 i)((o 0 1)(1 0 Vl)(v 1)) = Vl (.al 0 S?2 0 Z)((a 0 1)(1 0 Vi)(a 0 1)(1 0 V2))

Let V1,2 =(o 01)(10V1)((r 0 1)(10V2). We easily get: V1, 2V1, 2 = V1, 2V1*2 = Pµ1 ® PA2 01

and then deduce that Pµ1 ® Pµ2 is the projection on the essential space of µ1 X112.

1.5.6 Lemma. With the hypothesis of 1.5.2, let yl and 112 be two representations of M*, having respectively V1 and V2 as generators. Let t be in Hom(pl, 92), and if: Aµ1 Aµ2 a morphism such that 4 o y1 = µ2; then, we have:

(tol)Vi=V2(tol)

(i)

(ii)

(4i 0 i) (Vi) = V2

-

Proof. For all w in M*, we have: 0 W)((t 0 1)V1) = t(i 0 W)(V1) = tYl(W) = 92(W)t = (i 0 w)(V2)t = (i 0 w)(V2(t 0 1))

from what (i) is easily reached. We have also: (i 0 W)((,P 0 i)(Vi)) = iP(i 0 W)(V1) = q3 0 Al (W) = Y2(w) = (i 0 UJ)(V2)

which allows to complete the proof.

34

1. Co-Involutive Hopf-Von Neumann Algebras

1.5.7 Proposition. With the hypothesis of 1.5.2, let p be a non-degenerate representation of M* on the finite-dimensional Hilbert space C", {Ei}i 0 such that i(x, w) j < c jjp(w) j for all w in M*. It exists some 9 in C*(µ)* such that, for all w in M*, we have: (x) w) _ ((w), 9) _ (7 (w), 9 0 )) _ (ir(9 0 jC), w) which implies:

x=7r*(eop) and completes the proof of (i). Now, for Q in (Aµ) * and w in M*, we have:

(ir(s)(Q),w) = (*(f1),w) therefore, by Kaplansky's theorem, we have: w E M*, Il/2(W)ll : 1} = 111211

38

1. Co-Involutive Hopf-Von Neumann Algebras

From (i), we get that (s,4)(Q) belongs to B(µ) and that which completes the proof. 1.6.3 Proposition. Let G be a locally compact group; the enveloping C* -algebra

and W*-algebra of L1(G) are noted respectively C*(G) and W*(G), and will be called the enveloping C*-algebra and W*-algebra of G; let us write 7rG the continuous unitary representation of G associated to the universal representation 7r of L'(G); 7rG will be called the universal representation of G.

(i) For a non-degenerate representation p of L1(G), if µG is the corresponding continuous unitary representation of G, we have, for all t in G: sµ7rG(t) = µW

(ii) Every element f in 7r*(B(p)) C LOO(G) is almost everywhere equal to a continuous function.

(iii) ([46]) For every bounded continuous function f on G, and every representation y (including iz = 7r) of G, let us put: Ill il,. = sup{

IG'(s)w(a)ds 7 W E

L'(G)7 Ily(w)ll < 1 1

then 7r*(B(p)) (resp. 7r*(B(Ha(G))) appears as being the subset of all functions f in Cb(G) such that Ilf IIµ (resp. Ilf II,) is finite. It is a complete normed vector space, denoted Bp(G) (resp. B(G)). It can thus be identified to the dual of the C* -algebra generated by ,t(L1(G)) (resp. the dual of C* (G) i. e. the predual of W * (G)).

Proof. By a straightforward application of 1.3.10, we get (i). Let 9 be in (C*(/.t))*; it can be written as a linear combination of positive elements. It implies that 7r*(9 o /t) = p*(9) is a linear combination of positive definite elements of L°O(G), which, according to 1.3.11, completes the proof of (ii) and (iii) is just an application of 1.6.2. 1.6.4 Lemma. Let H be a co-involutive Hopf-von Neumann algebra, µ' and it" two representations of M*. We put s' = s pi and s" = s pil. We have:

x till) = µl x /1"

(i)

(S (9 i)S,XA = 3µIXµii (i ® .s") (y' x 70 = ttl X /2'I (Z 0 s II )8p'x,r = sµ1X p11

(s' 0 s")(ir x (s1

®3II)37CX7r

7r)=µ' x/1"

= sµiXµii

1.6 Fourier-Stieltjes Algebra

39

Proof. Let w be in M*, 1l' in (Ai)* and 1l" in (A,AFI)*. We have: ((s1 (9 i)(ir x

fl ®,1l") = (ir 0 p""(w),s*(RI) *(s*(f/)On ), = ((r x ®12")

w)

_ (ir*s(12')ji(Q"),w) 0 f2"), w)

x ((µi

by 1.4.3 by 1.6.1 (ii) by 1.4.3

x p1i)(w),12- ® 12")

from what, by linearity, continuity and density, (i) results; the proof of (ii) is identical and leads to (iii), thanks to the decomposition:

S/ ®S// = (s'Oi)(iOs")

.

1.6.5 Proposition. Keeping the notations used above, S7rx7r is a one-to-one normal morphism from W*(]H[) to W*(H) 0 W*(1EI) such that: (i ®s7rx ir)S7rx-7r = (s7rxjr

®i)s7rx,7c

If every representation has a generator, then Slcx7r is a co-product in the sense of 1.2.1.

Proof. Let w be in M*. We have: (i ® s,rx,r)s7rx,r(7r(w)) = (i 0 s,rx,r)(7r x 7r)(w) = (7r x (7r x 7r))(w) = ((ir x 7r) x 7r)(w) = (Sirxir 0 i)(ir x ir)(w) = (sirx, 0 i)s,rx7r(-7r(w))

by 1.6.1(ii) by 1.6.4(ii) by 1.4.3 by 1. 6.4 (i) by 1.6.1 (ii)

which by continuity and density implies the equality looked for. Considering the unit representation 1, we have also: (Si 0

(Si 0 i)(a x 7)(w) = (1 x n)(u) = u(w)

which implies, by continuity and density: (Si ®i)SWxW = i

by 1.6.1 (ii) by 1.6.4 (i) by 1.4.6

40

1. Co-Involutive Hopf-Von Neumann Algebras

from what it results that S7rxlr is one-to-one, which completes the first part of the proof. If every representation has a generator, then 7< x ,7r is non-degenerate (1.5.5) and s,r x 7r (1) = 1, which ends the proof.

1.6.6 Proposition. We define an antirepresentation *r of M*, by putting, for every w in M*: *(w) = 7r(w o K)

Then s;r is an involutive normal antiautomorphism of W*(H) which satisfies: Z>

7r*(sfir)* = K7r* S7rx7rSfr = S(Sfr 0 s7r)S7rX7r

Proof. Let w be in M. We have: s;rs*('7r(w)) = s*(7r(w)) = s*(7r(w o ic))

=-(woic) = 7r(w)

by 1.6.1 (ii) by definition by 1.6.1 (ii) by definition

by continuity and density, we get that s;r is involutive. Let d? be in (W*(H))*; we have: W) = (7r(LO), (Sfr)*('?))

(Sfr(7r(LV))' S2)

by 1.6.1 (ii)

(ir(woi),Q) _ (ir(Q),w o w) (Kr* (fl), W)

which proves (i). Let Sit, ,f12 be in (W*(H))*. We have: (sirxirs*(ir(w)), 01 (9 122)

= (S,,i.*(W), Sl 0 Si2)

by 1.6.1 (ii)

= (s,rx,rir(w o ic), Si (9 52 ) = (7r x 7r(w o rK ), i1 (9 52 )

by 1.6.1 (ii)

=((7rx7r)*(Si052),wolc) = (rc(7r*(Q1)7r*(Si2)),w) _ (Ic7r*(52)Ic7r*(Si1),w)

_ (lr*(S*)*(52)lr*(S*)*(Si1),w)

by 1.4.3

by (i)

1.6 Fourier-Stieltjes Algebra

_ ((ir x r)*((s;r)*(f22) 0 (s, _ ((7r X 7r)(w), (s*)(Q2) 0 (s*)*(Qi)) _ (s,,.x,rir(w), (s*)*(Q2) 0 (s*)*(Qi))

41

by 1.4.3

by 1.6.1 (ii)

_ ((sir 0 s*)s,rx, 7r(w), Q2 (& Q1)

_ (c(s* ®s*)s,rx,r7r(w), fl1 (3 Q2)

which, by continuity and density, completes the proof of (ii).

1.6.7 Theorem. Let H be a co-involutive Hopf-von Neumann algebra; let us suppose that every representation of the Banach involutive algebra M* has a generator; then the triple (W*(H), ssrx,r, s;r) with the definitions of 1.6.5 and 1.6.6, is a co-involutive Hopf-von Neumann algebra. If H is abelian, then W*(H) is symmetric; if H is symmetric, then W * (]HI) is abelian. Proof. The first part of the theorem is given by 1.6.5 and 1.6.6. Moreover, if M is abelian, we have S7r x 7r = 7r x 7r (1.4.3 ), and then s,r x,r = ss,r x,r; if IH[ is symmetric, M* is abelian (1.2.1) and then W * (H) too.

1.6.8 Corollary ([44]). Let G be a locally compact group, 7rG its universal r,;presentation, W * (G) its enveloping W * -algebra. There exists a unique morphism I' from W * (G) to W * (G) 0 W * (G), and a unique antiautomorphism ic in W * (G) such that, for all t in G:

r(nc(t)) =7c(t) 0 ac(f) K('7rG(t)) = 7rG(t)*

Moreover, (W*(G), r, n) is a symmetric co-involutive Hopf-von Neumann algebra, which we shall call the Ernest algebra of G. Proof. Using 1.4.7 and 1.6.3 (i), we have, for all t in G:

s,rx,ri'G(t) = iG(t) 0 G(t) s*7rG(t) = 70-1) and so, in particular s,r x,c (1) = 10 1, which ends the proof. 1.6.9 Theorem. Let H be a co-involutive Hopf-von Neumann algebra; then, the space B(H) = (W*(H))* is canonically equipped with a structure of semisimple involutive Banach algebra with unit of norm one. It will be called the Fourier-Stieltjes algebra associated to H. The mapping x7r* is a faithful

representation of this algebra in M; it will be called its Fourier-Stieltjes representation.

42

1. Co-Involutive Hopf-Von Neumann Algebras

Proof. By transposing the mappings ss,rx, and s;j., we get a product * and an involution ° on B(H). For (ll and (12 in B(H) and w in M*, we have: (Klr*(Q1 * Q2), w) = (ir(w o ti), (Ii * (12)

= (csx,.7r(o re), (ll 0 (12) by definition by 1.6.1 (ii) by 1.4.3

= ((-7r x 7r)(w o rc), f22 0 Q1)

= (lr*(Q2)lr*(Q1),w o rc) = (!7r*(Q1)!7r*(Q2),w)

Moreover, we have:

((ir(woi),Q°) = (s*(ir(w o x)*), ,fl) -

by definition

= (s*(ir(w° o 0, f2) = (7r(w° o K), i2) -

by 1.6.1 (ii)

= (ir(w°),Q) = ('7r*

(0),w°)-

= (,7r*(Q)*,w) Since 7r is non-degenerate, 7r* is faithful and B(H) is semi-simple. It results from the decomposition y* = 7r*(sµ)* that we have 7r*(B(IEII)+) = PR(H); we have seen in 1.3.7 that the unit of M belongs to PR(]HI), it follows that 7r* 1(1)

is a unit for B(H); it is easy to check that 117r* 1(1) = 1, which completes the proof. 1.6.10 Remarks. Since it results from 1.6.2 (i) that -7r* (B(]HI)) is stable by

ril, the fact used above that 7r* (B(H)+) = PR(H) implies that the image of B(H) by the Fourier representation is the set of linear combinations of elements of PR(IHI), that is, also, the set of elements which can be written µ* (,fl ), where µ is a representation of M* and (l is in (A). Since 7r* is faithful, we can deduce that B(H) is composed of those elements of the form (sµ)*((2). Thanks to 1.6.2 (i), we observe, at last, that PR(IH[) is stable by rc. 1.6.11 Theorem ([46] ). Let G be a locally compact group, and f a function on G. The following assertions are equivalent:

(i) The function f belongs to B(G), i.e. (1.6.3 (iii)), f is a bounded continuous function on G, and: sup

IG f (t)h(t)dt

I

h E L'(G), 117r(h)JI < 1 1 < +oo

where -7r is the universal representation of L1(G).

1.6 Fourier- Stieltjes Algebra

43

(ii) The function f is a linear combination of continuous positive definite functions on G. (iii) There exists a Hilbert space H, vectors 6,77 in H, and a continuous unitary representation 11G on H such that f (t) = (IG(t)e 177)

Moreover, B(G) is a subalgebra of the algebra Cb(G) of continuous bounded functions on G. Equipped with the norm:

IG f(t)h(t)at h E Ll(G), 11n(h)II

1}

B(G) is a Banach algebra, which can be identified with the dual of C*(G) (i.e. the predual of W * (G)), the duality being given, if f (t) = (,G(t)r I r7), and h is in C* (G), by: (f, h) = (p(h) 17l) (where µ denotes here the representation of C*(G) associated with jG).

Proof. Using 1.6.3 (iii) and 1.6.10, we see that B(G) is the image of the involutive algebra B(1H[a(G)) by its Fourier-Stieltjes representation. Then we have the implication (i) = (ii) by 1.6.10 and 1.3.11; (ii) = (iii) is a corollary

of 1.3.11, and (iii) = (i) is a straightforward calculation. The end of the theorem is just a re-writing of 1.6.3 (iii).

Chapter 2 Kac Algebras

This chapter deals with technical results about Haar weights, as they have

been studied by the authors in [36] and [136], and, independently, by Vainerman and Kac ([180] ). On a co-involutive Hopf-von Neumann algebra

(M, F, it), a Haar weight is a faithful, semi-finite, normal weight on M+, which is left-invariant with respect to F, i.e. such that: (i ® Sp)T(x) = W(x)1 for all x in M+

(in 2.5, we show, after Kirchberg, that this axiom may be weakened), and, roughly speaking, satisfies two other axioms involving i . The quadruple (M, T, x, p) is then called a Kac algebra. On L°O(G), the integral defined by a left Haar measure is obviously a left-invariant weight, and it is straightforward to see that it is a Haar weight (2.2.2 ). Thus, L°O(G), equipped with the convenient operations, is an abelian Kac algebra; we shall denote it K(G). Left-invariantness allows us to define the fundamental operator W, which is an isometry belonging to M ®,C(Hw), satisfying the so-called pentagonal relation (2..4.4 ):

(1OW)(cy®1)(1OW)(cr®1)(WO1) = (W ®1)(1 ®W) With the other axioms, we get that W is unitary (2.6.2 ), "implements" the co-product T, i.e. that we have, for all x in M (2.6.3 ): .P(X) = W(l 0 X)W* Moreover, the mapping: A : w --+ (w 0 i)(W*)

from M. to C(H,,) is a non-degenerate representation of the involutive Banach algebra M. (2.6.1 and 2.6.3); it will be called the Fourier representation of the (predual of the) Kac algebra.

2.1 An Overview of Weight Theory

45

Then, W and A appear both as generalizations to an abstract Kac algebra of the left regular representation AG of a locally compact group G; more precisely, for the Kac algebra Ka (G), the fundamental operator is the bounded continuous function s -+ AG(s)*, when considered as an element of L°O(G) ® £(L2(G)), and the Fourier representation is nothing but the left regular representation of L'(G) (2.4.7 and 2.5-4).

Another result strengthens the analogy with the case of locally compact groups: on a co-involutive Hopf-von Neumann algebra, the Haar weight (whenever it exists) is unique up to a positive scalar (2.8.6 ). The situation of locally compact groups is generalized, too, with the definition of square-integrable elements of M*, i.e. elements w of M* such that there exists a (unique) vector a(w) in HW such that, for all x in (2.1.6 ): (X*

)

w) = (a(w) I AV (x))

If the weight Sp is left-invariant, the set I,, of such elements is a left ideal of M* (2.4.5 ), and, if Sp is a Haar weight, we have, for all wl in M* and w2 in IV (2.4.6): a(w1 * w2) = A(w1)a(w2)

In the case of K. (G), we recover L1 (G) fl L2 (G), which is a left ideal of L1(G),

and the formula: .fi * .f2 = AG(.fi).f2

(fi E Ll(G), f2 E L'(G) n L2(G))

which links the convolution and the left regular representation of G. If K = (M, F, x, gyp) is a Kac algebra, so is K = (M, cf', ic, cp o ic), and

it is possible, too, to define a Kac algebra K' on the commutant M' of M (we suppose M is a von Neumann algebra on the Hilbert space H

[0, +oo] such that W (Ax) = Aco (x) for all A

in R+ and x in M+. We shall use the notations and constructions of the Takesaki-Tomita theory associated to weights (cf. [158], [14], [114], [16], [56] or [150]) hereafter briefly summarized: Let us define:

9 ={xEM+, '(x) [0, +oo], which satisfies m(Aw) = Am(w) for all A in R+ and w in W_- Given m, n in M+, A in R+ and x in M, it is straightforward to define m + n, Am and x*mx in M+, and it is clear that M+ is naturally imbedded in M+ . Moreover, ([59], cor. 1.6), for every m in M+, there exists an increasing sequence of elements of M+ converging up to m. Let H be an Hilbert space on which M is standard (1.1.1 (iii)); then, ([59], lemma 1.4), for every m in M+, there exists a closed subspace H' of H, and

a positive self-adjoint operator T on H' such that D(T) - = H' and: (6 E D(T 112))

IIT'I2e112

+00

(6

V

D(T1/2

))

2.1 An Overview of Weight Theory

53

Now, let N be a von Neumann subalgebra of M; an operator valued weight on M with values in N is a mapping E : M+ -' N+ such that:

E(m + n) = E(m) + E(n)

(m, n E M+)

(A E ][Y+, m E M+) E(Am) _ AE(m) E(a*ma) = a*E(m)a (a E N, m E

M may be considered as an operator valued weight on M with values in the von Neumann subalgebra C1 M . Faithfulness, semi-finiteness and normality of operator valued weights are defined exactly the same way as for weights. Those operator valued weights such that E(1M) = 1N are called conditional expectations; as we have then

E(M+) = N+, E can be extended to a positive linear map from M to N, which satisfies E2 = E. (ii) Let M be a von Neumann algebra, N a von Neumann subalgebra of N, V (resp. 0) a faithful semi-finite normal weight on M (resp. N). Then,

if at (x) = °t (x) for all x in N and t in R, there exists a unique faithful semi-finite normal operator valued weight E on M with values in N such

that So= OR If the restriction of Sp to N+ is semi-finite, we may take 0 = V N; the operator valued weight E is then a faithful normal conditionnal expectation which is associated to the projection P of HV on the closure of Ap( fl N),

that is, for any x in M, Ex is the unique element of N such that (Ex)P is equal to PxP. Moreover, for all x 91., we have ([160]): PelW(x) = Av(Ex)

Let A be a von Neumann algebra and w be a positive element in A*, such that w(1) = 1. We may then define a conditional expectation (i ® w) from M ® A to M ® C M, such that, for any faithful semi-finite normal weight on M, we have:

wo(i®w)=5P®W For any w in A*, we may define (i ® w) by linearity.

(iii) The tensor product of faithful semi-finite normal operator valued weights may be defined, using the tensor product of weights. Using this, it is possible to define a faithful semi-finite normal operator-valued weight i 0 from A 0 M to A ® C r-o A, where A, M are von Neumann algebras, and Sc a faithful semi-finite normal weight on A. Moreover, for any positive element w in M*, we shall have:

wo(i0c) =w®(

54

2. Kac Algebras

2.1.9 Lemma. For all w in M*, we have: (w

® i)( i®w) C 9'tw.

Proof. Let us suppose w is positive; for any X in 9fl:Ow , then (w ® i)(X) is positive and we have:

i)(X)) = W(i 0 0)(X) < +00

from which it results that (w ® i)(X) belongs to 9R+, and (i) follows, by linearity. Let us suppose now that w is positive and w(1) = 1. Then w ® i is a normal conditionnal expectation from M 0 M to M, and we have, for all X in M ®M: ((w 0 i)(X ))*(w 0 i)(X) < (w ® i)(X *X )

whence, if X belongs to SP(((w 0 i)(X))*(w 0 i)(X )) < W(w 0 i)(X *X) < +oo

by (i)

from which (ii) follows, by linearity.

2.1.10 Lemma. Let x be in 9

and y be in 9XLA,(.)®(P; then (x* 0 1)y(x ®1)

belongs to 9, w®Sp, and we have:

(w4,() 0 Sp)(y) = (SP 0 p)((x* 01)y(x ® 1))

Proof. For all a in M+, we have: WA,(x)(a) = Sp(x*ax)

and this equality remains true if a is in the extended positive part of M; in particular, for all y positive in M 0 M, if we replace a by (i 0 )(y), we get: WA,(2:)((i 0 WXY)) = V(X*(i 0 V)(Y)X) = P((i 0 W)(X* 0 ')Y(x (D 1))

which may be written, using ([59], 5.5): (w4j,(Z) 0 SP)(y) = (5P (9 W)((x* (9 1)y(x (9 1))

Then, both sides are finite if y belongs to 9R by linearity.

W

(X)®W, and the result comes

2.2 Definitions

55

,2.2 Definitions 2.2.1 Definitions. Let (M, r) be a Hopf-von Neumann algebra and Sp be a faithful semi-finite normal weight on M. We shall say that W is a left-invariant weight with respect to r if it satisfies the following property:

(i 0 )r(x) = P(x)1

(LIW)

(x E M+)

Then (p satisfies the weaker property: (HWi)

r(9zW) c 91iow

Let H = (M, r, x) be a co-involutive Hopf-von Neumann algebra, and (p be a faithful semi-finite normal weight on M. We shall say that Sp is a Saar weight on H if it complies with (HWi) and the two following axioms: (HWii) (i 0 0)((' 0 Y*)F(x)) = 14(i 0 W)(-r(Y*)(1 0 XM

(x,yE4YL,)

(left and right-hand sides of the equation in (HWii) make sense, thanks to (HWi)) (HWiii)

Kct =7!t

(teR)

2.2.2 Proposition. Let G be a locally compact group and Ha(G) = (L°°(G), ra,'ca) the associated abelian co-involutive Hopf-von Neumann algebra (1.2.9). Let us consider the faithful semi-finite normal weight Spa on L°O(G), which is the trace arising from the Saar measure on G by:

IG

f(3)ds

(f

E L°O(G)+)

It is a left-invariant and a Saar weight on Ha(G). Moreover, the weight SO is finite if and only if the group G is compact. Proof. We identify HWa with L2 (G), 9 a with L2 (G) n L°O (G), 9AWa with L'(G) n L°O(G); let f in LOO(G)+; since JG If 112(st)dt = fG If 112(s)ds, it is clear that (LIW) holds. The translation of (HWii) is: IG

(t)f(st)dt = J (s1t)f(t)dt

(f, g E L2(G) n L°O(G))

which results straightforwardly from the left-invariance property of ds. And

(HWiii) is trivial, Sp being a trace.

56

2. Kac Algebras

2.2.3 Proposition. Let H = (M, r, ic) be a co-involutive Hopf-von Neumann algebra, and Sp be a faithful semi-finite normal weight on M. If Sp satisfies (LIW) (resp. (HWi), (HWii), (HWiii)), then cp' satisfies the same property. Proof. Let x be in M'+, and So a left-invariant weight; then:

(i 0 p')((J 0 J)f(J5J)(J 0 J)) = J(i 0 )r(J5J)J = cp(J5J)1 = cp'(x)l

(i

So cp' is a left-invariant weight. Proving axiom (HWi), (HWii) and (HWiii) is just a straightforward cal-

culation of the same kind, using 91,i = J9p J. 2.2.4 Proposition. Let H = (M, I', rc) be a co-involutive Hopf-von Neumann algebra, and Sp be a faithful semi-finite normal weight on M. If Sp is a leftinvariant (resp. a Haar) weight on H, then cp o x is a left-invariant (resp. a Haar) weight on

Proof. In 1.2.10 (ii), we have seen that the application u(x) = Jrc(x)*J from M to M' is an H-isomorphism from W to H'. Furthermore, we have gyp' o u (x) = Sp o n (x) for all x in M+; so this Hisomorphism exchanges the weight W o ic with the weight W'. The result is then a corollary of 2.2.3. 2.2.5 Definitions and Notations. Let H = (M, r, rc) be a co-involutive Hopfvon Neumann algebra, and cp a Haar weight on H. We shall then say that the quadruple K = (M, r, rc, Sp) is a Kac algebra. By 2.2.2, for any locally compact group G, (L°°(C), ra, rya, cpa) is a Kac algebra, denoted K, ,,(G). For any Kac algebra K = (M, I', rc, W), we denote K' = (M', I'', rc', cp') and

K1 = (M, cr, rc, Sp o i) the Kac algebras associated respectively in 2.2.3 and 2.2.4. They will be called respectively the commutant Kac algebra of K, and the opposite Kac algebra of K. Evidently, we have Ka (G)' = Ka (G) and Ka (G)' = Ka (G°PP ).

Let K = (M, I', rc, cp) be a Kac algebra, and a > 0. It is clear that (M, I', rc, aSo) is a Kac algebra, which will be denoted alts. Let K1 = (M1, , Vi) and K2 = (M2, F2, K2, Sp2) be two Kac algebras. We shall say that K1 and K2 are isomorphic if there exists an H-isomorphism: I'1, id

u:(Ml,ri7ic1)-+(M27F2,K2) and a > 0 such that W2 o u = acpl. Clearly, then, u is implemented by a unitary U from H(1 onto HW2 defined for all x in 'Awl, by: UJLp1(x) = a_1/2A p2 (U W)

Therefore K and aK are isomorphic.

2.2 Definitions

57

2.2.6 Proposition. Let K = (M, r, x, W) be a Kac algebra and R a projection in the centre of M, such that:

F(R) > R 0 x x(R) = R We shall denote by KR the quadruple (MR, rR, KR,'PR) where, for all x in M: (i) MR is the usual reduced algebra; the canonical surjection M - MR will be denoted by r;

(ii) (iii)

.PR(r(x)) = (r 0 r)(-P(x)) KR(r(x)) = r(K(x))

(iv) the weight coR on MR is obtained by reduction from co on M as in ([16], def. 3.2.4).

Then KR is a Kac algebra, called a reduced Kac algebra of K, and r is a surjective H-morphism.

Proof. It is trivial to check that (MR, -PR, KR) is a co-involutive Hopf-von Neumann algebra and that r is an H-morphism. We have 9XV, = r (9J1 , ) which implies:

ra(9XnA) = ra(r(9Rw)) = (r 0 r)r(m,,) C (r 0

9Ri,&nA

Let x, y be in'7Lp. We have:

(i 0 'VR)((1 0 r(Y)*)rR(''(x))) ° (e 0 'PR)(T 0 r)((1 0 Y*)r(x)) (i 0 V)(r 0 r)((l 0 y*)-r(x)) _ (i 0 W)((R 0 Ry')r(Rx)) R(i o V)((1 0 Ry*)-P(Rx))

Rk(; 0 »)(r(RV')(1 0 rte)) tqi 0 V)(F(Ry*)(R 0 Rx))

= rqa 0 w)((R 0 a)r(v`)(1 0 x)) KR(i 0 WR)(r 0 r)((-P(y*)(1 0 x))

= kR(l 0 'OR)(r(T(Y)x)(1 (9 r(x)))

As rat = atR r, the axiom (HWiii) is trivially proved.

2.2.7 Proposition. Let K = (M, I', ic, Sp) a Kac algebra and M a sub von Neumann algebra of M such that:

58

2. Kac Algebras

iM ® M N

()r(1cf)c N

N

N

(ii) ic(M)= M

(iii)a'°(M)=M (tER) N

(iv) the restriction Sp I M+, which NwillN be denoted S ,N is a semi finite weight. M W e shall denote K the quadruple (M, r, R, ,P) where r and R. are respectively N N

the restrictions of F and ic to M; the canonical one-to-one morphism M -+ M will be denoted by j. Then, K is a Kac algebra, called a sub-Kac algebra of K, and j is a oneto-one IRE-morphism.

Proof. Everything is proved by restriction. In particular, using ([16], 3.2.6),

we have at = at J

In what follows, (M, r) is a Hopf-von Neumann algebra and V a faithful, semi-finite, normal weight on M, satisfying (HWi).

2.3.1 Lemma. For all w in M*, we have:

(ii)(w®i)r(9l,)c9t4,. Proof. It is clear that (HWi) implies r(971,) C 9721®,; so the proof results immediately from 2.1.9 (i) and (ii). 2.3.2 Definition. Thanks to 2.3.1, we can define an unbounded linear operator

£(w) on H. by: V(f(w)) = AV(91p) t(w)AV(x) = Aw((w 0 i)-P(x))

2.3.3 Proposition. For all w, w' in M*, we have:

i(w * w') = i(w Aw)

.

Proof. Let x be in 91,; we have: t(w * wJ)AV (x)

_ AV((w * w' ® i)r(x)) = A ((w ®w' ® i)(r ® i)r(x))

by 2.3.2

2.3 Towards the Fourier Representation

59

=An((w0 oi 0 i)(i 0 flf(x)) = AW((WI 0 i)(W 0 i 0 0(i 0 F).r(x)) AW((W' 0 i)F((w 0 i).r(x))) = e(w by 2.3.1 and 2.3.2 0 OrW) =Q(w )f(w)A,p(x) by 2.3.2

2.3.4 Lemma. For all w in M*, x, y in 91., we have: (t(w)A,(x) I A ,(y)) = (w 0 p)((1 0 y*)r(x)) Proof. We have, if w is positive and w (l) = 1:

(e(w)e(=) I Fe(y)) = (nv((w (9 .)(r(=)) I ne(v)) = W(Y*(w 0 OFW)

by 2.3.2

= (W 0 W)((1 0 Y*).r(x))

and so the result is proved, by linearity.

2.3.5 Definition. Let H = (M,1', ic) be a co-involutive Hopf-von Neumann algebra, and cp a faithful, semi-finite, normal weight on M, satisfying (HWi). Then, we define, for all w in M*, an unbounded operator A (w) by: A(w) = t(w o K)

We have then: V(A(w)) = nn(`nw)

a(w)A,o(x) = 4((w o K 0 i)P(x)) A(w * w') = A(w)A(w')

(x E 7to)

(W, W' E M*)

(A(w)A4,(x) I ma(y)) = (w o x 0 w)((1 0 v')r(=))

(w E Ma X, Y E NW)

2.3.6 Example. Let G be a locally compact group; let us consider the Kac algebra &(G) defined in 2.2.5; for all f in L1(G) the operator A(f) defined in 2.3.5 is equal to the restriction of AG (f) to L2 (G) fl L°O (G), where AG is the left regular representation of L1(G) (cf. 1.1.5).

Proof. By 2.3.5, we have, for f in L1(G), g in L2 (G) fl Lo' (G), tin G:

(A(f)g)(t) =

IG

f

(s)g(s-1 t)dt

= (AG(f)g)(t)

by 1.1.5

60

2. Kac Algebras

2.3.7 Example. Let us compute the mapping A' associated to the co-involutive Hopf von Neumann algebra El' and the weight Sp'. Let w be in. M*; we note

w' the element of M' defined, for all x in M', by:

(x,S) = (Jx*J,w) We easily get (w o rc)' = w' o ice, and w'(JxJ) = w(x) for all x in M. Then, we have, for x in gyp:

JA'(w')JA,(x) = JA'(wi )Aw'(JxJ) = JAW, (P 0 K11 0 i).r'(jXj)) 1

= JAW, MW o rc)' (& 0(1 0 T)P'(x)(J 0 J)) = AW((ZU -Orv 0 i)-F(X))

by 2.3.5

- A(6j-)AW(x)

and then: A'(WI) = JA(W)J

2.4 The Fundamental Operator W In what follows, (M, T) is a Hopf-von Neumann algebra and V a left-invariant weight with respect to T (so Sp satisfies (HWi)).

2.4.1 Lemma. For all x, y in gyp, we have: (9 0)((X* 0 ').P(Y*Y)(x 0 1)) =

Proof. By definition (2.2.1), we have:

(i 0 w)r(y*y) = V(y*y)1 whence, by ([59] 2.1 (3); cf. 2.1.8 (i)): V(Y*Y)X*X = X*(i 0 V)F(Y*Y)X *

= (i 0 VWX 0 l)-P(Y*Y)(x 0 1)) Then: W(X*X)W(Y*Y) = (V 0 w)((x* 0 l)-r(y*y)(x 0 1))

2.4 The Fundamental Operator W

61

2.4.2 Proposition. (i) There exists a unique isometry W such that, for every x, y

91., we have:

W(A,(x) ® jtW(y)) = Al,®,(r(y)(x ®1))

(ii) For all x in M, we have:

I'(x)W = W(1®x) (iii) W belongs to M ®,C(HV) Then, W is called the fundamental operator associated to (M, F, Sp).

Proof. By polarization and linearity it follows from 2.4.1 that, for all Si, x2, Y1, y2 in 91, we have:

0 1)r(y2y1)(x1 0 1)) = P(x2x1)'P(y2y1) which can be also written as:

(A®,(r(y1)(x1 ®1)) n ,®c,(r(y2)(x2 0 1))) = (A,(xi) 0 AS0(yl) I AV(X2) 0 A (y2))

p) is dense in H,, 0 H,,, there exists an isometry W in ,C(HW 0 H,) such that, for all x, y in, we have: As AW(91W) ® A,p(

W(Aw(x) 0 AW(y)) = AVOW(F(y)(x (9 1))

which is (i). Let x be in M, y, x in 91.. We have:

.P(x)W(A,(y) 0 A,(z))

-P(x) Avow (IV(z) (y 0 1))

by (i)

Aw(&w(-P(xz)(Y 0 1))

= W(AW(y) 0 Aw(xx)) W(l 0 x)(AW(y) 0 AV(z))

by (i)

By linearity, continuity and density, we obtain (ii). Let a, b be in gyp. We have:

(JaJ 0 JbJ)W(Aw(x) 0 AW(y)) = r(y)(x (9 1)(JA,(a) ® JA9p(b)) = r(y)(JaJA0(x) 0 JAW(b))

= r(y)(JaJ 0 1)(A,(x) 0 JAW(b)) This equality still holds, by continuity, when AW(x) converges to any vector e in H,, and when a strongly converges to 1. Therefore, we have:

(10 JbJ)W( 0 AV(y)) = -P(y)( 0 JAV(b))

2. Kac Algebras

62

which is valid for all y in 91, and

in Hg,. Let x' be in M' and let us replace

by x' in the above formula: (1

JbJ)W(xi 0 1)(C 0 iv(y)) = r(y)(x 0 1)(E 0 JnW(b)) = (X 0 1).r(y)(C 0 JAW(b)

= (x' 0 1)(1 0 JbJ)W(C 0 An(y)) which still holds when b strongly converges to 1, then we get:

W (x' 01)( 0 A(y)) =

(x'

01)W(6 0 AW(y))

By linearity, density and continuity we can conclude that W and

x' ® 1

commute, therefore W belongs to M 0 ,C(HI). 2.4.3 Lemma. We have, for all x, y in 91V and w in M* :

w((i 0 )(r(y*)(1 0 x))) = (A,(x) I (w 0 i)(W)A,(y))

Proof. Let a be in; we have: WA,(a)((i 0 P)(F(Y*Xl 0 x)))

_ (w 0 w)((a` 0 i)r(v')(a 0 x))

_ (A1,(a) 0 An(x) I AWOn(r(y)(a 0 1)))

by 2.1.10 by (LIW)

0 aW(=) I N'(nw(a) 0 ew(v)))

by 2.4.2 (i)

_

(AV W I (wA, (a) 0 0 (W) Aw W)

and so the result is proved by continuity and linearity arguments.

2.4.4 Corollary. Let (M,1') be a Hopf-von Neumann algebra, and ( a leftinvariant weight with respect to 1. Then: (i) for all w in M*, the linear mapping £(w) defined in 2.3.2 by:

An(X) -'

0 i)I'(T))

(x E NW)

is bounded; we shall still note £(w) its unique continuous extension to Hg,; it satisfies: £(w) = (w 0 i)(W) and I_< IIwII

(ii) the fundamental operator W satisfies:

(r 0 i)(W) = (10 W)(a 0 1)(1 0 W)(a 0 1)

2.4 The Fundamental Operator W

63

which can also be expressed as the so-called "pentagonal relation":

(10 W)(o ® 1)(1 0 W)(cr 0 1)(W 01) = (W 0 1)(1 0 W) .

Proof. Let w be in M; , x, y in'JIp; we have: (AW(x) I i(w)AW(y))

by 2.3.2

(AW(x) 14((w 0 i)-P(y)))

V(((w 0 i)f(y*))X) W((i 0 O)TWX1 0 X)) (A,(x) I (w 0 i)(W)AW(y))

by 2.4.3

By linearity, we see that, for all w in M* , £(w) is the restriction of (w 0 i) (W ) to A ,(')1,); so (i) is proved. If we apply (1.5.1 (i)), to the linear mapping w -+ £(w) = (i 0 w)(oWo) and to the Hopf-von Neumann algebra (M, cI'), we get:

(i 0 sr)(awa) = (awa (9 i)(1

)(iWa

a)

from which we deduce:

(1' 0 i)(w) = (10 wXU 0 1)(1 0 w)(a 0 1) The pentagonal relation comes then from 2.4.2 (ii).

2.4.5 Theorem. Let (M, T) be a Hopf-von Neumann algebra, and V a leftinvariant weight with respect to T. Then IV (defined in 2.1.6) is a left ideal in M*; moreover, we have, for all w in M,, and wo in I.: a(w * wo) = t(w)*a(wo)

and then: lw * wollSP :5 lIwlI llwolko

Proof. Let x

91,; we have:

(x*,w*w0) = (X'Co

Cc1o)-

by 1.2.1

= ((CA) 0 i)T(x),LOO)

= (((w ® i)I'(x))*,wo) = (a(wo) I A ,((w 0 i)I'(x))) = (a(w0) l e(w)AvW) = (C(am)"a(wo) I A,o(x))

by 2.3.2 by 2.4.4 (1)

64

2. Kac Algebras

which gives, by density, the first result; the norm inequality is just then a staightforward application of 2.1.6 (ii) and 2.4.4 (i).

2.4.6 Proposition. Let H = (M, I', ic) be a co-involutive Hopf-von Neumann algebra, and cp a left-invariant weight with respect to F. Then: (i) For all w in M*, the linear mapping A(w) defined in 2.3.5 by: A(w)Aw(x) =

(x E

A(p((w o re 0 i)r(x))

)

is bounded; we shall still note )(w) its unique continuous extension to H.; it satisfies: A(w) = (w o ,c ® i)(W)

and

IIA(w)II C 11wII

(ii) For all w in M* and all wo in I,, we have: a(w * wo) = A(w°)*a(wo)

(iii) I , fl i., is a dense involutive subalgebra of M*. (iv) The weight W satisfies (HWii) if and only if A is involutive; we have then, for all w in M* and all w' in I,,: a(w * w') = A(w)a(w')

(v) The weight cp satisfies (HWii) if and only if, for all w in M*, we have:

(wOi)(W*) = (woicOi)(W) . Proof. The assertion (i) is just a rewriting of 2.4.4(i). Thanks to 1.2.5, (ii) comes from 2.4.5. By 2.1.7 (ii), we see that I,, is a dense left ideal of M*; so is a dense right ideal of M*, and the algebraic tensor product I Q IV is a dense subspace of (M ®M)*. Let x be in M such that (x, w1 * w2) = 0 for all w1 i w2 in IV. We get (F(s), wi 0 w2) = 0 which therefore implies x = 0. Thanks to the Hahn-Banach theorem, it follows that the subspace generated by Iw' * I, is dense in M*. But, by 2.4.5, this subspace is included in Ip fl and so (iii) is proved. Using 2.3.5, we get, for any w in M*, x, y in 9lw:

(A,(x) I A(w°)A ,(y)) _ (A(w°)A,(y) I A,(x))

_ (w° o rc 0 )((1 0x)I'(y)) _ (w

x*)f(y))-

_ (w ®SP)(r(y*)(1 0 x)) Using again 2.3.5, we have: (AW(x) I A(w)*Aw(y)) = (w o x 0 p)((l 0 y*)F(x))

by 1.2.5

2.4 The

Fundamental Operator W

65

And, by density and continuity, we get (iv). If we apply 1.5.1 (ii) to the linear mapping w -4 A(w o r%) = (i (9 w)(cWa) and to the co-involutive Hopf-von Neumann algebra W, we get that A is involutive if and only if, for all w in M*, we have: (i 0 w o rc)(o*Wc) = (i 0 w)(oW*cr)

and so (v) comes from (iv). 2.4.7 Proposition. Let G be a locally compact group. (i) The function s -+ AG(S), considered as an element of Cb(G,,C(L2(G))), which can be identified to a subspace of ,C(L2(G))®L°O(G), is equal to cWGc, where WG is the fundamental operator of the Kac algebra Ka(G). (ii) The ideal IVa of the Kac algebra Ka(G) is L1(G)nL2(G); the mapping a is then the canonical injection from L1(G) n L2(G) into L2(G). Proof. Let 91, 92 be in Loo (G) n L2(G); by definition, we get, for s, t in G:

(W(g1 ® 92))(s, t) = 92(st)gl(s) where we identified L2(G) ® L2(G) with L2(G x G). By continuity, for all f in L2(G x G), we get:

(WGf)(S,t) = f(s, St) and:

(crWGaf)(s,t) = f(ts,t)

We identify L2 (GI L2(G)) with L2(G) ® L2(G) (or L2 (G x G)): OP in L2(G, L2(G)) will be identified to the function f : (s, t) -+ (t)(s). We have then:

(AG(F1)(t))(S) ='P(t)(ts) = f (ts, t) = (crWGof)(S,t) = (aWGo(t))(s) so we get (i); and (ii) is trivial. 2.4.8 Proposition. The fundamental operator W' associated to (M', I'', Sp') as defined in 2.2.3 is:

W, _ (10 J)W(J 0 J). Proof. It is an easy calculation, using the identification AV(JxJ) = JAW(x), for all x in 91W.

2.4.9 Lemma. Let (M, I') be a Hopf-von Neumann algebra, co a left-invariant weight with respect to r, W the fundamental operator associated by 2.4.2; let A be a von Neumann algebra, 0 a faithful semi finite normal weight on A,

66

X in

2. Kac Algebras

,and y in; then, (i 0 r)(X)(1 0 y 0 1) belongs to

and we have:

0 r)(X)(1 0 y & 1)) = (10 W)(1 0

0 Ap(y)) -

Proof. We have:

( 0V ® p)((1®y* ®1)(i ® r)(X *X)(1®y ®1)) = SP(y* (O ®z ®W)(i 0 r)(X *X )y)

_

_

(*(O

® co)(X *X )y)

(y*)(O

0 co)(X*X) < +oo

which gives the first part of the result. After polarization, we get, from the calculation above, the existence of an (9 F)(X)(1 0 y ®1)). isometry U sending A,,®,(X) 0 A,,(y) on Now, let xl be in 970, x2 be in 91W. We have: AjO®WOSp((i 0 r)(x1 0 x2)(1 0 y 0 1))

= A,,(xl) ® A W(r(x2)(y ®1)) by 2.4.2 = A,(xl) 0 W (Aw(y) 0 A,(x2)) = (10 W)(1® o)(A,(xi) 0 Aw(x2) 0 Ap(y)) Therefore U coincides with (1® W)(10 a) on all the vectors which are of the form A, , (x i) 0 AW(x 2) 0 A , (y ), which, by linearity, continuity and density, completes the proof.

2.5 Haar Weights Are Left-Invariant In the sequel, H = (M, r, ic) is a co-involutive Hopf-von Neumann algebra, and co a faithful, semi-finite, normal weight on M, satisfying (HWi) and (HWii); so, we can use the results 2.3.5 (but not 2.4 which is about leftinvariant weights, which may not satisfy (HWii)). 2.5.1 Lemma. For all w in M*, and x in 91,,, we have: AW(91w) C D(A(w)*)

and

A(w)*Aw(x) = A(w°)A,(x)

Proof. Let us assume that w is positive and w(1) = 1. We have, for all y in

(a(w)l(=) I Aw(v)) = (w 0 K 0 w)((1 0 v')r(=))

by 2.3.5

2.5 Haar Weights Are Left-Invariant

67

_ (w

0 X))) W((w (9 i).P(y*)x)

(AV(x) AV(w 0 i).P(y)) (A,,(x) A(w")AW(y)) by 2.3.5 and the hypothesis So, we get A(w)* D A(w°) which is the lemma.

2.5.2 Lemma. Let C be in 11,(91,p). The linear form vi on M*, defined by vo(w) = (a(w)e I C) is continuous, and IIii

11
0 such that: 'p2 o u(x) = a5p1(Rux)

Vx E M1 .

Proof. From 1.2.7 we get Ti (Ru) > Ru 0 Ru and x 1(Ru) = Ru. So, from 2.2.6, the quadruple K1Ru = (M1Ru , P Ru ) K1Ru , SP1Ru) is a Kac algebra. Let r be the canonical surjection M1 --> M1Ru 'We can define a bijective H-morphism

v from M1Ru to M2 by v(r(x)) = u(x). Clearly, 'P2 0 v is a Haar weight on K1Ru and then, using 2.7.7, there is a > 0 such that, for all x in M1Ru : '2 o v(x) = acolRu (x)

So, for all x in M: '2 0 u(x) = c02 o v(r(x)) = a'P1Ru (r(x)) = aSP1(Rux)

2.7.9 Corollary.Let K1 = (M1, I'i, i1, 'l) and K2 = (M2, I'2, K2, 'P2) be two Kac algebras, u an H-isomorphism from (Mi,rl,Ki) to (M2,r2,ic2). Then there exists a > 0 such that S02 0 u = acp , and so the Kac algebras K1 and K2 are isomorphic in the sense of 2.2.5.

Chapter 3 Representations of a Kac Algebra; Dual Kac Algebra

In this chapter, we shall use the notations hereafter: K = (M, F, ,c, gyp) will be a Kac algebra, A its Fourier representation, W its fundamental operator and M the von Neumann algebra generated by A. This chapter deals with the representations of the Banach algebra M*, following Kirchberg ([79]) and de Canniere and the authors ([21] ), and the construction of the dual Kac algebra, as found independently by the authors ([34]) and Vainermann and Kac ([180]).

This chapter begins with a Kirchberg's important result on Kac algebras: every non-degenerate representation of the involutive Banach algebra M* has a unitary generator (3.1.4 ). For the Kac algebra K. (G) constructed with L°O (G), one recovers the well-known result that every non-degenerate representation of L1(G) is given by a unitary representation of G. As a corollary, we get that, for any non-degenerate representation p, the Kronecker product A x p is quasi-equivalent to A (3.2.2 ); in the group case, that means that, for every unitary representation µG of G, the tensor product AG 0 PG is quasi-equivalent to AG, which is Fell's theorem ([48] ). When we choose p = A, we then get a coproduct T on M (3.2.2 ). A co-involution k on M is then defined, for all w in M*, by the formula: ik(A(w)) = A(w o rc)

The triple (M, T, k) obtained is a co-involutive Hopf-von Neumann algZbra. For locally compacts groups, that means (8.3.6) that the von Neumann algebra £(G) generated by the left regular representation AG has a co-involutive Hopf-von Neumann structure given by a coproduct P. and a co-involution is such that, for all s in G: Fs(AG(S)) = AG(S) ® AG(S) 1c8(AG(S)) = AG(S-1)

By predualizing the canonical surjection from the von Neumann algebra generated by M* onto M, one obtains an isometric, multiplicative and involutive morphism from the Banach involutive algebra M* in the Fourier-Stieltjes

84

3. Representations of a Kac Algebra; Dual Kac Algebra

algebra defined in Chap. 1, the image of which will be called the Fourier algebra of the Kac algebra, and is a self-adjoint ideal of the Fourier-Stieltjes algebra. For locally compact groups, we recover the situation of the Fourier algebra A(G) and the Fourier-Stieltjes algebra B(G) defined and studied by Eymard in [46]. Using the canonical Tomita-Takesaki construction, we define, starting from

the left ideal I. of M*, a left Hilbert algebra dense into the Hilbert space Hip, which generates the von Neumann algebra M and a faithful semi-finite normal weight WP on M (3.5.2 ), satisfying a Plancherel-type relation: (A(w)*A(w))

= If a(w)IP

for all w in I.

Moreover, the modular operator A, is affiliated to the centre of M, and is, in the sense of [114], the Radon-Nikodym derivative of the weight 50 with respect to the weight soon (1.6.7). We prove that this weight is a Haar weight (3.7.4 ), and we have so defined a dual Kac algebra K = (M, I', ik, cp), the fundamental

operator W of which is aW * a, and the Fourier representation A of which is given by icA* (where A. : M* -p M is obtained by predualizing A). As A is non-degenerate, A is faithful. ,o.

On ,C(G), the weight so constructed is equal to the Plancherel weight studied by Haagerup in [58]. So, Ks (G) = (r(G), I's, K, , W9) is another

example of a Kac algebra (3.7.5 ), which is symmetric and will be studied in Chap. 4. Another essential result about Kac algebras is the following: characters

on M* (that is, elements x of M such that x 54 0 and P(x) = x ® x), are unitaries, verify x(x) = x*, and, with the weak topology of M, form a locally compact group, called the intrinsic group of the Kac algebra (3.6.10). See also (1.2.2), (1.2.3) and (2.6.6).

3.1 The Generator of a Representation 3.1.1 Lemma. With the definition of 1.3.6, we have: PR(K) ni 91* C A* (M*)

Proof. Let x be in PR(K) f1 s31 . It follows from 1.3.6 that there exists a Hilbert space 71, a non-degenerate representation y of M* on H and a vector 6 in ?-1, such that, for all w in M* : (X7 W) = WWX 10

3.1 The Generator of a Representation

85

As the space II fl I is norm dense in M* by 2.4.6 (iii), the algebra µ(II fl I ) is dense in p (M* ). Thanks to Kaplansky's theorem, it exists a sequence {Wfl}nEN of elements of IV such that I< 1 and that (W) strongly converges to 1. Let us consider the linear forms on M* defined by: (w E M*)

!Pn(w) = (x,w n * w * wn)

We have, by hypothesis on x: !Pn(w° * w) = (x, (w * wn)° * (w * wn)) >_ 0

Therefore 4 n is positive definite. Moreover, since x belongs to by 2.4.5, wn * w * wn belongs to Iv,, we get:

91* and since,

by 2.1.6 (ii) by 2.6.1 (v) by 2.5.3

4'n(w) = (a(wn * w * wn) = (A(wo * w)a(wn) I

= (A()a(wn) IA(wn)11V(x*)) (A(w), rIa(Wn),A(Wn)AV(x*))

Let us put On = fla(Wn),a(Wn)Ap(x*). We have then Ifin(w) _ (A(f1),w), and therefore: IPn = A*(fn) E A*(M*) From 1.6.10, we get the existence of x (I in B(K)+ such that x = 'r* (x f2). Let us also consider the decomposition A* = -x*(sa)* (1.6.1 (ii)). We have:

(x fl - (sA)*(Qfl),lr(w))l = I(ir*(xQ) I(x - 4n, W) I I (x, w) - (4n, W) I I (X, W) - (X, Wo * W * Wn) I

J(Y(W) I

(Y(Wn)*Y(W)Y(Wn)6 16)1

_ I(,1(w),Q f' flN(W., )E) 1 :5 JIM(W)II jj

ti(Wn)611 11 + IL(Wn)611

by ([571 Proof 2.11) < 211y(w)II 2117r(W)II 11611116

F(Wn)611

Therefore, thanks to Kaplansky's theorem, we get:

I- (s)(Q)II < 2IICI1 IC - i(wn)CII n --+oo 0 )

As, by 1.6.2 (ii), (SA)* is an isometry, its image is closed in B(K), therefore x Sl belongs to it. Which is to say that there is an element ,fl in M* such that x,fl = (SA)*(Q) which implies that X = ir*(xf2) = A*(sl).

86

3. Representations of a Kac Algebra; Dual Kac Algebra

3.1.2 Lemma. (i) Let x be in,, fl 9lcpo,, and w in I,,; let us write 77 = a(w). Then

belongs to Yt

,

_ 4(x),

and we have: A(w)Ap(x(x*))

(ii) The set {i7 E M*, A*(Sl) E 91*} is dense in M. Proof. For all w' in M*, we have:

(A*w') = (A(S)I) = (A(x) I A(w')*a(w)) = (A,(x) I A(w'O)a(w) = (iL,0(x) a(w'O * w))

= (x*,wO * w)= (ic(x),w° * w') = ((w° ® i)I'i(x),w')

by 2.5.3 by 2.6.1 (v) by 2.1.6 (ii)

Therefore: ((w° ® i)I'r,,(x))* _ (w o x ®i)I'(r,.(x*)) As KK(x*) belongs to

by 1.2.5

belongs to 91, by 2.3.1 (ii), and 2.3.5 gives

,

the completion of (i). By using 2.1.7 (ii) and 2.7.5 (iv), (ii) is an immediate corollary of (i). 3.1.3 Proposition. Let y be a representation of M*; let us suppose that the set {9 E (Aµ)*; µ*(9) E 91* } is dense in (Aµ)*; then, y is quasi-equivalent to a subrepresentation of A.

Proof. Let 0 be in (A)t such that p* (9)* belongs to 91.. Therefore, by lemma 3.1.1, we get the existence of an element (1 in M* such that: A*(Q) = µ*(e) Then, we have, for all w in M* :

(/A(w),9) = (j(O),w) which can be written as well: (Sp(W(W)), 0) = (S,\(7r(W)), J?)

(A(w),Q)

3.1 The Generator of a Representation

87

By density, for all x in W * (K), we shall have: (S/A(X), 0) = (SA(x), Q)

Let us assume s,\ (x) = 0; then we shall have (s,(x), 0) = 0 for all 0 satisfying

the above hypothesis; by linearity and density, it implies that sµ(x) = 0; therefore Ker s,\ C Ker sµ; so, there exists a morphism 4$ from M to Ap such that 4$(A(w)) = µ(w) and the lemma is proved.

3.1.4 Theorem. Any representation p of M* has a generator. Proof. Let w be in M*, Q in (Aµ)*; we have: (A x p)*(w ® ,'l) = A*(w)p*(A2)

As sl, is a right ideal, we see, using 3.1.2 (ii), that the representation A x p satisfies the hypothesis of 3.1.3. So, A x p is quasi-equivalent to a subrepresentation of A, and, by 1.5.4 (i) and (ii), we get the existence of a partial isometry U in AA x µ ® M C M 0 A p ® M such that, for all w in M*: (A x y)(w) = (i 0 w)(U)

UU"=U*U=Paxµ0 1 where PAxµ is the projection on the essential space of A x p. Now, let w be in M* , w(1) = 1, and 0 in (Aµ)*; we have: W)(U)

(0 X p)(W),

D)

by 1.4.3 Therefore:

i)(U)

A*(W*"*)p*(S2)

P 0 i)(aW*a)Y*(S2) (W"* 0 i)(oW"o(1 0

and, by linearity and density:

(i 0 n 0 i)(U) = aw*a(i 0 p*(fl)) So:

10 µ,(.R) = oWo(i 0 R (& i)(U) (i 0 D 0 i)((a 0 1)(10 CrWcr)(a 0 1)U)

by 2.6.1 (i)

88

3. Representations of a Kac Algebra; Dual Kac Algebra

Therefore, we have:

(10 /1(w), w ®17) = (1)(/1(Si),) _ (2' 0 Si 0w)((a ®1)(1 ® aWa)(a ®1)U) and, eventually:

10 Y(W) = (i 0 i 0 WWU 0 1)(10 awa)(U 0 1)U) Let x be in C (HW), by multiplying this equality (valid for all w in M*) by x ®1

on the left and on the right, we get that (a ®1)(1® aWa)(a ®1)U commutes with x 0 10 1, therefore it belongs to M ® A, ®M f1(,C(H) 0 C 0 C)', i.e. to C 0 Aµ 0 M, and there is a partial isometry V in A,, 0 M such that:

(a & 1)(10 awa)(U 0 1)U = 10 V and then, for all w in M* : Y (LO) = (i 0 W) (V)

Then, we easily get:

10 V*V=U*U=PPxp01 and so we deduce that the projector PA x µ may be written 10 Q, where Q is in AW Moreover, we get :

10 vv' = (a 0 1)(1 & owo)(o 0 0 0 0 1)(10

i) 0

0 1)(10 0 1)(a 0 1)(10 aw*a)(a

1)

And so we get that V * V = VV* = Q ®1 and the theorem is proved. 3.1.5 Corollary. (i) Let jl and 112 two non-degenerate representations of M*; then /21 x /22 is non-degenerate. (ii) The triple (W*(K), ss, x,r, s;r), with the definitions of 1.6.5 and 1.6.6, is a co-involutive Hopf-von Neumann algebra; it is symmetric if K is abelian, and abelian if K is symmetric.

Proof. The assertion (i) is a direct corollary of 1.5.5 and 3.1.4, as 7r is nondegenerate and (ii) is a trivial application of 1.6.7 and 3.1.4.

3.2 The Essential Property of the Representation A

89

3.,2 The Essential Property of the Representation A 3.2.1 Lemma. Let µ be a non-degenerate representation of M. with a generator V. For all w in M*, we have:

spa x µ)(w) = v(i 0 a(w))v*

.

Proof. It results from 1.5.5 that the generator U of A x µ is equal to: (c 0 i)(i 0 oW*o)(1 0 V) _ (Cr 0 1)(1 0 a)(1 0 W*)(1 0 cT)(o 0 1)(1 0 V) _ (Cr 0 1)(1 0 a)(1 ® W*)(V ®1)(1 0 ar)(a 0 1) Therefore, we have, for all w in M* :

t(a x µ)(w)V O(i0

Lo)((l 0 U)(1 0 W*)(V 0 1)(10 U))V u)(1 0 W*)(V 0 1)(10 U)(V 0 1)) 0

w')(=Or)(v)(l0 a)) (+0 i 0 w)((1 0 o)(i 0 W*)(1 0 W)(1 0 a)(v 0 i) (1 0 0)(1 0 W')(1 0 a)) i 0 w)((V 0 1)(1 0 oW'o)) = V(1 0 (i 0 LO)(oW'o)) = V(1 0 A(te))

by 1.5.1(; by 2.6.4

by 2.6.1 (i)

which completes the proof.

3.2.2 Theorem and Definitions. Let µ be a non-degenerate representation of M* with a generator V. Then: (i) The mapping which sends any element x of M to V(1 0 x)V* is a oneto-one normal morphism from M to Ap ® M which shall be denoted by %y,; (ii) The representations A and A x µ are equivalent; moreover, we have, for all w in M*: ryp(A(w)) = t(a x µ)(w)

(iii) We have: 'Yµ sa = csA x p

(iv) The mapping ya is a coproduct on M; it shall be denoted by 1' and by transposition it induces a product *^ on M*. For all x in M. we have: F(x) = aW*cr(1 0 X)oWa

90

3. Representations of a Kac Algebra; Dual Kac Algebra

(v) For all u in the intrinsic group of K, and w in M*, we have: yu(A(w)) = A(u w) Proof. As V is unitary, it is enough to check in which space belongs V(1 Ox)V *

for any x in M. It results from 3.2.1 that for all w in M*, we have: Yµ(A(w)) = t(A x µ(w)) E Ap 0 M

which by density and continuity gives (i) and (ii) on our way; the assertion (iii) is straightforward, (iv) results immediately from the associativity of the Kronecker product and (v) is the application of (ii) and 2.6.6 (iii). 3.2.3 Lemma. Let /11,112 be two non-degenerate representations of M*, with, respectively, generators V1, V2; then: (i) For all x in M, and t in Hom(/11, /22), we have: (t 01)'5

(x) = 7"' µ2 (x)(t (9 1)

(ii) For any morphism if: Aµ1 -+ Aµ2, such that 4P o pl = 112, we have:

(Oi)5',1=

7µ2

Proof. Let w be in M*. We have:

(t

(t 0 1)4;(A x µ1)w) = s(a X 92)(w)(t 0 1) = 7µ2(a(w))(t 0 1)

by 3.2.2 (ii) by 1.4.5(1) by 3.2.2 (ii)

by continuity and density, we get (i). Let x be in M; we have: z)'Yµi(x) = (4i 0 z)(Vi(l 0 x) V1) = V2(1 0 x)V2

=

7,02 (X)

by 3.2.2 (1) by 1.5.6 (ii) by 3.2.2 (i)

which completes the proof. 3.2.4 Lemma. Let y a non-degenerate representation of M*, W^ in M* and in (Aµ)*. We have:

A* (W 0 w)o7µ) _ A*Cw)µ*C)

3.2 The Essential Property of the Representation A

91

Proof. Let w be in M*. We have:

(A*((Q®')o#5',),c4.)

=(Aw®,1l)

by 3.2.2 (ii) by 1.4.3

which completes the proof.

3.2.5 Lemma. Let y1 and 112 be two non-degenerate representations of M*. We have: '7141 xµ2 = (c ®i)(i ® yµ1 )yµ2

Proof. Let w be in M*, fll in (Au1)*, (22 in (Aµ2 )* and w in M*. We have:

(siX

2(/\(W)) i J?1 (9 122

- (A X Ill x 112(w), w 0 f2l 0 02)

_ _ (A((Q1 0 w)

)(112)*(Q2),w)

by 3.2.2 (ii) by 1.4.3 by 3.2.4 by 1.4.3

_ ((A x 92)(w) (171 0 w) o ryµl .f12) _ (c(A X µ2)(w) (S22 0 ,(ll (9 cv) o (i (9 'YPi )) = (5',2 (A(w)), ((12 (9 ,fll (9 w) o (z D10 S22 0 ^tIA 1 )71A 2

by 3.2.2 (ii)

which completes the proof by linearity, density and continuity. 3.2.6 Proposition. Let p be a non-degenerate representation of M*. We have, with the notations of 3.2.2:

(oi)i'=(iot). Proof. We have: by 3.2.2 (iv) by 3.2.3 (ii) by 3.2.2 (ii) by 3.2.3 (ii) by 3.2.5 by 3.2.2 (iv)

92

3. Representations of a Kac Algebra; Dual Kac Algebra

3.2.7 Fell's Theorem ([48]). Let G be a locally compact group, AG the left regular representation of G, µG a unitary representation of G. The representation IIG ®AG is then quasi-equivalent to AG. More precisely we get, for any

s in G: V(1 (9 AG(s))V* = PG(s) 0 AG(s)

where V is the unitary in C(fl) 0 L°O(G) defined by the continuous bounded function s -+ PG(s).

Proof. Let us apply theorem 3.2.2 to the Kac algebra Ka (G) defined in 2.2.5. We have seen that V is the generator of the non-degenerate representation p associated to PG (1.5.10), that M is then the von Neumann algebra £(G) generated by the left regular representation A of L1(G) (2.5.4). So the morphism

y satisfies: yp(X) = V(1 0 X)V*

-ym(A(f)) = (A x y)(f)

(X EC(G))

(f E L' (G))

From this last relation, we deduce, using 1.4.7, for the unitary representations of G, associated to y o A and cA x p respectively: 'Yµ(AG(s)) = PG(s) 0 AG(S)

3.2.8 Corollary. Let G be a locally compact group, ,C(G) be the von Neumann algebra generated by the left regular representation AG. There exists a unique

normal injective morphism I's from £(G) to £(G) 0 £(G) such that, for all s in G, we have: I's(AG(s)) = AG(S) ® AG(S)

and I's is a coproduct (in the sense of 1.2.1) on C(G).

3.3 The Dual Co-Involutive Hopf- Von Neumann Algebra 3.3.1 Proposition and Definitions. The mapping from £(H 0) onto itself defined by x --+ Jx*J for all x in £(H,) is an involutive anti-automorphism

of £(H,). The restriction of this mapping to 1 ! is an involutive antiautomorphism of M in the sense of 1.2.5. It shall be denoted ic. Moreover, the involutions rc and hl are linked by the following relations: A(w o rc) A*(w o Ic)

(w E M* ) (W^ E M*) .

3.3 The Dual Co-Involutive Hopf-Von Neumann Algebra

93

Proof. For all x in At, let us put k(x) = Jx*J. Let w be in M*. We have: k(A(w)) = JA(w)*J = A(w o ic)

by 2.5.5 (ii)

which altogether provides the first equality and ensures, by continuity, that for all x in M, k(x) belongs to M; the involutive character of k is trivial. For all w in M*, we have:

(A(2),wo x) _ (A(w o x), W^)

_ (k(A(w)),Z) _ (A(w),2 o k)

_ (A(2' o

by the first equality

w)

which completes the proof.

3.3.2 Theorem. The triple (M, I', h) is a co-involutive Hopf-von Neumann algebra. It will be called the dual co-involutive Hopf-von Neumann algebra of K.

Proof. Let w be in M*, W1, W2 in M*. We have:

(I'kA(w),2'i ®w2) = (I'A(w o K), w1 (9w2)

=

((A x A)(w o x), W2 (9 w1)

= (A(c22)A(2i),w o /C)

by 3.3.1 by 3.2.2 (iv) by 1.4.3

= (*(w1 o by 3.3.1 0 k), w) = ((A x A)(w),(w1 ®w2) o (k0 k)) by 1.4.3 by 3.2.2 (iv) = (c(k 0 k)I'a(w), ('2'i ®w2)) Therefore we have:

I'ka(w) = c(k 0 k)I'A(w) and we can complete the proof by continuity. 3.3.3 Proposition. The mapping sa is an ]H[-morphism from (W*(K), Ss7rx7r, s*)

to (MI F, ic). If K is abelian, (M, I', I) is symmetric, and if K is symmetric, (M)I', ic) is abelian. Proof. Let w be in M*. We have: I'sA7r(w) = yas'\Ir(w)

= csaxalr(w) = c(A x A)(w) = c(sa 0 sa)(7r x ir)(w) _ (s,\ ® sa)csirxirlr(w)

by 3.2.2 (iv) by 3.2.2 (iii) by 1.6.1 (ii) by 1.6.4 (iii) by 1.6.1 (ii)

94

3. Representations of a Kac Algebra; Dual Kac Algebra

and, we get :

FS,\ = (S,\ 0 sa)cs,x, And: kA(w)

=A(woic) = sa7r(w o K)

= svr(w)

by 1.6.1 (ii) by 3.3.1 by 1.6.1 (ii) by definition of * (1.6.6)

= sAs;r7r(w)

and we get: s,\ = s,\ s;r

At last, since A is non-degenerate, we have s,\(1) = 1, by 1.6.1 (iii); as s,\ is one-to-one, it completes the proof, together with 3.1.5 (ii). 3.3.4 Definition and Notations. Let us denote C,\(K) the C*-algebra generated by the Fourier representation A of K. By 1.6.1 (i), we may identify its dual (C(K))* with a closed subspace B,, (1K) of B (IK) . More precisely, to each

element 9 of (C (K))*, one associates the element 9 o A of B(K) (where A denotes the restriction of s,\ to C*(K); ,cf. 1.6.1 (i) and (ii)). By 1.6.2 (ii), the mapping (s,\)* is an isometry from M* into B(K), the image of which is contained in BX(K). By transposing 3.3.3, (s,\)* is then an isometric Banach algebra morphism, the image of which will be denoted by A(K) and called the Fourier algebra associated to K. Every element of A(K) vanishes over Ker s X; conversely, let 9 be in B(K), such that (x, 9) = 0 for all x in Ker s,\. We can define a linear mapping w on M by writing: (SA(z),w) = (z,9) (z E W*(K))

In fact, w appears as the composition of the restriction of 9 to the reduced algebra W * (IK)supp s,\) with the canonical isomorphism between W * (K)supp 8a and M. Therefore w is ultraweakly continuous and belongs to M*, and we have 9 = (s, )* (w). So, an element of B(K) belongs to A(K) if and only if it vanishes over Ker s,\.

3.3.5 Proposition. The mapping xA* is a non-degenerate faithful representation of M* in M. Its generator is W and we have:

(i) (ii)

(i ®1')(W) _ (W ® 1)(10 C)(W ® 1)(10 a) (i ®w o k)(W) _ (i ® w)(W*)

(w E M*) .

Proof. By 1.6.1 (ii) we have xA* = x7r*(s,,)*, it then results from 1.6.9 and 3.3.4 that it is a faithful representation. By 2.6.1 (i), for all w in M*, and all

3.3 The Dual Co-Involutive Hopf-Von Neumann Algebra

95

w in M*, we have: (A* (W^ ), w o x) = (A(w o x), w) = (w ®w)(W )

therefore: ICa*(w) = (i ®w)(W)

and we get (i) and (ii) through a straightforward application of 1.5.1 (i) and (ii).

3.3.6 Theorem. Let G be a locally compact group, £(G) be the von Neumann algebra generated by the left regular representation AG. There exists a unique

normal morphism I's from £(G) to £(G) 0 £(G), and a unique normal antiautomorphism r., in £(G) such that, for all s in G.: te(at(s)) = AG(S) 0 AG(S) KS(AG(s)) = AG(S_ 1)

Then, (,C(G), I's, tc,) is a symmetric co-involutive Hopf-von Neumann algebra; we shall denote it 1,(G). It is the dual co-involutive Hopf-von Neumann algebra associated to the Kac algebra Ka(G). Moreover, the morphism sA from the envelopping W * -algebra W * (G) to £(G) such that we have, for all

s in G: SA(G(S)) = AG(S)

is an H-morphism from the Ernest algebra of G (cf. 1.6.8) to Hs(G).

Proof. The existence of x, is the only non-trivial result; applying 3.3.1, we see there exists a co-involution k on (C(G), I', ), defined in 3.2.8, such that,

for all f in L'(G): (A(f )) _ A(f 0 Ica)

From 1.1.3, we have, for any s in G, f in L 1(G) : 1

(f 0 rva)(S) = f (S - ),AG(S-

1)

and so, we have:

(JG f(5 G(5)dS)

-

IGf(s-1),AG(3-1)AG(s)ds = IG f(s)(s1 )ds

We then get, for any s in G:

K(at(e)) = ac(a-1)

96

3. Representations of a Kac Algebra; Dual Kac Algebra

3.3.7 Theorem. Let K = (M, I', ,c, gyp) be a Kac algebra, K' the commutant Kac algebra. Then the dual co-involutive Hopf von Neumann algebra (M', 1^7 IC'^) is equal to (M, cI', IC).

Proof. Using 2.5.9, we see that the Fourier representation A' generates M;

so M'^ = M. By 2.4.8, the fundamental operator W' associated to K' is W' = (J 0 J) W (J 0 J). The coproduct P'" is, then, using 3.2.2 (iv), such that:

P'"'(x) = (J ® J)cW*o(J 0 J)(1® x)(J 0 J)crWar(J 0 J) = (k ® = SP(x)

by 3.3.1 and 3.2.2 (iv) by 1.2.5 applied to (M, P, k)

As) by the identification of H,p with H, pi, the associated antilinear isomor-

phism J, and J ,, are equal, we see, by 3.3.1, that rc' = k and the theorem is proved. N

N

N

3.3.8 Proposition. Let K = (M, P, I, Sp) be a Kac algebra, and K = (M, P, k, gyp)

be a Kac subalgebra in the sense of 2.2.7. Let us denote j the canoniN

cal imbedding from M into M. which is an IHt-morphism. There is then a canonical surjective H-morphism r from (M, I', k) to (M", P"', R) such that r(AAw)) = A(w o j), for all w in M*, where A is the Fourier representation of K.

Proof. Let usNcall I the isometry from H0 to H. defined, for all element x in

91, nMby:

(i)

As j is an H-morphism, the application w -+ w o j is multiplicative and involutive from M* to (M)*. This application is surjective because, for any a, -y in Hip, we have wy,a o j = wI* y,I* a. Let x in w in M*. We have:

j o k 0 )r())

I,X(w o

=

by 2.3.5

N

o IC o j 0 i)P(x)) = A ,((w o re o i))PU(x))) = A(w)AWU(x)) = A(w)IA ,(x)

by (i)

therefore:

la(w o j) = A(w)I

(ii)

Let us put, for x in M, r(x) = I*xI. We have r(1) = 1 and r(a(w)) = A(wo j), because I is an isometry.

3.4 Eymard Algebra

97

N

Let now w be in (M^)*. We have: N

(j(L(2)),w) = (*(c2)),w o j) (5(w oj),w) _ (r(A(w)), w) _ (A*r*(w), w)

then:

N

A*r*(W) = j A*(w)

As A* is injective, it can be then easily deduced that r* is involutive and multiplicative, and therefore that r is an H-morphism.

3.4 Eymard Algebra 3.4.1 Lemma. The set BA(K) is a self-adjoint part of B(K). Proof. For all x in M and w in M*, we have: (x*, w) = (x,w° o lc)

and, by 3.3.1:

by 1.2.5

I° r) II = I= IBy

using 1.6.2 (i) we then see that 7r*(BA(K)) is stable by involution, and, thanks to 1.3.4, we get the result. 3.4.2 Proposition. (i) Any norm-one positive element of BA(K) is the limit, for the ar(B(K), C*(K)) topology, of norm-one positive elements of A(K). (ii) The space BA(K) is the o(B(K), C*(K)) -closure of A(K). (iii) The space BA(K) (resp. 7r*(BA(K))) is composed of the elements of the form (s,)(Q) (resp. M* (Q)), where p is a representation of M* weakly contained in A. in the sense of [47], and Q an element of (All)*.

Proof. Any element of BA(K) vanishes on Ker (cf. 1.6.1(ii)). By ([25], 3.4.2 (i)) any norm-one positive element of BA(IK) is thus the limit, for the cr(B(K), C*(K)) topology, of elements of the form (SA)*(i7), where 11 is norm one positive in M*; at last 3.3.4 allows to deduce (i). It results from (i), by linearity that BA (K) is contained in the closure, for the ar(B(K), C* (K)N) topology, of A(K); on the other hand, since B,\(K) is the annihilator of Ker A, it is cr(B(K), C*(K)) closed, which completes the proof of (ii).

3. Representations of a Kac Algebra; Dual Kac Algebra

98

Let p be a representation of M* weakly contained in A and 5? in (AM)*.

By ([25], 3.4.4), (s )* (fl) is the a(B(K), C* (K)) limit of elements of the form (sA)*(w) where w belongs to M* , i.e. of positive elements of A(K). By belongs to BA (K); by linearity this using (ii), we can conclude that (sµ )* conclusion still holds for any Sl. To prove the converse, let us note that those elements of the form (sµ) *

with p weakly contained in A and 51 is in (A,)* compose a vector space. Indeed if p (resp. p') is weakly contained in A and 5 (resp. fl') belongs to (A,,)* (resp. (Ai)), it is easy to check that:

(s)*(Q) + (s')(Q') = (s,,A')*(Q ® ST) and that p ® p' is weakly contained in A. By linearity, it is therefore enough to consider x in BA(K)+. Let y be the positive linear f o r m on C (K) such that x = y o A (cf. 3.3.4). The Gelfand-Naimark-Segal construction allows to associate to y a triple (?-(, 6,,u). Then p o A is a representation of M* and we have: ((Y 0 A)* ('20, W)

WA(WN 10 (Y' A M)

_ (yo5,ir(w)) (X, -7r(W))

(7r* (X), W)

Therefore, we have: 7r* (x)

o A)*

and: X=

since it is clear that p o A is weakly contained in A, it completes the proof. 3.4.3 Lemma. The product of B(K) is a(B(K), C*(K)) separately continuous on the bounded parts.

Proof. Let ei be a bounded family of elements of B(K) converging to an element 9 in B(K). For all 9' in B(K) and w in M*, we have: (ir(w),9*9') _ (r* (Oi * 8'), w) by 1.6.9 _ (ir(w .

(9')) , O j)

which converges to: (ir(w i*(9')), 9) = (ir(w), 9 * 9')

thanks to the same computation.

3.4 Eymard Algebra

99

Since the 8i * 8' are bounded by I O' l I sup (8i I I which is finite, by the

density of 7r(M*) in C*(K), we get that 8i * 8' converges to 8 * 8' for c(B(K), C*(K)). The left multiplication is dealt with in the same way.

3.4.4 Theorem. The sets A(K) and BA(K) and are norm-closed self-adjoint ideals of B(K). Specifically, BA(K) is an involutive Banach algebra which we shall call the Eymard algebra associated to K. Proof. Let cw be in M*, 8 in B(K) and w in M*. We have:

(ir*(8 * (sA)*(w)),w _ (A*(.')ir(8),w)

by 1.6.9 and 1.6.1 (ii) _ ((A x ir)*(w ® 8), w) by 1.4.3 _ ((A x 7r)(w), w ®8) _ (%A(w), 8 ® w) by 3.2.2 (ii)

_ (A(w),(8®)o%) _ (A((8®) o'Y7r),w) From what we obtain that: 0 * (S,\)*(^) = (S,\)*((O 0 ^) 0 ^ )

By 3.3.4, it follows that A(K) is a left ideal of B(K). As A(K) is self-adjoint and norm-closed, the first part of the theorem is secured. Now, let 8 be in BA(K)+1. By 3.4.2 (i), 8 can be cr(B(K), C*(1K)) approximated by norm-one positive elements of A(K). Applying the first part of this proof, as well as 3.4.3 and 3.4.2 (ii), we find that, for all 8' in B(K), 8' * 8 belongs to BA(K). By linearity, we can conclude that BA(IK) is a B(K)-left ideal; since, by 3.4.1 and 1.6.1 (i), BA (]K) is norm closed and self-adjoint, the proof is completed. 3.4.5 Proposition. The restriction to BA(1K) o f the Fourier-Stieltjes representation o f B(K) is the transposed o f the mapping kA from M * to C (K) (once BA(K) is identified with the dual of C *(K)).

Proof. Let 8 be in (C (K))* and w in M. Then by 3.3.4, 8 o A belongs to BA (1K) and we have:

(icir(8o),w) _ (7r(woK),8o) _ (A(woi),8) _ (kA(w))),8) which completes the proof.

by 3.3.1

3. Representations of a Kac Algebra; Dual Kac Algebra

100

3.4.6 Eymard's Theorem ([46] ). Let G be a locally compact group, and f an element of B(G). The following assertions are equivalent: (i) There exists a Hilbert space H, vectors , 77 in H, and a unitary representation ,G of G on H, weakly contained in the left regular representation, such that, for all t in G:

f(t) = GIOX 17l) (ii) We have:

If (t)h(t)dtI, h E L1(G), IIA(h)II < 1

sup

GI

< +oo

The space of such functions is noted Ba(G); it is a closed ideal of the FourierStieltjes algebra of G, and will be called the Eymard algebra of G. Moreover,

if f is in BA(G), its B(G)-norm is equal to: sup

I

G

h E L1(G), I< 1

The space B,\(G) can be identified with the dual of C *(G) (the C*-algebra generated by the left regular representation of L1(G)), the duality being given, i f f (t) = (,G(t)e I 'i), and h in C (G), by:

(f, h) = (1 77) where µ denotes again the associated representation of C *(G) (recall that µ is weakly contained in A). Moreover, for every w in £(G)*, the set:

A(G) = Is -+ (A(s1),w)

(s E G)}

is a norm-closed ideal of BA(G). It will be called the Fourier algebra of G, and its dual is ,C(G). Proof. Using 3.4.2 (iii), we see that property (i) characterizes 7r*(BA(Ka(G))); by 1.6.3 (iii), it is the same for property (ii).

So, we get BA(G) = 7r*(BA(Ka(G))) = Ka7r*(B.\(Ka(G))) and all other properties of BA(G) come then from 1.6.3 (iii) and 3.4.4. Let us now consider ica 7r* (A(Ka (G))) . Using definition 3.3.4, it is the set of all elements which may be written, for all fl in £(G)*: Ka7r*(sa)*(f2) = KaA*(S?)

In 1.3.10, we have seen that A*(fl) is the function s -+ (AG(s), fl); by then iaA*(fl) is the function s (AG(S'), fl).

3.5 Construction of the Dual Weight

101

So, we get A(G) = ica 7r* (A(Ka (G)) ), and all properties of A(G) come from 3.4.4.

3.5 Construction of the Decal Weight In that paragraph, we consider the set Z = a(Ip fl i3). 3.5.1 Proposition. Let w, w1 be in I( fl I,. The formulas:

(i)

a(w)T a(wf) = a(w * w') a(w)#

(ii)

= a(w°)

allow us to equip B with a structure of left Hilbert algebra, dense in H.. Let us denote by *r the left multiplication of Z. We have, for all w in I , fl 10:

(iii)

ir(a(w)) = A(w)

and the von Neumann algebra generated by *r(Z) is equal to M.

Proof. (a) We have seen in 2.4.6 (iii) that I, fl I3 is an involutive subalgebra of M. As a is a bijection from I, fl I3 to , we see that 93, equipped with T and # is an involutive algebra. (b) Let be in H,, orthogonal to 93; by 2.4.5, we have, for all L01, L02 in

I:

0 = (a(wi * w2) I ) _ (A(wi)*a(w2) I ) _ (a(w2) I A(w1)e)

by 2.4.6 (ii)

because of the density of a(I0) in Hp (2.1.7 (ii)), it implies A(w1) = 0 for all w1 in I,; because of the density of I. in M* (2.1.7 (ii)), it implies, for all w in M*, A(w)e = 0, which, in turn, because of A being non-degenerate (2.6.3 (i)) implies 0. Therefore Z is dense in H.. (c) For all w1 fixed in I, fl I3 the mapping a(w) -+ a(w1)T a(w) is continuous from 93 to 93. In fact, we have: a(w1)T a(w) = a(w1 * w) = A(w1)a(w)

by definition by 2.6.1 (v)

(d) For all wl, w2, w3 in I, fl I3, we have: (a(W1)ta(w2) I a(w3)) = (A(wi)a(w2) I a(w3)) = (a(w2) I A(wi)a(w3))

by (c) by 2.5.3

= (a(w2) I a(w1)T a(w3))

by (c)

= (a(w2) I a(wl)#Ta(W3))

by definition

3. Representations of a Kac Algebra; Dual Kac Algebra

102

in H, orthogonal to 93T93. We have, for all wl, w2 in I, fl

(e) Let

o = (a(wi)Tm(w2) If) _ (A(wfla(W2) If)

_ (a(w2) I a(wj)F)

by (c) by 2.5.3

By (b) it implies A(wl )e = 0, since a(I, fl 1) is dense in H by (b); by continuity, it implies A(w)e = 0, for all w in M*; therefore because of A being

non-degenerate, it implies e = 0; so,T (f) Let w be in I, fl I and x in giv f

is dense in H.. ¶fl ,01, . We have:

(A (x) a(w)#) = (A,(x)Ia(w°)) = (x*,wo)_ = (IC (x), w)

= (a(w) I A ,(rs(x* )))

because

cpoK =

by definition by 2.1.6 (ii) by 1.2.5 by 2.1.6 (ii)

x(91,). Therefore, the mapping # has an adjoint, the restric-

tion of which to A,(

fl Ovo,,) is the mapping Ap(x) -- Ap(K(x*)). By

2.7.5 (iv), this adjoint mapping is densely defined, therefore 0 is closable. Following ([158], def. 5.1; cf. 2.1.1 (iii)), if we remark that (iii) has been

proved in (c), and that, thanks to 2.4.6 (iii), *(Z) generates M, we have completed the proof. 3.5.2 Definitions. We shall denote B' the right Hilbert algebra associated to 93, 95" the achieved left Hilbert algebra, and cao the maximal modular subalgebra

of B" (cf. 2.1.1(iii)). We shall still note T and # (resp. ) the product and the involution on 93" A

A

(resp. 95'). A

A

We shall note S and F, the closures of # and b, with respective domains denoted to D# and Db . In particular, we have, for all x in flpo1 : FAW(x) = Av(r(x*)) If in Hp is left bounded with respect to Z, we shall still note u1 ) the "left multiplication" by ([14], def. 2.1). 3.5.3 Definitions. We shall note cp the faithful, semi-finite normal weight on M canonically associated to ([14], th. 2.11), and call the dual weight associated to K. For all w in I , fl II, by 3.5.1 (c), A(w) belongs to 7^W B) and therefore to fl OZ . Moreover, for wl and w2 in I, fl Imo, we have: SP(A(w2)*A(wl)) = (a(w1) 1 a(w2))

3.5 Construction of the Dual Weight

103

To the weight S we associate the Hilbert space H. and the canonical oneto-one mapping Ao : Ho. We shall note 21 the left Hilbert algebra associated to Vp, i.e. 11(91, fl fir), which is isomorphic to B" (2.1.1 (iii)). More precisely, the mapping which, to every C in B", associates the vector: A

can be uniquely prolonged into a unitary operator from H(2 to HO still denoted by F. It will be called the Fourier-Plancherel mapping and will allow us to identify HO and H., and, through this identification of H., we

have 93" ='. Using the definition 2.1.6 (ii), we shall note a instead of a(. 3.5.4 Proposition. For all w in I., a(w) is left-bounded with respect to 95, and we have:

(i) *(a(w)) _ A(w)

(ii) a(w) = 1iW(a(w)) (iii) for all a, ,y in 21', yTab is left-bounded with respect to 95 and we have:

(7Tat) = A(wy2a) . Proof. Let w1 be in Ip, w2 in I, fl I,,,

in 93'. We have:

w1)

by 2.6.1 (v)

_ 1(a(w2 * w1))e because w2 * w1 belongs to Ip fl Io _ \(w2 * w1) by 3.5.1 (c)

_ 1(w2)a(w1) As 1 is in the closure of A(I, fl Io) = *(Z), we have: i'(e)a(w1) = A(w1)C

which yields (i), and (ii) immediately, then (iii) follows by applying 2.1.7 (i).

3.5.5 Remark. Thanks to 2.1.1 (iv), we know that M is in a standard position in H; so, by 1.1.1 (iii), every element of M* can be written M for some vectors a,# in H.. This element shall be written wa,Q

3.5.6 Corollary. (i) The algebra ,C(G) is in a standard position in L2(G). (ii) The predual ,C(G)* is equal to the set { Sl f,g ,C(G), f, g E L2 (G) } .

104

3. Representations of a Kac Algebra; Dual Kac Algebra

(iii) For all f in L2(G), let us put f(s) = f (s-1)- for all s in G. The set of all If * g, f, g E L2(G)} is the Fourier algebra A(G) defined in 3.4.6. Proof. The assertions (i) and (ii) are just applications of 3.5.5 to Ka(G). In 3.4.6, A(G) has been defined as the set of functions s -+ (AG(S), w) , for all w in ,C(G)*. But we have: (AG(S-')7 W^f,g) = (AG(S -1)f 19) =

IG.f(st)9(t)dt

- JG f

(t)9(s-it)dt

f(t)9(t-is)dt

= G

(f * 9)(S)

And so, (ii) implies (iii).

3.6 Connection Relations and Consequences 3.6.1 Proposition (Connection relations). The operations T and T are linked by the following relations: (i) For any a, -y in 2L' and b in H., we have: (W(a ® /3) I ^10 b) = (/3 I (^'Tab)b)

(ii) For any a, 'y in 2t' and 0, b in B', we have: (W(a 0 0) 17 ®b) = (/3t8b I yTab)

(iii) The set 93'T93' is included in AW(91,,) and, for any a, 'y in HW and b in TB', we have:

(W(aO/3)f706) = (lr(/3t6)al7). Proof. Combining 2.6.1 (iii) and 3.5.4 (iii), we get (i). Therefore, with the hypothesis of (ii), we have: (W(a 0 6) 17 0 a) = (13 I * (b)(7Ta")) = (n

_ ('(6)/3 which is (ii).

I 7Ta') (/3t6b I'YTab)

3.6 Connection Relations and Consequences

105

It can also be written as follows: A

A

(W(a ®/3) 1'Y ®b) = (/3t6b jir'(ab)'Y) = (ir'(a)(/3t6) I 'Y))

It follows that:

II'(c)(/3t6)ll

suv{J(w(a 0 p)

0 a)j, ry E W, 11711 s 1} < IllilIVbll

Thus, /3T8 is left-bounded with respect to 2t (cf. 2.1.1(iii)), and we can write: A

(W(a 0 13)1 'Y 0 b) = (ir(/3t61')a 17)

which, by continuity, still holds for any a, y in Hgp. This completes the proof.

3.6.2 Lemma. (i) The set B'TB' is included in Aw('tv fl'Yt,,c). More pre93', we have: cisely, for /, 6 ir(/3Tbb) = A*

and:

)*

(wb,a A

A

rz(7r(#TbI )*) = 7r(bt /3b )

(ii) The space A,('Yt7p fl qi o,c) is a core for F.

Proof. Let /, b be in 93', a, y in Hip. We have:

(A*(Z's,)*al7) _ (A*(^b,a),w'Y,«)

_ (W(a®/3)1y®b)

by 2.6.1(iii)

_ (7r(#Tbb)a y)

by 3.6.1 (iii)

from what follows the first equality. For w in M*, we have, then:

(ic(ir(flt6)),w) _ (A*

w°)

= Ca(w)wQ,b

by 1.2.5

106

3. Representations of a Kac Algebra; Dual Kac Algebra

Therefore, we have: A

A

n('r(/3T8b)*) = A*(wa,b)* = ir(oti36)

by the first equality; as As, by ([158], p. 17),

the proof of (i) is completed. 'TB' is a core for F, (ii) is immediate. 0K

3.6.3 Lemma. Let x be in M. (i) For any a in Db, xa belongs to Db, and we have:

Fxa = n(x)*Fa A

A

(ii) For any /3 in DO, x# belongs to D#, and we have: Sx# = IC(x)*S#

Proof. Let a be in A9,(w n 9iw°,c), w in I, n I.0, x in M. We have: (xa I a(w)#) _ (xa I a(w°))

by 3.5.1 by 2.1.6 (ii)

_ (KA(xa),w) _ (ic(xA ' (a)), w}

_ 1(Fa))*, w} _ (A(,c(x*)Fa)*, w)

by 3.5.2 (i)

_ (a(w) I ,c(x*)F^a)

by 2.1.6 (ii)

Therefore xa belongs to Db and:

Fxa = ,c(x*)Fa

As 4(m. n

is a core for F by 3.6.2 (ii), we have proved (i). Let a be in Db , 0 in D# and x in M. We have: A

A

(x# I F"a) _ (/3 I x*F^a) _ (,C3 F^ic(x)a) _ (ic(x)a I ,5`"P) _ (a I ,c(x*)S^3)

which completes the proof.

by (i)

3.6 Connection Relations and Consequences

107

3.6.4 Proposition. The modular operator A =A, is affiliated to M'. Proof. Let a be in Bo and x in M. We have:

xAa = xFSa = Eic(x*)c = FSxa = Axa

by 3.6.3 (i), because Sa belongs to Db by 3.6.3 (ii)

as 93Q is a core for A, we have xA C Ax, which completes the proof.

3.6.5 Corollary. For all t in R. we have:

Proof. For all x in M, and t in R, we have: Fat (x) = UW*(crt (x) 0 1)Wo _ or

W*(Ditxa-it

_ crW*(Qit 0

by 3.2.2 (iv)

0 1)Wa

1)(x (D

1)(A-it 01)Wi

Now, by 2.6.1 (iv), W belongs to M ® M and by 3.6.4, Alt belongs to M', therefore, we have:

i'cr°(x) _ o(oit 0 1)W*(x 'Alt

(10

®1)a' )aw*(X (D 1)Wo(1 (0 A-it) 01)W(A-it

_ (10 At)I'(x)(1 0 A-It )

by 3.2.2 (iv)

®Qt r'(x) which completes the proof.

3.6.6 Corollary. For any x in M, we have: (i) (ii)

,(x*)=JxJ (.1 o J)W(3 0 J) = W*

.

Proof. Let a be in 93o and x in M. We have: " xJa = xa12Sa

= Al/2xSa = Al/2S,c(x*)a = Jn(x* )a

by density, we get (i). By (i) and 2.6.2, we get immediately (ii).

by 3.6.4 by 3.6.3 (ii)

108

3. Representations of a Kac Algebra; Dual Kac Algebra

3.6.7 Theorem. The modular operator A is affiliated to the centre of M; moreover, it is the Radon-Nikodym derivative of the weight W with respect to the weight cp o k, in the sense of [114] (cf. 2.1.1 (v)).

Proof. By 2.7.6 (i), we have atVOIC = 0 t for all t in R. The theorem 5.4 of [114] gives then the existence of a unique positive self-adjoint operator h, (cf. 2.1.1 (v)). Let x be affiliated to the centre of M, such that co o # = qtpor,,; then A ,(x) belongs to D(0/2), and we have: in fl IIh1V2AW(x)112 =

o K(x*x) = I=

INow, lI1'iL(x)lI2

= I=

by 3.5.2 (i)

by 3.6.2 (ii), we see that A ,(m. fl gqcpo.%) is a core for F, thus also for a-1/2; on the other hand, 3.6.4 implies that h1/2 and a-1/2 commute.

Using the same arguments as in ([23] lemma 23), we can conclude that h1/2 = A-1/2, and so h = A-1. The operator A is therefore affiliated to the centre of M and y o gyp, which completes the proof. 3.6.8 Corollary. For all t in R, we have: /LV 1 = c

tk

.

Proof. By 3.6.7, for all t in R, A' belongs to the centre of M. We then have for all x in M:

kCr (x)=Jot(x)J = = 'A a it JxJAit = A t k(x)Ait JAitxA-it J

by ([14], 4.10) by 3.3.1

which completes the proof.

3.6.9 Lemma. Let x in M. x 54 0, such that F(x) = x 0 x. Then, we have, for all w in M*:

t)

A(w)x = xA(x(x) - w)

xA(w) -a(s w)x A(x(x) - w) = x(x)A(w)x

3.6 Connection Relations and Consequences

109

Proof. Let y be in 91W. We have: A(w)xdW(J)

_ .x(w)4(xJ) = Av((w o K 0 i)l(xy))

by 2.3.5

= AP((w 0 rt. 0 i)((x 0 X).P(Y)))

= XAW((w 0 10 x (9 OFM)

= XAW(((t4x) w) o k 0 z)r(y)) XA(x(x) ' w)Asv(y)

by 1.2.5 by 2.3.5

which yields (i), by continuity. Taking the adjoints in (i), one gets, using 2.5.3: x* A(w°) =

w)° )x* WOW _ A(x* .

by 1.2.5

and, changing w to w°, x to x* (which satisfies the same hypothesis), we get (ii).

Let us now assume that w is in I., and let w' be another element of I.. We have:

A(rc(x) w)Ja(w') = V (Ja(w'))a(K(x) w) = JA(w')Jic(x)a(w) = JA(c.gi)x*ja(w)

= Jx*A(,c(x*) wJa(w) = r%'(x)JA(r4x*) w')Ja(w)

= rc(x)f'(Ja(rc(x*) w'))a(w) = rc(x)A(w)Ja(rc(x*) w') = K(x)A(w)JK(x*)a(w') = rc(x)A(w)xfa(w')

by 3.5.4 (ii) and 3.5.1 (c) by 3.5.4 (i) and 2.1.7 (iii) by 3.6.6 (i)

by (i) applied to w' and x by 3.6.6 (i) by 3.5.4 (i) and 3.5.1(c) by 3.5.4 (i) by 2.1.7 (iii) by 3.6.6 (i)

By continuity, we get: A(K(x) w) = rc(x)A(w)x

for all w in I,, and by continuity again, for all w in M*, which is (iii).

3.6.10 Theorem. Let K be a Kac algebra. The intrinsic group of K is equal to the set of characters on M*, that is the set of all x in M, such that x 0

and F(x)=x®x.

Proof. Let x be a character on M*, that is, x belongs to M, and is such that x 0 0 and r(x) = x 0 x. As rc(x) satisfies the same hypothesis, we have, for

110

3. Representations of a Kac Algebra; Dual Kac Algebra

allwinM*: xn(x)A(w) = xA(rc(x) - w)rc(x) = A(w)xrc(x)

by 3.6.9 (ii) applied to ic(x) by 3.6.9 (i)

So, by continuity, xK(x) belongs to M', and, then, by 2.7.2 (i), it is equal to a scalar a. But then, for all w in M*, we have: A(w)x = xA(rc(x) - w) = x,c(x)A(w)x

by 3.6.9 (i) by 3.6.9 (iii)

= aA(w)x

By continuity, we get x = ax, and, as x 54 0, we have a = 1. So xr%,(x) = 1, x is invertible, and the theorem is proved. 3.6.11 Proposition. Let G be a locally compact group, Ka(G) the abelian Kac algebra associated to G in 2.2.5; the dual co-involutive Hopf-von Neumann algebra associated to Ka(G) is (,C(G), F,, rc9) (cf. 3.3.6), and the dual weight

(on .C(G) is the Plancherel weight cp8 studied in [58], associated to the left Hilbert algebra ,C(G) of continuous functions on G with compact support.

Proof. By definition 3.5.2, the weight (cpa)" is associated to the left Hilbert algebra a(I. n i3), that is, by 2.4.7 (ii) and 1.1.2 (ii), the set:

If E L1(G) n L2 (G); the function s -+ f '(s) = J(s)ZG(s') belongs to L1(G) n L2 (G) }

equipped (by 3.5.1 and 1.1.2 (ii)) with the usual convolution product and the involution °. We have 1C(G) C a(I , n I3), and the operations on 1C(G) being the restrictions of those on a(Iw n I,p). Both generate the same von Neumann algebra,C(G). Thanks to 3.6.7 and 2.2.2, it appears that the modular operator associated to (cpa)" is the Radon-Nikodym derivative of the left Haar measure with respect to the right Haar measure, that is the modular function Z1G. It is also the modular operator associated to cos. Therefore, using ([114], prop. 5.9), we have: S03 = (cpa)"

3.6.12 Theorem. Let G be a locally compact group; the set of continuous characters of G (i. e. continuous multiplicative functions from G to C, except the function 0), is a locally compact abelian group, which is the intrinsic group of the Kac algebra Ka(G).

Proof. The intrinsic group of Ka(G) is, by 3.6.10, the set of all f in L°O(G)

such that f 0 0 and I'a(f) = f ® f (i.e. f(st) = f (s) f (t), a.e.)

3.7 The Dual Kac Algebra

111

Let now g in 1C(G) such that (g, f) 34 0; we have:

(g, f) f (s) = (JG (tf (s) = JG f (st)g(t)dt = JG f (t)g(s-1t)dt We then see that f is almost everywhere equal to a continuous function, and we get the result.

3.7 The Dual Kac Algebra 3.7.1 Lemma. Let x, y in 910; then I'(y)(x 01) belongs to A

and we have:

-(I'(y)(x 01)) = aW*o(A,(x) 0 AO(y))

Proof. Let X31, 02 and 6 in B'; let w be in I,, and x in 910. We have:

((1 0 ''(6))aW*a(A,(x) 0 a(w)) I Al 0 02) _ (cTW*o(A,(x) 0a(w)) I01 0A2T8 )

_

(W*(a(w) 0 A,(x)) I #2 T"'6" (& 01) (A(fl a(w)Atbb)A,(x) I Al)

by 2.6.1 (iii)

_ (A(w . (/32t6)*)A(x) I /31)

by 2.1.7 (iv) by 3.6.2

= (A(w ' A* (wb,A2)) A , (x) I A1)

= (A(w

A*(t`'b,02 ))I wAQ(x),A1)

(A*(2'A,(),,31), w ' A* (wb,A2 ))

_ (A(Z'6,2 )A*(^

, w)

(A* (``'AC,(x),Al * wb,A2 ) w)

by 3.3.5

(A(w), "A- (x) 1#, * wb,02 )

_ (I'(A(w))(A,(x) (9 8) I Al (9 02)

So, by linearity and density, we have:

(10 Let 61 be in

(*'(Si) 0

0 a(w)) = t(A(w))(A,(x) 0 6) V. We have: i'(6))aW*c(A,(x)

0 a(w)) _ (*'(o1) 01)(r'(A(w))(A,(x) 0 6) = ®6) = 1'(A(w))(x ®1)(61 ®6)

112

3. Representations of a Kac Algebra; Dual Kac Algebra

Then, we can deduce that crW*a(A,(x) 0 a(w)) is left-bounded with respect to the weight cP 0 , i.e. that it belongs to A0®0(910; so I'(\(w))(x (9 1) belongs to 91000 and:

0 a(w)) Let i/i the weight on M defined for y in M+ by: O(Y) = P 0 P"O(W 0 l)-P(Y)(x (9 1))

We have, for all w in M* : (A(w)*A(w))

_ (5P 0 )((x*01)r(A(w))*r(A(w))(x ®1)) _ IIAO(x) 0 a(w)112

by (i), since W* is isometric by 3.5.2 (ii)

_ W(x*x)V(A(w)*A(w))

The weight b is therefore semi-finite. Moreover Sp ® S is normal as well as I'

and the mapping Y -* (x* (9 1)Y(x (9 1) from M 0 M into itself. Therefore is normal. For all t in R and y in M+, we have: O(Ut (y)) _ (SP 0c)((x* 0 1)t(ct (y))(x 0 1)) _ P (9 SP)((x 0 1)(i 0 at )I'(y)(x 0 1))

by 3.6.5

_ (SP 0 S)((x* 01)r(y)(x 0 1)) _ ib(y)

Therefore 0 is o °-invariant, we can apply the proposition 2.1.5 to the von Neumann algebra M, the two weights c (x * x) and tb, with E = A (Ip ), because, by 2.1.7 (ii), A(,(E) = a(I,) is dense in HO. Therefore we have, for all y in M+: S

b(y) = c (x*x)SP(y)

that is, for all x in 910 and y in M (SP 0 S)((x* 0

(Y) (x ®1)) = S (x*x)V^(y)

By polarization, for x 1, x2, y1, Y2 in 910, we shall have: (SP 0 S )((x2 0 1)I'(y2yi)(xi 0 1)) = S (x2x1)5P(y2y1)

Therefore, there is some isometry U in C(Ho 0 Hg,) such that, for all x, y in 910, we have:

3.7 The Dual Kac Algebra

113

A1,®c(r(y)(x ®1)) = U(Ao(x) 0 An(y)) By (i) and by density, we see that U = crW * cr, which completes the proof. 3.7.2 Proposition. The weight 5P is left-invariant.

Proof. Let x be in 'A-, y in 9t . By 2.1.8 (i), it exists a sequence {afl}flEN of positive elements in M which are monotonely converging up to (i ® S)(I'(y)). We have:

= lim T (an, wA (x) )

= 1 m T (anAc(x) I A. (x))

= lim Tv(x*anx) n

= SP (x*((i 0 S F*"' W)

_ V((i 0 )((x* 01)I'(y)(x (9 1)))

by ([59], 2.1(3))

_ (SP 0 )((x' ® 1)P(y)(x 0 1))

by 3.7.1

= c (x*x)SP(y)

_

by ([5917 5.5)

(i)

I A,(x))

H' C H, and a Now, we know by 2.1.8 (i) that is exists a closed subspace positive selfadjoint operator T on H' such that: 'D(T)- = H' and:

(i ®SP)(r(y))(w) = IITV2eII2 (i 0 SP)(I'(y))(we) = +oo

E D(T 1/2)) (

It follows from (i) that D(T112) contains 11v( therefore H' = HH.

D(T1/2))

), which is dense in Hp;

Moreover, for all 6 in AW(9L), we have:

I= (e(y)e I )

jj(AY)1/2jgjj2

Thanks to the unicity of the polar decomposition, it follows: T1/2 = SP(y)1/21 or:

T=Ay)1

by (i)

114

3. Representations of a Kac Algebra; Dual Kac Algebra

Therefore (i 0 c) (P(y)) is bounded and is worth ^(y)l, which completes the proof.

3.7.3 Theorem. The quadruple (M, T, R-7 W^) is a Kac algebra. Its Fourier

representation is A = xA*; its fundamental operator is W = aW*cr (once Hip and H0 have been identified). Proof. We got in 3.3.2 that (M, T, k) is a co-involutive Hopf-von Neumann algebra and in 3.7.2 that Sp is a left-invariant weight on it. We can therefore apply 2.4 to (M, P, gyp); the fundamental operator associated to (M, P, Sp) appears to be o W * a, by 3.7.2. Applying 2.4.6. to (M, P, ic, c) we can then define a bounded linear mapping A(2), for all w in M* , by: A(w) = (Z' o I 0 i)(crW*a) _ (i 0 w o k)(W*) Using 2.6.1 (ii) and 3.3.1 we get: A(w) = A*( W"" o Ic) = rcA*(w)

Using 3.3.5, we see that A is a faithful non-degenerate representation; so, by 2.4.6 (iv), the weight Sp satisfies (HWii); it satisfies (HWiii) by 3.5.3, and then (M, P, Ic, Sp) is a Kac algebra.

3.7.4 Definition. The Kac algebra K = (i121, F7 Ic, c) will be called the dual Kac algebra of K. 3.7.5 Theorem. Let G be a locally compact group. Then the quadruple (C(G), P,, rC,, 5P,), where ,C(G) is the von Neumann algebra generated by the left regular representation AG of G, P,, rc, and W, have been defined respectively in 3.2.8, 3.3.6 and 3.6.11, is a symmetric Kac algebra, denoted K,(G), and we have: K8(G) = Ka(G)^

Its fundamental operator is equal to the function s -+ AG(s) (s E G), considered as an element of £(G) ® L°O(G). Its Fourier representation A(w) is defined as being, for all w in £(G)*, the function s -+ (AG(S),tZ') (s E G); the Fourier representation A of K,(G) is then surjective on the Fourier algebra defined in 3.4.6 and 3.5.6 (iii).

Proof. By 3.7.3 and 3.6.11, we see that (L(G), P,, 1c,, W.) is the dual Kac algebra of Ka(G). Therefore, we get, by 2.4.7 (i) that o.W*a is the function s --+ AG(s), when considered as an element of C(G) 0 L°O(G). By 3.7.3, it is the fundamental operator of K, (G). By 2.6.1 (i), applied to 1K, (G), we get then the Fourier representation of K, (G).

3.7 The Dual Kac Algebra

115

3.7.6 Proposition. Let K1 = (M1, I'i, xl, cot) and K2 = (M2, I'2, K2, W2) be two Kac algebras, U a unitary H.1 --> H(,2 which implements an IREisomorphism from K1 to K2 (cf. 2.7.9 and 2.2.5). Then U implements an H-isomorphism from K1 to K2, too.

Proof. Let u an ]H[-isomorphism from M1 to M2, such that SP2 o u = awl (cf. 2.7.9) and let U be defined by (cf. 2.2.5): U11p1(x) = a-1/242 (U W) Then, f o r a n y w in I, . 2 , w o u belongs to I.1, and we have:

Ual(w o u) = a1/2a2(w) So, we get, using 2.6.1 (v), for all w' in M2*:

UA1(w'o u)al(w o u) = Ua1(w'ou*wou) = Ua1((w'*w)ou) = a1/2a2 (w' * w) = al/2A2(w')a2(w) = A2(w')Ual(w o u)

from which we get that the application u defined, for x in M2, by:

u(x) = U*xU is a von Neumann isomorphism from M2 to Ml such that, for all w' in Ml*: u(A2(w')) = Al(w' o u)

From that, by predualizing, we get u*, a Banach space isomorphism from M1* to M2*, such that, for all w1 in M1*: ,\2*u*(wl) = uA2*(w1)

and, as A2* is injective (3.3.5), we see that u* is multiplicative and involutive, and, so, u is an ]8[-isomorphism. Moreover, we have then, for all w in 1602: S l(Al(w o u)*)11(w o u)) = 11al(w o u)112

by 3.5.2 (ii)

= aIIa2(w)112

= from that, we deduce, by 2.7.9, that W^ 1 o u = a^ W2.

by 3.5.2 (ii)

3. Representations of a Kac Algebra; Dual Kac Algebra

116

3.7.7 Proposition. Let K be a Kac algebra, W its fundamental operator, A the modular operator associated to the dual weight gyp. Then: (i) We have:

W(1®A)W* =,A^ OA (ii) For all t in R, ait belongs to the intrinsic group of K. (iii) For all t in R, w in M*, we have: A(w o Ra=t )

o

(iv) For all x in M+, we have: (SP ®i)(r(x)) =

SP(x)4-1

(v) Let x be in 9Zv fl 9Zcpo,. and i in 93o. Then (i 0 w,7)F(x) belongs to 9

fl 91,po,, and we have:

(x)

A W((' ®w,)(I'(x))) =

Proof. The fundamental operator of the dual Kac algebra K is uW*o (3.7.3); so (i) comes from 2.7.6 (vii) applied to K. From (i), we get W(1 (&a:t )W * = .' ®Qi; so (ii) comes from 3.2.2 (iv). By 3.6.6 (i) we have K(z 't) = JaztJ = so, 3.6.9 (iii) applied to A-it yields (iii). Let w be in M* +. We put w' = w(LX ) in the sense of ([114], prop. 4.2). By 3.6.7 and ([16], 1.1.2 (b)), we have:

Therefore, for all x in M+, by ([114], prop. 4.2), we have: (SP 0 w)(I'(x)) = lim (p o . 0 w')((a 0 i)CF(x)) C--,0 where: We have:

W(1 0 (1 + CIA))W* = 10 1 + eW(1 O'A^)W*

= 10 1 + E( 0 (i +

Ea)-')w*

=

(1®1+(AeA)y1

3.7 The Dual Kac Algebra

117

and:

(A 0 A)'C = (A 0 A)W(i 0 (1 + CA)-l )W

= W(i OA^)W*W(l 0 (1 +,eA)-')W*

= w(i Oa^(l + = r(aE)

Ea)-l)yy*

where:

therefore, we get:

lim(cp o ic 0 w')f(a&x) (v 0 W)(r(X)) = C-->o by 2.2.4

lim SP o

= SOW UJ(a-1)

by 3.6.7

which is (iv). We assume 117711 = 1. Then (i 0 w,,) is a conditional expectation and we have:

(i 0 w7)(r(x*x))

(i 0 w77)(r'(x*))(i 0 w7)(f (x)) which implies:

SP((i ® w77)(f(x*))(i 0 WO (I'(x))) : (SP 0 w7)(r'(x*x))

=I

therefore (i (9 w,,) (F(x)) belongs to

((i 0

by (iv)

. Furthermore, we have:

(i 0 w7 o ic)(rc 0 ,)f(x*) _ ("71 o

o

which, by hypothesis, and by 2.3.1 (ii), belongs to 9l,; therefore (iOw77)(f(x)) belongs to and by 3.5.2 (i), Ap((i ®w77)f(x)) belongs to Db, and we have:

EA (x,xy) and (x, y) (y-1, xy).

So, if the function f belongs to the abelian von Neumann algebra L' (G, m), the function (s, t) -+ f (st) on G x G belongs to L°O(G x G, m®m) (identified to LOO (G, m) ® L°° (G, m) ), and the function s -+ f (s -1) on G belongs to L' (G, m). We have then defined a coproduct P and a co-involution ic on L°O(G, m). Moreover, the measure m defines a normal semi-finite faithful trace on L' (G, m)+, which is left-invariant with respect to the coprod-

4.2 Takesaki's Theorem on Symmetric Kac Algebras

135

uct F. The fundamental operator associated by 2.4.2 is defined, for all f in L2 (G x G, m ®m ), x, y in G, by: (Wf)(x, y) = f (x, xy)

Let us now consider, for all s in G, the unitary operator µ(s) on L2(G, m) defined, for all f in L2 (G, m), t in G, by: 1 WS)f )(t) = f W t)

By ([94], lemma 7.4), y is injective. A straightforward calculation gives that the function s -+ µ(s)* on G, if it is considered as an element of the tensor product LOO (G, m) 0 ,C (L2 (G, m) ), is equal to W, and therefore, we get that

W is unitary, and that (here x is an automorphism because L°O(G, m) is abelian):

(K ®i)(W) = W* which, by 2.4.6 (v), proves that (L°°(G, m), F, ic, m) is an abelian Kac algebra. Moreover, for all s in G, we have:

WRWs) 0 1)W = µ(s) & N(s) and, by 3.6.10 and 3.2.2, µ(s) belongs to the intrinsic group G' of L°O(G, m)".

In fact, by 4.2.3 (i) and 4.2.5 (iii), there exists an isomorphism U from L2 (G, m) to L2(G') such that, for all v in G': UvU* = AGI (v)

and moreover, such that x -- UxU* is an isomorphism from the Kac algebra (L°°(G, m), 1 ', i , m) to K, ,(G') and, therefore, from the dual Kac algebra (L' (G, m), F, K, m) "* to K, (G').

So, (U 0 U)W(U* 0 U*) is the fundamental operator of Ka(G'), and, by 2.4.7 (i), (U ® U)W*(U* 0 U*) is the function v -+ AGI (v) on G', and (U ® I) W * (U* ® I) is the identity function on G', considered as an element of L°O(G') ® L°O(G, m)". So, for all w in (L°°(G, m)")*, we clearly see that U(i ® w)(W)U* is the function v -+ (v, w) on G' and, as (i 0 w)(W) is the function s --> (µ(s), w) on G, we infer that the isomorphism f -> U* f U from L°O(G') to L°O(G, m) is just the composition by y. Using f = XG'-µ(G), we see then that G'- µ(G) is of Haar measure 0. So, if v belongs to G' - µ(G), v j(G) C G' - p(G) is of Haar measure 0, and so is p(G), which is impossible; so ji is surjective, and the theorem is proved.

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4. Duality Theorems for Kac Algebras and Locally Compact Groups

4.3 Eymard's Duality Theorem for Locally Compact Groups In that paragraph G will denote a locally compact group. We shall apply the preceeding paragraph to K, (G). 4.3.1 Lemma ([44]). The left regular representation AG is a one-to-one homeomorphism from G to the intrinsic group of K,(G). Proof. Let s be in G. Then, by 3.3.6, AG (s) belongs to G(K, (G) ); the mapping AG is also clearly one-to-one.

Now let {AG(s)} denote a net converging to the identity 1 = AG(e), where e is the unit of the group G. We wish to show that s converges to e. Proceeding by way of contradiction, we suppose it does not. Then there

exists a subnet, say sn of sv and a compact neighbourhood V of e such that sn V, for all n. Choose a compact neighbourhood U of e such that UU 1 C V. Then, for each sn, we have (xu denotes the characteristic function of U and µ the Haar left measure on G): (AG(sn)Xu I Xu) =

Xs1zuxudf = y(Snu nu) = 0 G

Thus :

I((AG(Sn) -1)Xu I_ (Xu I XU) _ 11(u) > 0

Thus AG (sn) does not converge weakly to 1, which contradicts our first assumption and completes the proof. 4.3.2 Theorem. The left regular representation AG is a bicontinuous bijection from G onto the intrinsic group of K,(G).

Proof. Let denote Go the intrinsic group G(K,(G)). By 3.7.5, the Fourier representation A of K, (G) is the mapping, defined, for all w in .C(G)*, by: S -+ (AG(S)*,w)

which belongs to L' (G). By 4.2.4 applied to K, (G), there exists a unitary U from L2(G) on L2(Go) such that UA(w)U* be equal to gw, which is the mapping defined, for all t in Go, by: t -+ (t*, w}

which belongs to L°°(G0). Thus, we have:

U*gwU=gwoAG

4.3 Eymard's Duality Theorem for Locally Compact Groups

137

and, by continuity, for all f in LO° (Go) :

U*fU=foAG By selecting f =

(*)

XGO_AG(G), we see that f = 0, and Go - AG(G) is a zero-

measure set. Let so E Go and so AG(G); we have soAG(G) C Go - AG(G); therefore s oA G (G) is a zero-measure set as well as )tG (G) by left-invariance, but this is impossible by (*). Therefore AG(G) = Go, which completes the proof.

4.3.3 Corollary (Eymard's Theorem [46]). Let G be a locally compact group.

Let us recall (cf. 3.4.6 and 3.5.6 that the Fourier algebra A(G) has been defined as the set f f * 9, f , g E L (G)} (where f (s) = 1(s - 1) for all s in G, f in L2(G)), equipped with the norm: fG °(s)f(s)ds Then, the spectrum of A(G) is G; so every character on A(G) is involutive. Proof. By 4.3.2, the spectrum of L(G)* is equal to AG(G). By 3.4.6 and 3.5.6,

there is an isomorphism between C(G)* and A(G), which, to each w f,g in ,C(G)*, associates the function f * g, linked by:

(f*)(s) = (AG(s),fg) So, we see that the spectrum of A(G) is the set {s', s E G}, that is G. 4.3.4 Corollary. Let G1, G2 two locally compact groups, u an H-morphism from Hs (G1) to H9 (G2 ); then there exists a continuous group homomorphism a from G1 to G2 such that u(AG1(s)) = -G2 (a(s)), for all s in G1. The image a(G1) is a closed subgroup of G2. If u is injective, then a is injective; if u is surjective, then a is surjective too.

Proof. It is clear that u sends the intrinsic group of Hs(G1) into the intrinsic group of 1HIs (G2), and that the restriction of u is a continuous homomorphism

of groups. So the existence of a comes directly from 4.3.2. Then, the subset {x E £(G1); I's(x) = x 0 x} is a closed subset of the unit-ball of ,C(G1), and therefore, is compact for weak topology. Its image par u, that is: {AG2 (a(s)), s E G2} U {0}

by 3.6.10 and 4.3.2, is then also compact for the weak topology of C(G2). So {AG2 (a(s)), s E G1 } is locally compact, and by 4.3.1, a(G1) is a locally compact subgroup of G2, and so is a closed subgroup of G2.

138

4. Duality Theorems for Kac Algebras and Locally Compact Groups

For every w in C(G2)*, we have: gw(a(s)) = (AG2 (a(s)*), w) = (UAG1(s),w) = (AG1(s)*,WoU) = g(w o u)(s)

by 4.2.1 by definition of a by 4.2.1

And:

gwoa=g(w0u) Therefore, gw o a = 0 implies w o u = 0, by 4.2.1 (v), and, so, if u is surjective,

it implies w = 0. Using 4.3.3, we get then that a(G1) is dense in G2, and so that a(G1) = G2. If u is injective, a is trivially injective.

4.3.5 Corollary. Let G1 and G2 be two locally compact groups; then, the following assertions are equivalent: (i) There exists a bicontinuous isomorphism u : G1 --- G2 (ii) There exists an ]H[-isomorphism from Ha(G2) onto 1H,, (G1). (iii) There exists an H-isomorphism from 1H1B(G1) onto 1HI8(G2).

Proof. We have (i) = (ii) because the application f -+ f o u is an I8Iisomorphism from Ha(G2) onto 1HIa (G1), (ii) = (iii) by 3.7.6, and (iii) = (i) by 4.3.2. 4.3.6 Corollary ([163]). Let G be a locally compact group.

(i) Let (M, I', i) be a sub co-involutive Hopf-von Neumann algebra of 18(G), such that there exists a Haar weight So on (M, F, K). Then, there exists a closed subgroup G' of G such that M is generated by {AG(3), S E G'} and is isomorphic to ,C(G').

(ii) Let K be a sub-Kac algebra of K8(G); then there exists G', open subgroup of G, and an H-isomorphism from K onto K8(G'). (iii) Let Kp be a reduced Kac algebra of Ka(G); then there exist G', open subgroup of G, and an H-isomorphism from Kp onto Ka(G'); moreover, we have:

P = XG' Proof. Let K = (M, I', K, cp); as K is a symmetric Kac algebra, by 4.2.5 (i), it is isomorphic to K8 (G(K) ). But it is clear, by 4.3.2, that: G(am) = M n {aG(s), s E G}

Let us put:

G'={sEG; AG(s)EM}

4.3 Eymard's Duality Theorem for Locally Compact Groups

139

Then, G' is a closed subgroup of G, and M is generated by {AG(s), s E G'}. Moreover, by 4.3.2, AG G' is a bicontinuous isomorphism from G' onto G(K), and by 4.3.5, (M, 1', i) is H-isomorphic to 1HI3 (G'), and we have (i). With the hypothesis of (ii), let E be the conditional expectation from ,C(G) to M obtained by 3.7.9 (i). For any s in G, we have:

l(Eac(s)) =

EAG(e)OEAG(s)

So, by 3.6.10, either EAG(s) = 0, or EAG(s) is in the intrinsic group of K, and is therefore unitary, which implies EAG(s) = AG(s).

So, the subset Is E G; AG (s) 0 M} is therefore equal to the subset Is E G; E AG (s) = 0} and is closed. Then G' = Is E G; AG (s) E M} is an open subgroup of G. So (ii) is proved and (iii) is proved using (ii) and 3.7.9 (ii).

4.3.7 Proposition. Let G be an abelian locally compact group, G the abelian locally compact group of all continuous characters of G (cf. 3.6.12), which will be called the dual group of G. Then: (i) The group G is the spectrum of L1(G), and the Gelfand representation of L1(G) is given by the Fourier transform: .f(X) =

VX E G, b'f E L1(G)

JQH()-ds

(ii) There exists a Haar measure dX on G. and an isomorphism U from L2(G) to L2(G) defined, for all f in L1(G) n L2(G), by:

Uf=f called the Fourier-Plancherel transform, such that the mapping x --> UxU* is

an isomorphism from the dual Kac algebra K3(G) to La(G), such that, for all f in L1(G): UAG(f)U* = J -

Proof. By 3.6.12, the set of continuous characters of G is the intrinsic group of Ka(G); as G is abelian, Ka(G) is symmetric, and, so, G' is the spectrum of the abelian Banach algebra L1(G), and the Gelfand transform, taking a coherent definition with 4.2.1, will be given by:

9f (x) = (x*, f ) = for all f in L' (G), X in G, which is (i).

fG

f (s) (X, s) - ds

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4. Duality Theorems for Kac Algebras and Locally Compact Groups

Using Takesaki's theorem (4.2.3 and 4.2.4), we see that there are a Haar measure dx on G, and an isomorphism U from L2(G) to L2(G), defined by U f = f for all f in L1(G) fl L2 (G), such that the mapping x -+ UxU* is an isomorphism of the dual Kac algebra 11(3 (G) to 10 (G). By 4.2.3 (ii), we get UAG(f )U* = f, for all f in L1(G), which ends the proof. 4.3.8 Pontrjagin's Theorem ([121]). Let G an abelian locally compact group, G its dual group, as defined in 4.3.7. Then the group G^^ is isomorphic to G. Proof. The isomorphism defined in 4.3.7 (ii) sends the intrinsic group of K8(G)

onto the intrinsic group of Ka(G), that is onto G^^. For all f in L1(G), we have, using 4.3.7 (ii): .f(s)(X's-l)ds

IG

=AX) = JGf(s)UAG(s)U*ds

from which we can deduce that UAG(s)U* is the function on G: x --+ (x, S-1)

Using Eymard's theorem (4.3.2), we get that these functions are all the characters on G, and so that the group G^^ is isomorphic to the group G.

4.4 The Kac Algebra li(3(G) It is now possible to describe the various objects associated to the Kac algebra K8 (G) by the general theory.

4.4.1 Proposition. (i) The enveloping C*-algebra C*(K8(G)) is the algebra Co(G) of continuous functions on G, vanishing at infinity; the canonical representation of C(G)* into C*(K8(G)) is then the Gelfand transform 9w(s) = (AG(8)*,w) for all w in £(G)*, s in G. (ii) The Fourier-Stieltjes algebra B(K8(G)) is the algebra M1(G) of bounded measures on G, and its Fourier-Stieltjes representation is the left regular representation of M1(G); an element x of £(G) is positive definite representable (in the sense of 1.3.6) if and only if there exists a (unique) positive bounded measure m on G such that: IG AG(s)dm(s) (iii) The enveloping W*-algebra W*(K3(G)) is the dual M1(G)* of M1(G). This Banach space, which is a W* -algebra, being equal to the bidual of C0(G),

4.4 The Kac Algebra K8 (G)

141

has then a structure of co-involutive Hopf-von Neumann algebra, given by:

(r(e), m1 0 m2)

m1 * m2) (is(9),m) _ (e, m°)

(0 E M1(G)*, m1, m2 E M1(G)) ME M1 (G)) (0 E M1(G)*)

where * is the multiplication of M1(G), and ° its involution, and where 8 is

defined by (0, m) = (9,rYi), with m(f) = (ffdm)- for all f in C0(G). (iv) The canonical imbedding (sA)* from L1(G) to B(K8(G)) = M1(G) is the usual imbedding from L1(G) to M1(G). Proof. By 4.3.3, the enveloping C*-algebra of C(G),, is the algebra of continuous functions on G, vanishing at infinity, and the canonical representation of L(G)* is its Gelfand transform. So (i) results from 4.3.3. As B(K8(G)) is the dual of C*(K8(G)), we deduce, from (i), that B(K8(G)) is equal, as a Banach space, to M'(G). Let us compute its Fourier-Stieltjes representation ice7r*; if m is in M1(G),

f, g in L2 (G), we shall have:

(lcslr*(m),wfg) = (lr(wf,g o tc), m) g(wf,g o tc)dm IG

IG

(KG(s)>wf,s)dm(s)

by 3.7.5 and (i)

And so, we have: AG(s)dm(s) JG We can deduce from it that the multiplication and the involution of B(K8 (G)) ,c87r*(m)

=

are the usual ones on M1(G), which gives (ii), with the help of 1.6.10. Then (iii) is a straightforward application of 3.1.5 (ii)).

Let f be in L' (G), and m = (sA)*(f). As A(f) is JGAG(s)f(s)ds, and, which is, by (ii), equal to by 1.6.1 (ii) and 3.7.3, equal to fG )G(s)dm(s), we see that m is the measure f (s)ds, which gives (iv).

4.4.2 Proposition. (i) Every non-degenerate representation y of ,C(G)* is given by a spectral measure Pµ on G, with values in Hµ, as defined, for example in ([105], IV, §17.4), such that, for all w in £(G)*:

µ(w)- Jc(ac(s)*iw)dPN(s) (ii) Let IL be a non-degenerate representation of ,C(G)*, Pµ its associated spectral measure on gµ, an element of (Aµ)*; then K8µ*(Sl) is the image by the left regular representation of the bounded measure d(P/(s), (1) on G.

4. Duality Theorems for Kac Algebras and Locally Compact Groups

142

(iii) Let µ1, 112 be two non-degenerate representations of ,C(G)*, Pµ17 Pµ2

their associated spectral measures. The spectral measure associated to the Kronecker product µ1 xµ2 is the convolution product of Pµ1 and Pµ2 defined by:

IC

.f(s)d(Pµl * Paz)(s) =

IGxG

.f(st)d(Pµz(t)(9 pill (s)).

Proof. By 4.4.1 (i) and 1.6.1 (i), there is a representation µ of C0(G) on ?-lµ such that µ o c = p; by ([105], IV, §17.4), the representation µ is given by a spectral measure on G with values in ?.lµ, which gives (i). We have then: (K. JL* A, W)

(W 0 rt..), fl)

=

J (aG (s)moo K)d(Pr(9), R) fc(ac(9),w)d(P. (s), n)

and therefore:

k8µ*(,fl) = J

Si)

G CG

which gives (ii). Let now f1 be in (Aµ1)*, Sf22 in (Aµ2)*; we have: ((µi x µ2)(w), J21 (9 02)

by 1.4.3

(111* (J?1)112* 022), W)

_ (IC8Y2*(.(l2)KSµ1s(fl1),w o rca)

s)d((P,42, n)* (P1, n))(s)

= IG(G = IG(

by (ii)

GH*

)w)d((t'NZ,.fl) * (F,41, f2))(s)

therefore, the measure d (Pµ1 x µ2 , D10 522) is the convolution product of the measures d(Pµ2 , ,f22) and d(Pµ1, Si); which gives (iii).

4.4.3 Theorem. Let P be a spectral measure on G with values in R. Then, there exists a unitary U in ,C(G) ®,C(f) such that, for all w in ,C(G)*, , is in H: (UI W 0 WC17) = fG (AG (S) *, w) d(P(s)

We shall write:

U=

AG(s)* ®dP(s) G

.

177)

4.4 The Kac Algebra K8 (G)

143

Proof. Let p be the non-degenerate representation of C0(G) associated to the spectral measure P, i.e. such that, for any f in C0(G):

y(f) = IG'(s)dP(e) Let us put v = p o 9; then v is a non-degenerate representation of L(G)* such that, for any w in £(G)*, we have: v(w) = frG. (AG (S) *, w) dP (s)

By 3.1.4, there exists a unitary U in ,C(1-() ®,C(G) which is the generator of v, and is such that, for all , 77 in 1-h, w in £(G): (UV I WC17 0

W) = Mw) 177)

So U = cUvcr satisfies the theorem. 4.4.4 Proposition. Let G be an abelian locally compact group, G its dual group, in the sense of 4.3.7. For any m in M1(G), let us define the Fourier transform

ofmby:

- IG(X,s)-dm(s)

m(X)

(x E G)

Then:

(i) For any m in M1(G), we have: AG(s)dm(s)) U* = m

U G

(ii) (Bochner's theorem) Every positive definite function on G is the Fourier transform of a unique positive bounded measure on G. (iii) (Stone's theorem) Every unitary representation p of G is given by a spectral measure Pµ on G, with values in 1-(µ, such that, for all s in G, we have:

µ(s) =

f(xs)-dF(x).

Proof. For any f in L1(G) (which is an ideal of M1(G)), (i) has been proved in 4.3.7 (ii). So, using the non-degeneracy of the representations, (i) is proved

for anyminM1(G). As L' (G^

has a bounded approximate unit, we see, using 4.4.1 (ii), that

every positive definite element in L°°(G) is of the form U(fG AG(s)dm(s))U*, with m in M1(G)+. So (ii) is proved, using (i) and 1.3.11.

144

4. Duality Theorems for Kac Algebras and Locally Compact Groups

Let us consider the non-degenerate representation of L1 (G) obtained from p by 1.1.4. By 4.3.7 (ii), L' (G) is isomorphic to £(G)*, and, by 4.4.2 (i) and 4.3.7 (i), there is a spectral measure Pµ on G, with values in ?-(j,, such that, for all f in L1(G), we have:

µM =

jfG

f(x)dP/A(x)

from which we get (iii).

4.5 Characterisation of the Representations and Wendel's Theorem Let K = (M, I', re, co) be a Kac algebra. Let A be a von Neumann algebra. In this paragraph, Q will denote a normal, one-to-one morphism from M to A ® M such that

(a 0 Or - (2 (& r)a Q(1M) = lA 0 1M By 3.2.6, for any non-degenerate representation p of M*, the algebra All and the morphism yµ fulfill these conditions.

4.5.1 Proposition. We have, for all x in M^ + and t in R: (i (9 5P)/3(x) = SP(x)l A A

(i 0

IP)#

t

Proof. Let x in M+. We have: (i 0 5P)/9(x) ®1M = (i 0 i 0 V^)(i 0 I')/3(x) = (i ®i ® w)(/3 ®i)I'(x)

by 3.7.2

by hypothesis

_ /3((i ®5P)I'(x))

= = S (x)(1A 0 1M)

which brings (i). Let t in R. We have:

((i ® at

®t

by 3.7.2 by hypothesis

4.5 Characterisation of the Representations and Wendel's Theorem

t

145

by hypothesis

® i ® ot)(i ® F"')# by 2.7.6 (ii) applied to K

_

by hypothesis

_ (13®i)(iOoT)I' _ (P ® i)(ot ® i)I'

by 2.7.6 (ii) applied to K

From what follows that (i ®Ut )P ® i and 3 ° 0 i coincide on F"' (M*"* ); as it is obvious that they coincide also on C 0 M, thanks to 2.7.6 (iv) applied to K, they will coincide on M 0 M, which completes the proof.

4.5.2 Proposition. Let ?k be a faithful, semi-finite, normal weight on A (to simplify, we shall suppose A C £(H p )). Then: (i) for all x in'L,, y in 91, the operator fl(y)(x 0 1) belongs to 910®o, and there is an isometry U in HO 0 H such that: U(A,(x) 0 AW(y)) = A, ,®c(f3(y)(x ®1))

(ii) U belongs to A 0 M. (iii) for all z in M, we have: 13(z)U = U(1 ®z) Proof. We have: (0 0 A)((X* 0 J)P(Y*Y)(X 0 1)) = O(X*((i 0 ^)O(Y*Y))X) =+G(%*T)W(Y Y)

by 4.5.1 (i)

and so fl(y)(x 0 1) belongs to,®.. Let X1, X2 in 910, y1, Y2 in 91o ; polarizing the preceding equality, we find: (0 ® SP)((x2 0 1)@(y2)/3(yl)(xl (9 1)) =

O(x2x1)SP(y2y1)

which can also be written as follows: (AO®cp(la(yl)(xl 0 1)) I A pOV(9(y2)(x2 ®1))) = (A,,j(xi) 0 Al o(yl) I Ao(x2) 0 A (y2))

which completes the proof of (i). Let be in 2L , 77 in at', x in TO and y in 9l,. We have: 0 7**r"(77)) U (AO (x) 0 Ao (y))

0 7**r'(?7))AOOo(8(y)(x 0 1))

3(Y)(x 0 1)( 0 77) 0 1)(Ao(x) (9 77)

by (i)

146

4. Duality Theorems for Kac Algebras and Locally Compact Groups

For all vector ( in H, by density, we shall have:

(ir'()

0 A,(y)) =

0 1)(C ®rl)

And, by having 7r' (e) converging to 1, we get: (10 7"r"(?7))U(( 0

0(y)(C 0 t7)

Let z in A'. We have:

(1 0 '(rl))U(z( 0 A,(y)) = /3(y)(z( ®rl)

= (z 01Ay)(( 0 i)

by hypothesis

= (z ®1)(1 (D ir'(r))U(( (9 AO (y))

by the same calculation as above So, by density and linearity, we get: 7'(77))U(z 0 1)

By having lr'(q) converging to 1, we get:

U(z ®1) = (z ®1)U and so U belongs to A 0 £(H). Let x in y in k n gi -, i in 2t0, such that 11rill = 1. We have: U(do(x) 0 J1\(wo_114n)JdO(y)) = U(Aw(x) 0 AW((z 0 wn)r(y)) = A+GOW(Q((z 0 wn)r(y))(x

1))

= nO(&O(((z 0 i Own0 0 i)I'(y))(x 0 1)) A00w(((z 0 i 0

-

AO(O

W((('

by 4.1.3 (iv) by (1)

by hypothesis

0 17)(i & P^)#(Y))(x 0

P (Y)) (X Jwo_,1,n)J)AOOW(,Q(y)(x (2) 1))

by 4.1.3 (v)

J/\(wo-1/4,,)J)U(Ao(x) 0 da(y))

by (i)

By linearity and density, we get: U(1 0 JA(wo_1/4,7)J) _ (1 ® J\(wa-1/4,7)J)U

4.5 Characterisation of the Representations and Wendel's Theorem

147

By density of A-1 42to in H, and by polarization, we get, for all w in M* :

U(1® JA(w)J) = (10 JA(w)J)U and, by density of A(M*) in M, we can conclude that U belongs to C(HO)®M, which completes the proof of (ii). Let x in Tto y in 910. We have: /3(z)U(A,p(x) 0 Ao(y)) = /3(z)A,p®,(/3(y)(x 0 1)) = A,p®,(,Q(zy)(x 01))

by (i) by (1)

= U(A,,(x) 0 A-(zy)) = U(1® z)(A,p(x) ® A,(y)) By density, continuity and linearity we can complete the proof of (iii).

4.5.3 Lemma. With the notations of 4.5.2, let X be in 910®0, Y in 910®o, x in MO, and y in 910. Then: and we have: (i) The operator (/3 0 i)(X)(x 010 1) belongs to

AO®W®o((fl 0 i)(X)(x 0101)) = (U 01)(Ao(x) 0 Ac®lp(X )) (ii) The operator (,l3 0 i)I'(y)(Y 0 1) belongs to 91V,®oO0 and we have: A

(1®cr)(10W*)(U®1)(1®c,)(A ®0(Y)®A0(y))

Proof. We have:

(000,O(X*(i 0 Po 0)(# 0 000 i)(X*X)X) =+V(=`((w0 V)(X* XXV 0 V)(X*X) < +00

by 4.5.1 (i)

which leads to the first part of (i). By polarization, for any X1, X2 in '10(g0 and X1, x2 in 91,x we shall find:

i)(Xi)(x 0101)) I Ap® _ (A,(x1) 0

((fl 0 i)(X2)(x 0101))) A (x2) 0 A (X2))

From what we can deduce the existence of an isometry of H,1,0 H ®H which

sends A,,(x) 0 A ®,-(X) on 110®,®0((/3 0 i)(X)(x 01 01)). It comes from 4.5.2 (i) that this isometry coincides with U ®1 on the elements of the form

148

4. Duality Theorems for Kac Algebras and Locally Compact Groups

AV) (x) ® AV (y1) 0 A, (y2) (where Y1, y2 are in 91.); therefore, by linearity and

continuity it is equal to U ®1, which completes the proof of (i). We have:

)((Y*

j) F (Y*Y)(Y 0 1))

0 0 i 0 0"')(# 0 0 F (Y*Y)Y) 0 )(Y*#(i 0 V) F (Y*Y)Y) W 0 )(Y*Y) O(Y*Y) A,

A

by 3.7.2

W

< + 00 which gives the first part of the proof of (ii). Using the same technique, through polarization, we get an isometry of 0 i)1,(y)(Y ®1)). Let H, ®H ® H sending A,,®,(Y) 0 A,(y) on x in N, y1, y2 in 9l ,. We have: A,,®

Sp((l3 0 i)(I'(yl))(x 0 Y2 ®1)) ((i 0 P)(/Q(y1))(x 0 Y2 ®1))

by hypothesis

= 0 r')(,a(yl)(x 0 1))(x 0 Y2 ®1)) (10 i)(1 0 W *)A,®c(f3(yl)(x 0 1)) 0 AO(y2)) = by 2.4.9 applied to cp and 3.7.3

(1 ® o)(1 ® W*)(U 01)(A,,(x) 0 A,(yl) ® A,(y2)) by (i)

This isometry does therefore coincide with (10 a)(1 0 W *) (U 0 1)(10 a) on those vectors of the form A,, (x) 0 A,, (y2) 0 A (yl ), which, by linearity, density and continuity, completes the proof of (ii).

4.5.4 Proposition. With the notations of .4.5.2, the isometry U satisfies:

(i 0 -r)(U) = (10 a)(U 0 1)(10 a)(U 0 1) . Proof. By 2.6.4, we have:

(i 0 4;-r)(U) = (10 a)(1 0 W)(1 0 a)(U 0 1)(10 a)(1 0 W*)(i 0 cr) Let x in 910 yl, y2 in

We have:

(10 W)(1 0 a)(U 0 1)(10 a)(1 0 W*)(Ao(x) (9 A - (yi) 0 A - (Y2)) (10 W)(1 0 cr)(U 0 1)(Ap(x) 0 AWOW(F(Y1)(Y2 0 1)))

by 2.4.2 (i) applied to K and 3.7.3 (10 W)(1 0 a)Atko -W(gW - (((# 0 i)(^ F(Y1)(Y2 0 1))(x 0 10 1)) by 4.5.3 (i)

4.5 Characterisation of the Representations and Wendel's Theorem

149

= (10 W)(1 0 a)AOOVOV((# 0 i)F(Yl)(fl(Y2)(x (9 1) (9 1)) = (U (9 1)(1 (9 0 1)) 0 by 4.5.3 (ii) (y1)) by 4.52(i) = (U 0 1)(1 0 a)(U(n+G(x) 0

= (U 0 1)(10 cr)(U 0 1)(10 a)(Ap(x) 0

0 AW(Y2))

therefore, we have:

(10 W)(1 0 Or)(U 0 1)(10 U)(1 0 W*) = (U 0 1)(1 (D q)(U 0 1)(10 a) which completes the proof.

4.5.5 Proposition. With the notations of 4.5.2, the isometry U is unitary. Then, by 2.6.5 and 4.5.4, it is the generator of a non-degenerate representation of M* which shall be denoted by it. Moreover, we have:

A=Aµ and

/9==yµ.

Proof. Let P the projection UU*. We have:

(i 0 F)(P) = (U 0 1)(i 0 c)(P 0 1)(U* 0 1)

by 4.5.4

1.

5.5 Isometries of the Preduals of Kac Algebras In what follows, we consider two Kac algebras K1 = (Mi, rl , tq, V1) and K2 = (M2, I'2, K2, SP2) and T, a linear multiplicative, bijective isometry from M2* to Ml*. We shall note £ its transposed, which is then linear, bijective, ultraweakly continuous isometry from M1 to M2.

5.5.1 Lemma. (i) The operator £(1) belongs to the intrinsic group of K2, and so:

(e(1)) _ £(1)*

(ii) Let w in M2* . We have: £(1)* .wo = (e(1)* .w)°

(iii) The mapping £p being ultraweakly continuous, we can consider its M1*. We have, then, for all projections P in the transposed (ep)* : M2* centre of M2 and for all w in M2*:

(tp)*(w) = T(t(l)*P - w)

.

5.5 Isometries of the Preduals of Kac Algebras

177

Proof. Let w, wl in M2*. We have:

(r2(e(i)),w w = (1, T(w wl)) = (1, T(w) T(i.i ))

by hypothesis

= (r1(1), T(w) 0 T(w )) (10 1 T(w) 0 T(wl)) _ (1,T(w))(1,T(w')) 1

therefore, by linearity, density and continuity, we have:

T2(t(l)) = 41) 0 Al) since, by 5.4.2 (i), we have £(1) 0 0, it completes the proof of (i), using 2.6.6 (i). Let now x be in M2. We have: (x) i(1)* w°) = (xE(1)*,wO) (K2(x*)K2V(1)),W)(IC2(x*W1)*,W)(rl-2 (x *), i(l) * - W)

(x, (AW - W)O)

which completes the proof of (ii). Let now y be in M1. We have: (y,(p)*(w))

=

(ep(y),w) - (t(y)e(1)*P,w)

(t(y), t(l) * P - w) = (y, T(t(l) * P

which completes the proof.

5.5.2 Proposition. With the notations of 5.4, for all x in Mi, we have: r2 rc24 = 4kl

by 5.5.2 (i) by 5.5.4 (ii)

So, Li is an IHI-isomorphism from Kl to K2.

Let us assume that Sa = 1. Then the mapping L1 is involutive, multiplicative and linear by 5.4.2 (ii), normal and bijective from Ml to M2 by construction. Finally, it verifies: L1(1) = 1

r24 = (L (9 Li)I'1 4t'1 = 1 ic1

by 5.4.2 (i) by 5.5.2 (ii) and 12.10 (i) by 5.5.4 (ii) and 1.2.10 (i)

So, Li is an IH[-isomorphism from K1 to K. Thus, (ii) is proved and (i) was proved in 5.5.1 (i). 5.5.6 Theorem. Let K1 = (M1,1'1,ic1,cp1) and K2 = (M2,1'2,ic2,p2) be two Kac algebras. Let u be a normal isomorphism from Ml to M2 such that: U(1)

1

I'2u = (U 0 u)rl Then u is an IHI-isomorphism from (Mi, I'1, m1) to (M2, I'2, r2).

Proof. Let us apply 5.5.5 (ii) to the transposed mapping u* : M2* -+ Ml,,; with u being multiplicative, we are in the first case of 5.5.5 (ii); as u(1) = 1, we get that u is an i-isomorphism.

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5. The Category of Kac Algebras

5.5.7 Corollary. Let K = (M, I', x, co) be a Kac algebra. If the quadruple x and Spo is pro(M, r, n,,, apt,) is also a Kac algebra, then we have n,, portional to W.

Proof. We just apply 5.5.6 to the identity map of M. 5.5.8 Lemma. With the hypothesis and notations of the second case of theorem 5.5.5 (ii), we have, for all w in M2*: VI1)*(W') = T(t(l)* - w) .

Proof. Let x in Ml. We have:

(x,(t'1)(w')) = (t'1(x),w') = (L1(x),w) = (x,(ei)(w)) = (x,T(t(1)*

. w))

by 5.5.1 (iii), which completes the proof. 5.5.9 Corollary. Let K1 = (Mi, I'1, i1, Spi) and K2 = (M2, I'2, r.2, (P2) be two

Kac algebras. Let us assume that there is a linear multiplicative, bijective isometry T, from the Banach algebra M2* on the Banach algebra M1*. Then K2 is El-isomorphic to either K1 or to K1. More precisely, if £ denotes the transposed mapping of T, we have: (i) The operator £(1) belongs to the intrinsic group of K2; (ii) There is an H-isomorphism 15 from K2 onto K1 or to Ki (in the first case Li is a von Neumann algebras homomorphism from M1 to M2, in the second case it is an anti-homomorphism) such that, for all w in M2*, we have: q5(*7^1(J)(A2(W))) = Ai(Tw)

where 741) denotes the automorphism of M2 implemented by £(1) (cf. 3.2.2).

Proof. Let us assume that we are in the first situation of 5.5.5 (ii). Then £1 is 1H[-isomorphism from K1 on K2. The dual ]H[-isomorphism (cf. 5.3.2) .Li from K2 to K1 is defined, for all w in M2*, by:

i02M)

A1W1)*(W))

= ai(T(2(1)* w))

by 5.5.1 (iii)

Now, we have:

ii

(A2

il (A2 (41)

= Ai(T(w))

The theorem stands for

_ Li .

by 3.2.2 (v)

by what is above

5.5 Isometries of the Preduals of Kac Algebras

183

Let us assume now that we are in the second situation of 5.5.5 (ii). Then is is an H-isomorphism from K1 on K. The dual il-isomorphism L1 from K2 on K1 will also be, straightforwardly, an il-isomorphism from K2 on Ki K. By 5.3.2, it is defined for all w' in (M2)*, by:

tl(A2(w

Al W,1 )* Pl))

with the help of lemma 5.5.8, it can as well be written: i'l (A2 (w)) = A1(T(t(1)* . w))

As before, we can deduce from it that the theorem stands with '1 = Li .

5.5.10 Corollary. Let K1 = (M1,r1,iq,so1) and K2 = (M2,f2,ic2,p2) be two Kac algebras. Let us assume that there is a linear, multiplicative, bijective isometry T from the Fourier algebra A(K1) to the Fourier algebra A(K2). Then K1 is IEI-isomorphic to either K2 or K2. More precisely, up to the canonical isomorphims (cf. 3.3.4) between (Mi)* and A(Ki) (for i = 1, 2), if £ denotes the linear mapping from M2 to MI transposed of T, we have:

(i) The operator £(1) belongs to the instrinsic group of K1 (ii) There is an H-isomorphism lfi from K1 onto K2 or KZ (in the first case £1 is a von Neumann homomorphism from M2 to M1, in the sec-

ond case it is an anti-homomorphism), such that for all 9 in A(K1), we have: (7t(i) (ic1ir1*(9))) = ,c2ir2*(T (e))

Proof. It is nothing but corollary 5.5.9 applied to the pair (K2, K1). 5.5.11 Wendel's Theorem ([198]). Let G1, G2 be two locally compact groups, T a linear, multiplicative isometry from L1(G1) to L1(G2). Then, there exist: (a) a character X on G2

(b) a bicontinuous isomorphism a from G2 to G1 such that, for all f in L1(G1) and almost all s in G2:

(Tf)(s) = x(s)f(a(s)) Proof. Let us apply 5.5.5 to Ka (G1) and Ka (G2 ); as Ka (v2) is commutative, there is only one case; the intrinsic group of Ka (G2) is the set of characters on G2 (3.6.12), and the H-isomorphism between Ka(G2) and Ka (G1) comes from an isomorphism of G1 and G2 (4.3.5).

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5. The Category of Kac Algebras

5.5.12 Walter's Theorem ([194]). Let G1i G2 be two locally compact groups, T a linear, multiplicative isometry from A(Gi) to A(G2). Then there exist:

(a) an elements in G1 (b) a bicontinuous isomorphism a from G2 on G1 or G1PP such that, for all t in G2 and f in A(Gi) :

(Tf)(t) = f(s Ia(t)) Proof. Let us apply 5.5.10 to Ka (Gi) and Ka (G2 ); the intrinsic group of K8 (G1) is the set {)tG(s), s E Gi } (4.3.2), and the H-isomorphism between Ka(Gi) and Ka(G2) comes from an isomorphism of G1 and G2 (4.3.5). The result comes then from the fact that K (G2 )' = Ka (G2PP) (2.2.5).

5.6 Isometries of Fourier-Stieltjes Algebras Let K1 and K2 be two Kac algebras. In what follows, we consider a multiplicative, isometric, linear bijection T from B(K2) to B(K1) (as defined in 1.6). Its transposed mapping £ which sends W * (Ki) into W * (K2) is then an ultraweakly continuous, isometric, linear bijection. 5.6.1 Lemma. (i) The operator £(1) belongs to the intrinsic group of W*(K2). (ii) The operator 5A2 (L(1)) belongs to the intrinsic group of K2. (iii) As £1 is ultraweakly continuous, it is legitimate to consider its transposed (t1)* : B(K2) -- B(Ki). Then, with the notations of 5.4, we have, for all 9 in B(K2):

(L1)(9) = T(t(1)* 9) Moreover, if 9 is positive, so will be T(.e(1)* 9).

Proof. The proof of (i) is strictly analogous to 5.5.1 (i), because it results from 5.4.2 that £(1) is invertible; (ii) is just a corollary of (i). The proof of the first part of (iii) is absolutely identical to the one of 5.5.1(iii). Let us assume 9 to be positive. Let x be a positive element of W * (Ki ). We have

(x) (Li)(9)) = (L1(x), 9) > 0 because, by 5.4.2 (ii), Li is a Jordan isomorphism, which completes the proof.

5.6.2 Lemma. (i) Let K be a Kac algebra. We define a set Q by:

Q = {Q E W*(K); Q projection and Q 0 1, s,x,(Q) < Q ® Q}

5.6 Isometries of Fourier-Stieltjes Algebras

185

Then Q has a greatest element and:

max Q = 1 - supp s,\ (ii) With the constructions and notations of (i) associated to the two Kac algebras K1 and K2, we have:

fl(Ql) = Q2

-

Proof. Because A 54 0, it is clear that the projection 1 - suppsA is different from 1. Moreover, we have:

(sA 0 i)s,rx,r(1 - supp sA) = s'\x,r(1 - supp sA) = 4; y7rsA(1 - supp sA) = 0

by 1. 6.4 (i)

by 3.2.2 (iii)

Therefore s,x,r(1 - supp sA) belongs to Ker(sa 0 i) and: s,rx,r(1 - supp sA) < (1 - supp sA) 0 1 By, 3.3.3, we also have: s a s;r (1 - supps,\) = i n s A (1 - supp s a) = 0

Therefore:

s* (1 - supp S,\) < 1 - supps '\ by s;r being involutive, we get in fact: s;r (1 - supp S,\) = 1 - supps A

We then can write down: s,r x ,r (1 - supp S,\) = s,r x ,r s * (1 - supp s A )

= S (s* ® s*) s,r x,r (1 - supp s,\) < c(s;r(1 supp S,\) 1)

= 1®(1 - supp S,\) Finally, by using again (*), it comes:

s,rx,r(1 - suppsA) < (1 - suppsA) ®(1 - supps,\) therefore 1 - supp sA belongs to Q.

by 1.6.6 by by (*

186

5. The Category of Kac Algebras

Let Q be in Q. We have:

rsa(Q) = ryasa(Q) = ssaXa(Q) = s(sa sa)s7rx7r(Q) < 4;(Sa

=

S,\(Q)

< S,\(Q)

sa)(Q 0 Q) sa(4l)

by 3.2.2 (iv) by 3.2.2 (iii) by 1.6.4 (iii) by hypothesis

1

It then results from 2.7.3 (ii) that s.\(Q) is either equal to 0 or 1. Let us assume s.\ (Q) = 1; it is equivalent to Q > supp sA, which implies: Q + (1 - supp S,\) > 1 and:

sirxir(Q) +sirxir(1 - suppsa) > 1 ® 1 and then:

Q0Q+(1-suppsa)®(1-suppsa) > 1®1

And:

((1 - Q) 0 suppsa)(Q0 Q + (1 -

supps,\))((1 - Q) 0 suppsa) > (1 - Q) 0 suppsa

which leads to:

(1-Q)0supps,\ =0 which is impossible, Q being different from 1, and supp s,\ different from 0. Therefore we must have s,\ (Q) = 0, which is Q < 1- supp sA and it completes the proof of (i). Let now Q be in Q1. As L1 is a Jordan isomorphism, L1(Q) is a projector of W* (K2 ). Moreover it is not equal to 1, because L1(Q) = 1 would obviously be equivalent to Q = 1. Now, let 9 and 9' be two positive elements of B(K2). We have:

(s.2x.2(Li(Q)),9 0 9') (sir2xir2(L(Q)L(1)*),909) = = (S1r2 X7r2 (L(Q))(L(1)* 0 L(1)*), 9 0 9') = (Sir2xir2(L(Q)),L(1)* . 9 0 L(1)* . 9') = (L(Q), (L(1)* .9) * (L(1)* .9'))

= (Q, T((L(1)* . 9 * (L(1)* . 9'))) = (Q) T(L(1)* 9) * T(L(1)* 9'))

by 5.4.1 by 5.6.1 (i)

by hypothesis

5.6 Isometries of Fourier- Stieltjes Algebras

187

(S7r2 X 72 (Q), T(t(l)* - 0) 0 T(t(l) * -0 1 ) ) < (Q 0 Q, a(e(i) 0) 0 T(e(i)' 0')) by assumption

(QI=A (L)(0))(Q,Vi ).(01)) V, (Q) 0) V, (Q) 1

V1 (Q) (& 4 (Q)

7

1

by 5.6.1 (iii)

01)

0 (D 01

So, we get: sW2 x V2

(L1(Q)) < L1 (Q) ®L1(Q)

and therefore fl (Q) belongs to Q2-

Therefore we get Li (Qi) C Q2. As £1 is bijective, we could prove ti 1(Q2) C Q1 the same way, which completes the proof of (ii). 5.6.3 Proposition. With the above notations, we have:

(ii) (iii)

supp s 1\1) = 1 - supp S.\2 . (Ker s,\1) = Ker sA2

T(A(K2)) = A(K1)

Proof. Let us apply 5.6.2 to prove (i), considering that Li preserves the Let order. The ideal Ker 3A1 is generated by the projection 1 - supp sal.

x in W * (Ki ). As L1 is a Jordan isomorphism (5.4.2), we have:

Ll(x(1 - supp sa1) = (ei(x) i(1 - supp sat) + Li(1 - suPP sa1)L1(x)) = L1(x)(1 - suPP sae)

by (1)

As L1 is bijective, we get L1(Ker sa1) = Ker 8A2 . And L(1) being unitary and Ker sae a bilateral ideal, it completes the proof of (ii). Let 0 in B(K2); by (ii), T(6) vanishes over Ker 31\2 if and only if 0 vanishes over .(Ker sat) = Ker sae , which gives the result, thanks to 3.3.4.

5.6.4 Notations. The restriction of T to A(K2) satisfies the hypothesis of 5.5.10. There is thus an element u of the intrinsic group of K2 and an ]isomorphism

from K2 to K1 or Wi such that, for all 0 in A(K2), we have: 4i^ruK27r2*(0) = x1ir1*(T9)

Let us determine u more accurately; the mapping (s,\,),-,'T(S,\2 )* is an isometric linear bijection from (M2)* to (M1)* which shall be denoted by T.

5. The Category of Kac Algebras

188

we get u =N 1(1); and by Let £ : M1 -4 M2 its transposed. By 5.5.9, N transposing the relation T(sa2 )* _ (sA1 )T which defines T, we get that N N s,2. _ £s, , by definition of £ and £. Then, we have: U = Al) = 3A2 (t(1)) .

5.6.5 Lemma. With the above notations, we have, for all 9 in B(K2): (l(1))'121r2*(9) = tq7r1*T(9)

Proof. To simplify, we shall put y = Ysa2

In 5.6.4, the above relation has

been proved for 9 in A(K2). Now let w in M2*. Let us recall that (sA2 )*(w) is the generic element of A(K2) and that A(K2) is a bilateral ideal of B(K2) (3.4.4). By applying 5.6.4, we then find: (7(1c21r2*(9+(saz)*(W)))) = wtni.(T(B * (sa2 )* (w)))

or:

(7'c27r2*(O))(7'c2ir2*(sA2)*(w)) _ (lciiri*T(9))(lcilri*T(sA2 )*(w))

and, by using 5.6.4 again: (7!c27r2*(9))(7/c2ir2*(Sa2)*(w)) _ ('c17r1*T(9))(4i'Y'c21r2*(Sa2 )*(w))

which, by 4.6.9 (ii), can also be written: (4i,ytC272*(O)kP'YA2(w))

= (iciiri*T(9))(i'$t2(w))

by having A2(w) converging to 1, we complete the proof. 5.6.6 Theorem. Let K1 and K2 be two Kac algebras. We assume that there exists a multiplicative, isometric, linear, bijective mapping T from the FourierStieltjes algebra B(K2) on B(1K1). Then, there exists an 1H[-isomorphism from

K2 onto K1 or K . More precisely, if £ stands for the transposed of T, we have:

(i) The operator 3A2 (L(1)) belongs to the intrinsic group of K2. (ii) There is an IH[-isomorphism 45 from K2 onto K1 or Ki (in the first case £1 is a von Neumann algebra homomorphism from W*(Ki) to W*(K2), in the second case it is an anti-homomorphism) such that, for all 9 in B(K2), we have:

y8a2

(j(1))1C21'r2*(9) = rcl7ri*T(9)

5.6 Isometries of Fourier-Stieltjes Algebras

189

Proof. It is enough to put 5.6.1 (i) and 5.6.5 together.

5.6.7 Corollary. Let K1 and K2 be two Kac algebras. Let isomorphism from W*(Ki) onto W*(K2) such that:

be a normal

-9 72 X 72 W = (W 0 W) s,,.l x Il

(i.e. such that respects the canonical coproduct of W*(K1) and W*(K2)). Then, there exists an H-isomorphism 4$ from K2 onto K1 such that, for all w in M1*, we have: T/(,7ri (w)) = 7r2 (W 0 q5)

We have also: S,n2 W _ WS;nl

.

Proof. Let us apply 5.6.6 to the transposed mapping W* = B(K2) --- B(K1). As W is multiplicative, we are in the first case, furthermore, as W(1) = 1, there is an H-isomorphism 4$ from K2 onto K1 such that, for all 0 in B(K2), we have: 4$(!27r2*(O)) = ic1ir1*(W*(O))

Because !PK2 = 114$, it can also be written: -7rl*(Tl*(O))

Therefore, for all w in M1*, we have: (W,7rl*o*(O))

(T/(7rl (W)), 0)

(w, 4i7r2*(0)) (w o 43, 7r2*(0))

_

(7r2(c4.o4$),O)

which gives the first result. We can see that: s;r2W7ri(w) = s*27r2(w o 4$) = 7r2(w o 4$) _ 7r2(w o 0 o'2) = 7'2(w o K1 00)

_Wir1(woK1) _ W*1(w) _ Ws*l 7r1 (w)

which gives the second result, by the ultraweak density of 7r1(M1*) in W*(K1).

190

5. The Category of Kac Algebras

5.6.8 Corollary. Let K1 and K2 be two Kac-algebras, W a K-isomorphism from K1 to K2 (i.e. an IH[-isomorphism from W * (K1) to W * (K2 )). Then there exists an IHI-isomorphism u from (M1, I'1, rc1) to (M2, F2, 1C2) such that W is the extension of u. Therefore, K-isomorphisms are H-isomorphisms.

Proof. It is a particular case of 5.6.7. 5.6.9 Corollary (Johnson's Theorem [65] ). Let G1 and G2 be two locally com-

pact groups. Let T a multiplicative, linear, bijective mapping from M1(G1) to M1(G2 ). Then there exists: (i) a character x on G2 (ii) a bi-continuous isomorphism a from G2 to G1 such that for all measure p of M1(G1) we have:

Tp = xa-'(P) Proof. By 4.4.1 (ii), the algebra M1(G1) is the Fourier-Stieltjes algebra associated to the Kac algebra K8 (G1). Let us recall that, by 3.6.12, the intrinsic group of Ka (G1) is composed of the characters on G1. Therefore, by using 5.6.6, we see that there is a character xr on G1 and an IHI-isomorphism 4$ from K8 (G1) to K8 (G2) (because K3(G2) = K3 (G2)) such that, for all p in M1(G1), we have: AG2 (T y) = 4(OXI AG1(12))

We easily compute that for all p in M1(G1) we have: aX-cAG1 W» = aG1 WY)

On the other hand, by 4.3.5, there is a bicontinuous isomorphism a' from G1 to G2 such that, for all s in G1: 4i(AG1(s)) = AG2(ar(s))

By integrating, we find, for all µ in M1(G1): P(AG1(µ)) = AG2 (a'(p))

Going back to (*), we have: by (**) by (***)

KGs (Tµ) = oP(AGi Wµ)) = AGZ

and therefore:

T p= a/(re'IL) = (le

o

(aI

(X µ)

al-1)(aI

W)

We finally reach the result by writing X = x' o a r -1 and a = ar -1

5.6 Isometries of Fourier-Stieltjes Algebras

191

5.6.10 Corollary (Walter's Theorem [194]). Let Gi and G2 be two locally compact groups. Let T be a multiplicative, isometric, linear, bijective mapping

from B(G1) to B(G2). Then there exists: (i) an elements in G1 (ii) a bicontinuous isomorphism a from G2 to G1 or to G1 PP such that, for all t in G2 and f in B(G1), we have:

(Tf)(t) = f(s 'mo(t))

Proof. By 1.6.3 (iii), up to the Fourier-Stieltjes representations, we have B(G2) = B(1[ (Gi)) (i = 1, 2). Therefore, applying 5.6.6, we get the existence of an element u in G(K3 (G1)) and an IHI-isomorphism from Ha (G1) to Ha (G2) or ]HIa (G2 )' = 1HIa (G2PP ), such that for all f in B (G1), we have: Tf = P (flu (f ))

By 4.3.2, there exists s in G1 such that u = AG1(s). Then, we have for all f in L°O(G1) and almost all t in G1: 1 (fl,\,;1 (S) Y )) M = f (s - t)

on the other hand, by 4.3.5, it exists a bicontinuous isomorphism a from G2 to Gi or G1PP such that: (***) lp(f) = f 0 a

Going back to (*), we finally find, for all t in G1 and f in B(G1), that: (Tf ) (t)

(PAG 1 (,q) (f )) (a

f (s-la(t)) which completes the proof.

Chapter 6 Special Cases: Unimodular, Compact, Discrete and Finite-Dimensional Kac Algebras

Let K = (M, T, re, cp) be a Kac algebra, K = (Ar,I, k, c) the dual Kac algebra. We have seen that the modular operator A = aV is the RadonNikodym derivative of the weight Sp with respect to the weight SO o i (3.6.7 ).

So, it is just a straightforward remark to notice that W is invariant under if and only if is a trace. Moreover, the class of Kac algebras whose Haar weight is a trace invariant under x is closed under duality (6.1.4 ). These Kac algebras are called "unimodular" because, for any locally compact group G, the Kac algebra ]I (G) is unimodular if and only if the group G is unimodular. Unimodular Kac algebras are the objects studied by Kac in 1961 ([66], [70] ).

We show later another analogy with the group case, namely that if Sp a finite weight, then (M, T, rs, cp) is a unimodular Kac algebra (6.2.1); it is called "of compact type", because Ka(G) is of compact type if and only if G is compact. We prove then, after Kac ([67] ), that every representation of the involutive Banach algebra M, is the sum of irreducible representations (this leads, for compact groups, to the Peter-Weyl theorem) and that the Fourier representation is the sum of all (equivalent classes) of irreducible representations of M. (6.2.5 ). With the help of Eymard's theorem, this leads to Tannaka's duality theorem for compact groups (6.2.6 ). If K is such that the Banach algebra M. has a unit, then K is a unimodular algebra and K is of compact type (6.3.3 ). So, such Kac algebras will be called "of discrete type". Moreover ([67] ), the von Neumann algebra M is then the sum of finite-dimensional matrix algebras:

M=eC(Hi)

withdi =dim Hi sup{I(x*,w)I, x E M, lIxil < 1} because (1) = 1 = IIwII

Thus, it exists a linear mapping b from a(IV) to IV such that Ilbil C 1 and b(a(w)) = w for all w in IV. By density, it is possible to extend b to the whole

200

6. Special Cases

of H; moreover b is clearly an involutive algebra morphism from a(Iw fl to I. fl 1, which, by continuity, completes the proof. 6.2.5 Theorem. Let K = (M, I', ,c, cp) be a compact type Kac algebra, such that W(1) = 1. Then we have:

(i) The Banach algebra M* admits wA, (1) as unit. Therefore we have A(K) = B(K), and s,\ is a 1H[-isomorphism from W*(K) to (M, I', k). (ii) The element A(W) is a one-dimensional projection p of the centre of M. and we have, for all x in M: px = XP = WA,(1)(x)p

I'(x)(p 1) =pox P(x)(1 ®p) = x ®p Moreover, the unit wA,(1) is a homomorphism from M to C. (iii) For all w in M*, A(w) is a compact operator on H, and, if w belongs to IW, A(w) is a Hilbert-Schmidt operator on H. The Fourier representation A can be decomposed into a direct sum of irreducible finite-dimensional representations Therefore, for all i in I, there exist Hilbert spaces

Hi, with di = dim Hi < -boo, such that the algebra M is isomorphic to ®iEI,C(Hi).

(iv) We have, with the above notations:

F(p) = > di 1 iEI

j,k

k(ej,k) ®ek,j

is a system of matrix units of ,C(Hi). (v) The Haar weight 5P is equal to Ei diTri, where Tri is the canonical trace on ,C(Hi). It is also equal to the restriction to M of the canonical trace on ,C(H), and we have, for all x in M: where

WA,(1)(x) = 0(xp)

(vi) For all i in I and integers 1, m such that 0 < 1, m < d1, we have: di

L

k=1

k

'o

k

+n

where 61 m is the Kronecker symbol and {L}k=1,...,d1 is the orthogonal basis

of Hi corresponding to the matrix units es

jlk.

6.2 Compact Type Kac Algebras

201

Proof. For all w in M*, we have: (A P

(,\*(w)AW(1) I AW(1) _ (,1(w o r%,)11,(1) I AW(1))

by 3.7.3

_ (Aw((w 0 z)-P(1)) I

by 2.3.5

= w(1) therefore A(CA,(l)) = 1, and, thanks to A being injective, we see that wAW(1) is the unit of M*, which gives (i). The relations p *5o = cp and V° = co (6.2.1) imply that A (V) is a projection. As we have w*Sp = Sp*w = w(l)So for all w in M*, we get that this projection lies in the centre of M. Moreover, for all x in M, we have: o rt. 0 Z)r(x))

by 2.3.5

= P(x)Aw(1) = (A,(x) I AW(l))AW(l)

therefore A(V) = p is the one-dimensional projection on CA,,(1). Now, for x in M, e, ,q in H, we have: (xp 177) = ( I A,(1))(z7 I A,(1)) (xA,(1) I A W(1))

= (PI)(2?n(1)(x) therefore pxp = wA, (1) (x) p, and, as p is central, we get the first formula of (ii), from which it is easy to get that wAp (1) is multiplicative. Then, for all w in M. we have: c)(r(=))wz(n)

therefore we get the second formula of (ii). The last one can be proved the same way, therefore we have (ii). As, by 6.2.4, H is a complete Hilbert algebra, we know by ([24], 1 §8.5) that H = ®i E I pi H, where the family {pj} i E I is the set of the minimal projections

of the centre of M. We know that pi H, up to a constant c?, is isomorphic to the Hilbert-Schmidt operators algebra on a Hilbert space Hi, that M is isomorphic to ®iEI,C(Hi) and then that EiEI ciTri. By using the standard representation of M on ®i EI (H1 0 Hi ), we get that Sp is equal to

202

6. Special Cases

the restriction of the canonical trace on L(®i (Hi ® Hi )). The same holds for every standard representation of M, and in particular on H. The representation A is therefore the sum of irreducible representations w -+ A(w)pi, equivalent to representations Ai of M* on spaces Hi such that £(Hi) = Ai(M*)".

If {j}jEJ is an orthonormal basis of H, for all w in I., we can compute the Hilbert-Schmidt norm of A(w):

IIA(w)IIis=

_ j,k

I(A(W)ejlek)12

IJ)t2

j,kE

by 6.2.3 (iii)

= IIIa(w)112

=

IIa(w)112

=IIw!I, (y2 yl C I C ) i

j=1

by 3.2.2 (iv) by 1.4.3

by 3.7.3

by 1.2.5 and 6.1.1 (iii) by 6.1.1 (ii)

6.2 Compact Type Kac Algebras

203

di

ci

(Yi6j

dxy2j

6

i

j,k=1

i

di

ci L (YS!6j' I 6k)(y26k I 6j ilk=1

i

di

>c1 L Tti(ejkYDTri(e"jy' ) j,k=1

i

di

ci L (Tr (9 Tr )(ej,kyl ®ek)3y2) j,k=1

i

di

i

cz

=(SP®SP)

j ej,kyl

i

i

ek,jy2

j,k=1

di

_ (c ® c)

Ie,k ® ek, j (Yi ®y2)

ci 1

j,k=1

di

ci

_ ('2'yi 0wy2) i

1

j k=1

e j,ky1 ®ek, j y2

,

Thanks to 6.1.1 (i), by linearity and density, this implies: di

F(P) E ci L rv(e j j,k=1

i

®ekJ

So, the operator ci 1 ',k=1 k(ezilk) ® ek . is equal to P(p)(x(pi) ® pi) and, therefore, is a projection; by 1.2.11 (ii), it implies ci = di, which completes the proof of (iv). For all x in M, we have:

(xp) = wAV (1) (x)SP(p)

by (ii)

= WA, (1) (x)

by (iv)

which completes the proof of (v). We also have, for 0 < 1, m < di: i'(p)(l ®e12,,z) = di 1 > i (ej,k) 0 ek 'elm j,k

= di 1 E k(el,k) ®ek,m k

by (iv) (*)

204

6. Special Cases

Now, we can write down, thanks to 6.1.1 (vi):

i

k

k1

`%'v km, x

= di

2

('k(e1)* *W k

k,m

/\Wkc(ek 1)* ® Wet

= dz

x) 1

k,

k

,

m

I'(x))

2

= dz E (Sp ® SP)(r(x)(k(ek) l)* ® ek,m)) k

= di

®V)(F(x)I'(p)(1 0 e21)M))

by (***)

= di 1(cp 0 )(i'(xp)(1 0el,m)) = W A,(1) (x) di

®SP)(I'(p)(1 ®el,m ))

by (ii)

= di 1wA,(1)(x)SP((i 0 W )(I'(p)(1 0 el,m)))

= d-1

(1)(° rx, )((Z ®SP)((1 0 p)I'(el,m))) by (HWii) SO

= di 12A,(1)S ((i 0 S)((1®p)I'(el,m))) by 6.2.1 and 6.1.4 = di 1wA,(1) (x) IP^ ((i 0 5?)(el,m ®p))

by (ii)

= di 1wA,(1)(x)c (el,m)

by (v) by (v)

= 6l,rn AW A,(1)(x)

which completes the proof. 6.2.6 Theorem. Let K = (M, T, ,c, Sp) be a compact type Kac algebra such that

p(1) = 1. Then we have: (i) Every non-degenerate representation of M* can be decomposed in a direct sum of finite-dimensional irreducible representations. (ii) Every irreducible representation of M* is finite-dimensional and is equivalent to a component of the Fourier representation A. So, the Fourier representation is the sum of all the (equivalence classes of) irreducible representations of M. Let us note Irr this set. (iii) For y in Irr, let Hµ be a Hilbert space such that µ(M*) = L(Hp ), dP = dim Hp < +oo. The Hilbert space H is then isomorphic to the Hilbert sum ®,EIrrH, 0 Hµ, where the norm on H. is multiplied by dµ. be an orthonormal basis of Hµ. The vectors (iv) Let {di2A,(,i*(f1is,)*)} (E Irr, 0 < j, k < dµ) form an orthonormal basis k

6.2 Compact Type Kac Algebras

205

of H and for all a in H, we have: dµ

dM E pEIrr

,u)*)

I

-

j,k=1

Proof. Let µ be a non-degenerate representation of M* on a Hilbert space Hµ . With the notations of 6.2.5, we put: K; = {ii(b(e))ri, C E p;H, 77 E Hµ}- C Hµ

If i 54 j, it is easy to check that Ki is orthogonal to Kj. As H = ®iElpiH and b(H) is dense in M*, we get that Hµ = ®iEIKi. Let now 77 be in Ki. It is clear that the subspace /2(M*)77 contains µ(b(piH))i which is finite-dimensional, as dense subspace. Therefore the space K = {µ(M* )r7} - is finite-dimensional; it is also p(M*) invariant; therefore K has a subspace K' which is 1t(M*) invariant and such that the corresponding representation of M* is irreducible.

If we consider a maximal family Ka of two by two orthogonal, finitedimensional and p(M* )-invariant subspaces of H. such that the corresponding representations of M* are irreducible, we have necessarily Hµ = ®aKa which completes the proof of (i). Let us assume that /2 is irreducible and let us use the above arguments. There exists i in I such that H,, = Ki. Let q, 77' be in Hµ . We have, for all w in IV:

(/A(w)s' 1 ij') =

By continuity, we have, for all

(a(w) I1V(y*(S2n,n`))*)

in H, thanks to 6.2.4:

(,u(b(e))ri I ij') = ( I AV(L*(flf7,T71)*))

We then have: (/2(b(p))rl 177')

by (*) _ (p(b())r1 I ri') because q belongs to K=

Therefore, we get that: ASP (' * (S?17 j'0 *) = pi AV (p * (QQ7,,7,) *) E piH

and then ) (wA* (0 , )*) (which is equal to 6.1.1(ii) and 3.5.4 (i)) belongs to C(Hi).

,., Hi 0 Hi

(A,(p*(Q,,,,)*)), thanks to

6. Special Cases

206

For w in Ie fl n., we have: (7^r(A,(y*(fl,7)*)a(w) I a(w))

(AV(y*(flj7)*) I a(uj*w")) Ily(w,)77112 > 0

7r(Aw(µ* (Q, )* )) belongs to L(Hi)+, and therefore there exists such that fr(e) = ir(Ap(µ*(sl7)*))1/2 and for any w in IV, we have:

So,

(A()e I

)

_ (a(w)l

in p1H

(a(w) I -7^r'(6)*6)

_ (a(w) I A ,(µ*(Qn)*)) = (p(w)q 177)

As A, and µ are irreducible, we see that µ is unitarily equivalent to AZ, which completes the proof of (ii).

Thanks to (ii) it is possible to put a bijection between Irr and the set I defined in 6.2.5 (i); then (iii) can be deduced from 6.2.5 (iii). For µ E Irr, let eJ k be the matrix units associated to the basis }j=1,...,dµ . We have: 3)

where 'I

3

®'k

is the canonical trace on ,C(H,,). Therefore, by (iii), an orthonormal

basis of H is made of the following elements {d1"2A,(e), µ E Irr, 1 < i l k < dµ } .

For w in I,,, we then have: (a(w)

k

_ (p(w)e I 6k)

= Trµ(e3µ(w))

,

= dµ 1SP(e%A(w))

through the identification of µ to a component of A, by (iii) = d; 1(a(w) I A-(e' ,j)) An orthonormal basis of H is thus made of:

{di2A(p*(Q,)*), pEIrr, 1<j,kjEI diTri, where Tri is the canonical trace on ,C(H1). The unit of e of M* is a homomorphism and satisfies e(x) = 5P(xp) for

all x in M. If we use the standard representation of M on H = ®iEJ(Hi®Hi), then cp is equal to the restriction to M of the canonical trace on ,C(H). Proof. By 6.3.4 (iii), we know that (i) implies (ii). Let us assume (ii), and let us put co = >jE1 diTri. We have then cp(p) = 1. Let us put c(x) _ p(xp) for all x in M. Then, for all w in M*, we have:

(x,e*w) _ (r(x),e®w) _ (Sp ® w)(r(x)(p ®1))

_ (® w)(p 0 x)

by assumption

_ (x,w) Therefore e*w = w for all w in M*. We show that w*c = w the very same way.

Let

be the orthonormal basis of Hi associated to

We have:

i)=Skljdi 1

Therefore, using (* ), we find:

(i (9 p)I'(p) _

iEI

di

1

1:

ic(e

jj) = i (1) = 1

iEI

j,k

From (*) we can also get : J

k

o !. ® i)r(p) = di k

ekI.7

)F(p) =d, 1i(e' ,.7)

(**)

6.3 Discrete Type Kac Algebras

211

And then:

r(4 j)

k oK®i0i)(i0F)r(p) = di(fl

k o r.. ® i ® i)(r (9 i)r(P)

(***)

which yields:

(i 0 Sp)r(ek,j) = di(fl

_,

ik o r ®i)r(Ci ® W)r(p)) = dill

k

(1)1

= Sj,kdi = 5P(ek )1

By linearity, we get (i 0 Sp)r(x) = cp(x)1, for all i in I and all x in £(Hi), and by linearity and normality, for all x in M+. Therefore Sp is left-invariant,

and r(91Sp) c®p. Similarly, using (*), we find, for i1 in I, and 0 < m, I < dii: i di 1 1] K(e4,q)®e4,q

.1

(10 e,1)r(p) =C1®eml) iEI

P,q

di,1E r(eP,1)®em,P P

and then:

C1®ei,m)*r(ek,;) = di(fl i i o

®i ®i)CC1®1 0 At* 2m)(r 0 i)T(p))

=di(fl ij) k oK®i®i)(r(3 didi, 1

E(%

i)((10eit

by (**)

l)I'(P))

i o IC 0 i)r(IC(eit 1)) ®em,p

k

by the above

P

This implies: (i 0 cp)((l 0 ei1)t m)*r(ek ')) = di(fl iti k o x (9 i)r(K(eit 1)) it = di K(i ®flq )F(e M91) k

and, using (***):

i;= didi,K(fl

(i0co)((10em)*r(4,j))

, oK®i(&fl

i )(1'Oi)f(p) C****)

Using once more (*), we get:

r(p)(1 ®ek,j)

eP,q) ®eq,P (10 e

dig

it

P.q

= di 1 E K(ej, q) 0 eqj q

212

6. Special Cases

and then:

r(e"11M* )(10 ek ') =

r(em 1)(1 (9 ek, ') I

= dj,(Q =, i, O KO i ® i)((r 0 i)r(p)(1 ®1® 4,)) by (***)

m

dii(Q i, ;, o rc 0 i 0 i)(r 0 i)(r(p)(1 0 ek,; ))

= di,d- 1

(Q1,1i o r 0 i)r(r(ek,q)) ®eqj by the above computation

This implies: (i 0 p)(r(et/>rn)(1 ®ek,.7 '))

= di/(,fl ;1 ;1 O X. 0 i)r(lc(e4,j))

= didi,(Q jl = didi,(S21,

o

M 41

0

,

by (**) k

(9 SP)((1(9 elVm)F(e' 1j))

by

By linearity, we get, for all xii in ,C(Hii) and yi in C(Hi ):

(i 0 pP)(r(x )(1 0 yi)) = ic((i 0 SP)((' 0 x )r(yi)) Let W be the fundamental operator constructed in 2.4.2 (i), thanks to Sp being left-invariant, and A the bounded linear application constructed in 2.3.5 and 2.4.6 (i). Thanks to 2.4.3, the above formula may be written: (AW(yi) I A(w°)Asp(xii )) = (AW(yi) I A(w)* AW(xi1))

which, by linearity, density, and thanks to 2.4.6 (iv), leads to (HWiii). As Sp

is a trace, we see that (M, r, ,c, 'p) is then a Kac algebra and, as M* has a unit, it is of discrete type, which completes the proof.

6.3.6 Corollary. Let I be a set, and let (M, r, n) be a co-involutive Hopfvon Neumann algebra, with M being abelian and isomorphic to £°°(I). The following assertions are equivalent: (i) The set I can be equipped with a structure of discrete group and then

(M,1',i) is equal to Ha(I). (ii) There exists an element e in I such that, for all x in M:

r(x)(6E®1)_be®x r(x)(1 ®SE) = x 0 &

r(8) _ 11 ic(8i) ®Si iEI

(where Si stands for the characteristic function of {i} over I).

6.4 KreYn's Duality Theorem

213

Proof. It is a consequence of 6.3.5, 4.2.5 (ii) and 6.3.2.

6.3.7 Corollary. Let (M, I', i) be a co-involutive Hopf-von Neumann algebra,

such that M = ®iEI.C(Hi) with di = dim Hi < -boo, and F is symmetric. Then, the following assertions are equivalent: (i) There exists a compact group G such that (M, I', ,) ^r ]HI8(G). (ii) There exists a one-dimensional projection p in the centre of M such

that for all x in M, we have:

f(x)(p®1) = p® x

F(p)

- iEI di 1 1: n(ej,k) ®ek1j j,k

where the ek, j are matrix units for £(Hi ). Proof. It is a consequence of 6.3.5, 4.2.5 (i), 6.3.3 and 6.2.2.

6.4 Krei'n's Duality Theorem 6.4.1 Preliminaries and Notations. Let I be a set, and, for all i in I, let di be in N, Hi be an Hilbert space of dimension di, {e}1p mi, j,kp dkP (so that we

6. Special Cases

214

®,C(Hkp ))), and a unitary may identify ,C(Hi) ®,C(Hj) with Ui, j in C(Hi) ®,C(Hj) such that, for any yi in C(Hi) and yj in G(HQ) and ®p_1(C','iJ,k®

zkp in £ (Hkp) such as yi * Y j = ®p zkp , we have: n

Ui, j (yi (9 yj)U j = (1) (1G(Cmi,j,kp) 0 zkp) p=1

(iii) For every i in I, there exists i' in I such that, for any yi in C(Hi), yo belongs to £(Hii ). More precisely, we have di = dig (so that we may identify

,C(Hi) with C(Hii)) and that there exists a unitary V in C(Hi) such that, for any yi in £(Hi ), we have:

yio =

Vi*(yi)tVi *

where (y)t means the element of C(Hi) whose matrix in the basis {e} is the transposed matrix of yi. (iv) In the decomposition described in axiom (ii), the space Hi,, defined in (i) appears if and only if j is equal to the element i1 defined in (iii); moreover, we have then mi,ii,io = 1. (v) For all un I, we have:

(e,)°*e,,, = Sm,,peio

.

6.4.3 Theorem. With the notations of 6.4.1, let us suppose that there exist on D a coproduct TD and a co-involution XD, such that (D, I'D, KD, SP) is a discrete type Kac algebra. Then, A (considered as a subspace of D*) is a dense sub-involutive algebra of D*, which is a Krein algebra.

Proof. By 1.2.11 (i), for all i, j in I, there exist k1,. .. , kn in I, mi, j,kl , .. ., mi) j,kn

in N, such that didj = Epmi,j,kpdkp (so that we may identify

,C(Hi) ® £(Hj) with ®p_1(Cmi,1,kp ®,C(Hkp ))) and, and a unitary Ui, j in ,C(Hi) ®,C(H,) such that, for all x = ®kxk in D, we have: P

rD(x)(Pi ®pj) = Ui,j ((1r(cmi,j,kl)

(9 xkl )

1=1

Uz j

Therefore, for any (1i in C (Hi) * and (1i in £(H), we get :

(x,Q2*S1') =

(1nia,® xkl) 1=1

,

,j .

(Qi

fli)

which gives that A is a subalgebra of D*, which satisfies 6.4.2 (ii).

('k)

6.4 Krein's Duality Theorem

215

Using 1.2.11 (ii), we get that A is invariant under the involution of D*, and satisfies 6.4.2 (iii).

Let e be the unit of D*; as e is a homomorphism, its support pe is a dimension-one projection in the centre of D; therefore, there exists io in I

such that dio = 1, pE = pi0 and e = fko, where Co is a unit vector of the one-dimensional space Hio ; therefore A satisfies 6.4.2 (i). Moreover, in

the decomposition (*), the index io appears if and only if there exists a dimension-one projection p in C(Hi ® Hj) such that, for all x in D, we have I'(x)p = e(x)p. But then, we have:

P = E(PE)P = rD(pE)n = rn(nf)n(pi 0 P;) But, by 6.3.4, we have:

rD(PE)(Pi 0 pj) = 0 if j 0 i' rD (PE)P(Pil 0 Pi) = Pi where the dimension-one projection Pi has been defined in 1.2.11 (ii). So, such a projection p does not exist if j 54 i', and is equal to Pi (and therefore unique) if j = i'. Therefore A satifies 6.4.2 (iv); as 6.4.2 (v) is given by 6.3.4 (v), the result is proved.

6.4.4 Theorem [109]. With the notations of 6.4.1, let us suppose that A is a Krein algebra. Then, there exist on D a coproduct rD and a co-involution 'CD, such that (D, rD, r, D, SP) is a discrete type Kac algebra, and A (considered as a subspace of D*) is a dense sub-involutive algebra of D*. Proof. Let us consider that A is a subspace of D*; by 6.4.2 (ii) we have then,

for all x=®kxk in D,i,j in 1, 0rn = (ti 0 rn)rn and so TD is a coproduct over D.

By 6.4.2(iii), we have, for alliinI,0 ; = apse

and also: aq,B ° q,8

: * ` `q sP *

tJs r = ap,r

Oi

Sj

p

Then we have:

q,8

aq,8 E J°i1 p= * flgi

; = Eap,rno.

*fl 1

P

q

p

.

P

Thanks to 6.4.2 (v), we have: no

ap,r w k

p

1dP

=

a1,8 s168 8

26

Let V be the linear mapping Hi - Hi defined by (V6' I

For all x=$xi in M, r,s such that 1 L1 bi ®A ci, with bi and c2 in B, and satisfies ax = xa for all a in A, then we have, for x = Ei bi ®A ci and y = Ej dj ®A ej as above, the multiplication of endomorphisms, given by: xy = 1: biEA(cid j) ®A ei i, j

where EA is the conditional expectation on A, and a convolution operation defined by:

X * y = 1: bi dj ®A ejci = > bi yci i (the convolution is well defined since y commutes with A).

244

Postface

When B is the crossed product of A by K, then this recovers the bialgebra structure of K (up to completions) in an invariant way. In the general case, one can show that, under our hypothesis, B is the crossed product of A by a Kac algebra K with a cocycle twisted action, if and only if End(AB (&A BB) is

a factor. This type of result generalizes beyond our finite index assumption, and shows that the Kac algebras arise naturally by means of their actions, as symmetry structures associated to inclusions of von Neumann algebras. I can only hope thus that the authors will continue their beautiful monograph with a study of the Kac algebras in action. Paris, October 1991

Adrian Ocneanu

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[118] P. Podles: Quantum spheres. Lett. Math. Phys. 14(1987), 193-202. [119] P. Podles: Complex quantum groups and their real representations. RIMS Kyoto preprint 1991. [120] P. Podles and S.L. Woronowicz: Quantum deformation of Lorentz group. Comm. Math. Physics 130(1990), 381-431.

[121] L.S. Pontrjagin: The theory of topological commutative groups. Ann. of Math. 35(1934), 361-368. [122] H.I0. PemeTHxHH, JI.A. TaxTaaMAH, JI..1I. (Da

eeB (N.Yu. Reshetikhin, L.A. Takhtadzhan, L. D. Faddeev): KBaHTOBaHHe rpyna x anre6p JIH. AJIre6pa x aHa.nM3 1-1(1989), 178-207; translated in: Quantization of Lie groups and Lie algebras. Leningrad Math. J. 1(1990), 193-226.

[123] N.Yu. Reshetikin, V.G. Turaev: Ribbon graphs and their invariants derived from quantum groups. Com. Math. Phys. 127(1990), 1-26. [124] C. Rickart: Banach algebras, Krieger, Huntington, NY, 1974. [125] M.A. Rieffel: Some solvable quantum groups. Proceeding of the Conf. on Operator Algebras and their connections with Topology and Ergodic Theory II, Crasova, Romania, sept. 89 [126] M.A. Rieffel: Deformation quantization and operator algebras. Operator theory: operator algebras and applications I, A.M.S., Providence, RI, (1990), 411-423. [127] M. Rosso: Comparaison des groupes SU(2) quantiques de Drinfel'd et Woronowicz. Note C.R.A.S. Paris 304(1987), 323-326. [128] M. Rosso: Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra. Com. Math. Phys. 117(1988), 581-593. [129] M. Rosso: Groupes quantiques et modles a vertex de V. Jones en theorie des noeuds. Note C.R.A.S. Paris 307(1988), 207-210.

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[130] M. Rosso: Algebres enveloppantes quantifiees, groupes quantiques compacts de matrices et calcul differentiel non commutatif. Duke Mat. J. 61(1990), 11-40. [131] M. Rosso: Representations des grouper quantiques: Se m. Bourbaki no 744 (juin 1991).

[132] R. Rousseau and A. Van Daele: Crossed products of commutation systems. Math. Ann. 239(1979), 7-20. [133] S. Sakai: C*-algebras and W*-algebras, Springer-Verlag, Berlin, 1971. [134] J.-L. Sauvageot: Implementation canonique pour les co-actions d'un groupe localement compact et les systemes dynamiques generalises. Math. Ann. 270(1985), 325337.

[135] J.-M. Schwartz: Relations entre "ring groups" et algebres de Kac. Bull. Sc. Math. 100(1976), 289-300.

[136] J.-M. Schwartz: Sur la structure des algebres de Kac I,II. J. Funct. Anal. 34(1979), 370-406; Proc. London Math. Soc. 41(1980), 465-480. [137] M.A. CeMeHOB-TAH-IIIaHCxHM (M.A. Semenov-Tyan-ShanskiT): LITO Taxoe KJIacaHaJH3 H ero npMJI. 17-4(1983), 17-33; translated cHLlecxaA r-MaTpHiga?

in: What is a classical R-matrix? Funct. Anal. appl. 17(1983), 259-272. [138] A.J.-L. Sheu, J.-H. Lu and A. Weinstein: Quantization of the Poisson SU(2) and its Poisson homogeneous space. The 2-sphere. Com. Math. Phys. 135(1991), 217-232. [139] A.M. IIITepH (A.I. Shtern): 11BOMCTBeHHOCTb AJIA cxpeIUUeHHblx npOH3BeAeHH

anre6p c4OH HeMMaHa. HTOrM HayKH H Tex. Corp. npo6Jl. MaT. 27(1985), 3365; translated in: Duality for crossed products of von Neumann algebras. J. Soviet Math. 37(1987), 1396-1421. [140] E.K. CxnAHHH (E.K. Sklyanin): McToA o6paTHOM 3anaxiM pacceAHHA H KBaH-

TOBOe HeJIHHelHoe ypaBHeaxe IIIpeAHHrepa. lloxn. AKaA. Hayx CCCP 244(1979), 1337-1341; translated in: Method of the inverse scattering problem and the nonlinear quantum Schrodinger equation. Soviet Phys. Dokl. 24(1979), 107-109. [141] E.K. CKJIJIHHH (E.K. Sklyanin): KBaHTOBbIt BapHaHT McTOAa o6paTHOM 3aAa'IH pacceAHHA. B c6.: 11H(44)epeHImaJIbHaJ reoMeTpHA, rpynnbl JIH H MexaHHKa.

San. HayxiH. ceMHH. JIeHHHrpaA. OTAenn. MaTeM. 14HCT. HM. CTexaOBa (JIOMM) 95(1980), 55-128; translated in: Quantum version of the inverse scattering problem. J. Soviet Maths 19(1982), 1546-1596. [142] E.K. CIKJIAHHH (E.K. Sklyanin): 06 oAHoi anre6pe, IlopoxcAaeMol1 xBaApaTHLIHbIMH COOTHomeHHAMH. YCnexH MaTeM. Hayx, 40-2(1985), 214.

[143] E.K. CKJIRHHH, JI.A. TaxTa.ID HH, JI.11. 4 aAAeeB (E.K. Sklyanin, L.A. Takhtadzhan, L. D. Faddeev): KBaHTOBbIII McTOA o6paTHOM 3aAagH. I. TeopeT. H MaTeM. c4H3Hxa 40(1979), 194-220; translated in: Quantum inverse problem method I. The-

oret. and Math. Physics 40(1980), 688-706. [144] E.K. CKJAHHH, JI.11. 4a=eeB (E.K. Sklyanin, L.D. Faddeev): KBaHTOBOMeXaHHLIecKHII nOAXOA K BHOJIHe HHTerpHpyeMblM MOAeJIAM TeOpHH nOJIA. .ZIOKJI.

AxaA. Hayx CCCP 243(1978), 1430-1433; translated in: Quantuum-mechanical approach to completely integrable models of field theory. Sov. Phys. Dokl. 23(1978), 902-904. [145] A.C. Cotk6eJmMaH (Ya.S. Soibel'man): HenpHBo,zHMMe npeACTaBJIeHHA anre6pbl 4 yHKIuHI Ha KBaHTOBoi rpynne SU(n) H KJIeTKH IIIy6epTa..LIox i. AKaA. Hayx CCCP 307(1989), 41-45; translated in: Irreducible representations of the function algebra on the quantum group SU(n), and Schubert cells. Soviet Math. Dokl. 40(1990), 34-38. [146] A.C. Colt6enbMaH (Ya.S. Soibel'man): Anre6pa 4IyHxiXH11 Ha KOMnaICTHOM rpynne m ee npe.z cTaBJIeHHH. AJIre6pa m aHaJH3 2-1(1990), 190-212; translated in: The algebra of functions on a compact quantum group, and its representations. Leningrad Math. J. 2(1991), 161-178. [147] A.C. Co16enbMaH (Ya.S. SoYbel'man): COCTOAHHA FeJIb4laHAa-Hat MapxaCHrana m rpynna Be1JIA AJIA XBaHTOBOM rpyuimi SU(n). I yHXIX. aHaJIH3 H

252

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[157] M. Takesaki: A characterization of group algebras as a converse of TannakaStinespring-Tatsuuma duality theorem. Amer. J. Math. 91(1969), 529-564.

[158] M. Takesaki: Tomita's theory of modular Hilbert algebras and its applications. Lectures notes in Math. n° 128, Springer-Verlag, Berlin (1970). [159] M. Takesaki: Duality and von Neumann algebras, Lectures on Operator Algebras. Lectures notes in Math. n° 247, Springer-Verlag, Berlin (1972), 665-785. [160] M. Takesaki: Conditional expectations in von Neumann algebras. J. of Funct. Anal. 9(1972), 306-321. [161] M. Takesaki: Duality for crossed products and the structure of von Neumann algebras of type III. Acta Math. 131(1973), 249-310. [162] M. Takesaki: Theory of operator algebras I. Springer-Verlag, New York, 1979

[163] M. Takesaki and N. Tatsuuma: Duality and subgroups. Ann. of Math. 93(1971), 344-364. [164] JI.A. TaxTa.Lut szH (L.A. Takhtadzhan): HexoMMyTaTHBHble rOMOJIOrHH KBaHTOBbIX TopoB. aHaJIH3, 23-2(1989), 75-76; translated in: Noncommutative

homology of quantum tori. Funct. Anal. appl. 23(1989), 147-149. [165] JI.A. TaxTaAmHH, JIM. 'a=eeB (L.A. Takhtadzhan, L.D. Faddeev): KBaHTOBbIt McTo.n o6paTHO t 3aAagH H XYZ Mo neJlb Uet3eH6epra. Ycnexa MaT. HayK

34(1979), 13-63; translated in: The quantum method of the inverse problem and the Heisenberg XYZ model. Russian Math. Surveys 34(1979), 11-68. [166] T. Tannaka: Uber den Dualitatssatz der nichtkommutativen topologischen Gruppen. Tohoku Math. J. 45(1938), 1-12. [167] N. Tatsuuma: A duality theorem for locally compact groups I,II. Proc. Japan Acad. 41(1965), 878-882; 42(1966), 46-47. [168] N. Tatsuuma: A duality theorem for locally compact groups. J. Math. Kyoto Univ. 6(1967), 187-293. [169] V.G. Turaev: The Yang-Baxter equation and invariants of links. Inv. Math. 92(1988), 527-553. [170] J1. 14. BattHepMaH (L.I. Vainerman): Xapa1KTepH3anxHJI ,nBO#CTBeHHbIX K JIOKaJIbHO KOMIIaKTHbIM rpynnaM. I yHKIn. aHaJIH3 H ero IIpaJi. 8-1(1974), 75-

76; translated in: Characterization of objects dual to locally compact groups. Funct. Anal. appl. 8(1974), 66-67. [171] JI.M. Bat HepMaH (L.I. Valnerman):)) KoHeLIHble rHnepxoMIIJIesCHble CHCTeMbI

Kalxa. I yHKn. anaJIH3 H ero upHa. 17-2(1983), 68-69; translated in: The finite hypercomplex systems of Kac. Funct. Anal. appl. 17(1983), 135-136.

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253

[172] JI.14. BattHepMaH (L.I. Valnerman): I'HIIepKOMIIJIexCHble CHCTeMbI C KOMIIaKTHbIM H AHCKpeTHbIM 6a3HCOM. RoxR. AKaA. Hayx CCCP 278(1984), 16-19;

translated in: Hypercomplex systems with a compact and discrete basis. Soviet Math. Dokl. 30(1984), 305-308. [173] J1.14. Ba11HepMaH (L.I. VaTnerman): I'apMOHHLIeCKHtI aHaJIH3 Ha rHIIepKOMIIJIeKCHbIX CHCTeMax c xoMIIaKTHbIM H AHCKpeTHbIM 6a3HcoM. CIIeKTpaJIbHaa TeopHSI

onepaTopoB H 6ecxoHeLIHOMepHbitt aHaJIH3. 14H. MaT. AH YCCP KHeB (1984),

19-32; translated in: Harmonic Analysis on Hypercomplex systems with a compact and discrete basis. Sel. Math. Sov. 10(1991), 181-193. [174] JI.H. BatlHepMaH (L.I. Valnerman): IIpHHIXHII ABO1CTBeHHOCTH ARA KoHe4IHbIx

rHIIepsomnAeKCHbIX cHcTeM. I43B. By3oB.MaT. 29-2(1985), 12-21; translated in:

A duality principle for finite hypercomplex systems. Soviet Math. (Iz. VUZ) 292(1985), 13-24. [175] JI.14. BaIHepMaH (L.I. Valnerman): 1IBOIICTBeHHOCTb anre6p c HHBOJIOIiHett

H onepaTopbi o6o6ngeHHoro cABHra. I4TorH HayKH H Tex. MaTeM. Allan. 24(1986), 165-205; translated in: Duality of algebras with an involution and generalized shift operators. J. Soviet Math. 42(1988), 2113-2138. [176] J1.14. BaIlHepMau (L.I. VaInerman): 06 a6cTpaKTHOtt (jopMyne IIJIaHmepeAA H (4opMyJle o6paiieHHA. Yxp. MaT. ac. 41(1989), 1041-1047; translated in: Abstract Plancherel equation and inversion formula. Ukr. Mat. J. 41(1989), 891-896. [177] L.I. VaYnerman: Correlation between compact quantum group and Kac algebras. to appear in Adv. Soviet Math. [178] JI.14. BaItHepMaH (L. I. VaYnerman): HeyHHMOAysApHbie KBaHTOBaHHble rHuep-

KoMIIJIexcHble clcTeMbz..oxJi. Axai . Hays YCCP Cep. A. 1989, 7-10. [179] J1.I4. BaItHepMaH, r.14. Kali (L.I. Vainerman, G.I. Kac): HeyHHMOAyJIApHbie KOJIbIieBble rpynm I H anre6pbl XOII4a-(4OH HelIMaHa. .oKJI. AxaA. HayK CCCP 211(1973), 1031-1034; translated in: Nonunimodular ring-groups and Hopfvon Neumann algebras. Soviet Math. Dokl. 14(1974), 1144-1148. [180] JI.I4. BaItHepMaH, r.14. Kaii (L.I. Valnerman, G.I. Kac): HeyHHMOAyJIxpHbie KonbIXesble rpynmb H anre6pbi Xon4 a-4 off He1MaHa. MaTeM. c6. 94(1974), 194225; translated in: Nonunimodular ring-groups and Hopf-von Neumann algebras. Math. USSR Sbornik 23(1974), 185-214. [181] J1.14. BatlHepMaH, A.A. Kanio sHbdt (L.I. Vainerman, A.A. Kalyuzhnil): KBaHTOBaHHbie rHIIepxoMJiexCHbie CHCTeMbI. rpaHHXIHbie 3aAaxim ARK AH(4(pepeHn[HaJIbHI3IX ypaBHeHHlt. I4K. MaT. AH YCCP KHeB (1988), 13-29; to be translated

in Sel. Math. Sov. [182] J1.14. BaItHepMaH, r.J1. JIHTBHHOB (L.I. VaYnerman, G.L. Litvinov): IInaHmepensi H 4opMyJia o6pangeHHA AJIA onepaTopoB o6o6ceHHoro uiABHra.

Aoxs. AKaA. Hays CCCP 257(1981), 792-795; translated in: The Plancherel formula and the inversion formula for generalized shift operators. Soviet Math. Dokl. 23(1981), 333-337. [183] A.R. BaxcMaH (L.L. Vaksman): q-AHanorH KO3(4()HIgHeHTOB Kne6ma-ropAaHa H anre6pa (4yHKIxHtrt Ha KBaHTOBol rpyIIIIe SU(2). Rosn. AKaA. Hays CCCP

306(1989), 269-272; translated in: q-analogues of Clebsch-Gordan coefficients and the algebra of functions on the quantum group SU(2). Soviet Math. Dokl. 39(1989), 467-470. [184] JI.J1. BaxcMaH H JI.M. KoporoAcsHtt (L.L. Vaksman and L.I. Korogodskii): Anre6pa orpaHHqeHHbIX (PyHKIXHit Ha KBaHTOBO1 rpynne ABHxceHHt IIJIOCKOCTH H

q-aHaJiorH 4yHKIrH1l BeCCeJA. floss. AxaA. Hays CCCP 304(1989); translated in: An algebra of bounded functions on the quantum group of the motion of the plane, and q-analogues of Bessel functions. Soviet Math. Dokl. 39(1989), 173-177. [185] JI.JI. BascMaH H $I.C. Cott6enbMaH (L.L. Vaksman and Ya.S. Soibel'man): Anre6pa (pyHxn(HIl Ha KBaHTOBOtt rpynne SU(2). 'yHxu. aHaJH3 H ero IIpHJI. 22-

254

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c4epbi. Anre6pa m axaJH3 2-5(1990), 101-120. Translated in: The Algebra of Func-

tions on the Quantum Group SU(n + 1), and odd-dimensional Quantum Spheres. Leningrad Math. J. 2(1991), 1023-1042. [187] J.-M. Vallin: C*-algebres de Hopf et C*-algebres de Kac. Proc. London Math. Soc. 50(1985), 131-174. [188] A. Van Daele: Continuous crossed product and type III von Neumann algebras. London Math. Soc. Lecture Note 31, Cambridge Univ. Press, 1978. [189] A. Van Daele: A quantum deformation of the Heisenberg group. Preprint [190] L. Van Heeswijck: Duality in the theory of crossed products. Math. Scand. 44(1979), 313-329.

[191] J.-L. Verdier: Groupes quantiques (d'apres V.G. Drinfel'd). Sem. Bourbaki no 685. Asterisque 152-153(1987), 305-319. [192] D. Voiculescu: Amenability and Kac algebras. Proceedings of the Colloque International "Algebres d'operateurs et leurs applications a la physique theorique" Marseille (1977)

[193] M.E. Walter: Group duality and isomorphisms of Fourier and Fourier-Stieltjes algebras from a W*-algebra point of view. Bull. Amer. Math. Soc. 76(1970), 1321-1325. [194] M.E. Walter: W*-algebras and non-abelian harmonic analysis. J. of Funct. Anal. 11(1972), 17-38. [195] M.E. Walter: A duality between locally compact groups and certain Banach algebras. J. Funct. Anal. 17(1974), 131-159. [196] M.E. Walter: Dual algebras. Math. Scand. 58(1986), 77-104. [197] A. Weil: L'integration dans les groupes topologiques et ses applications. Act. Sc. Ind. no 1145. Hermann. Paris. 1953.

[198] J.G. Wendel: On isometric isomorphism of group algebras. Pacific J. of Math. 1(1951), 305-311.

[199] J.G. Wendel: Left centralizers and isomorphisms of group algebras. Pacific J. of Math. 2(1952), 251-261. [200] S.L. Woronowicz: Twisted SU(2) group. An example of non-commutative differential calculus. Publ. RIMS 23(1987), 117-181. [201] S.L. Woronowicz: Compact matrix pseudogroups. Comm. Math. Phys. 111(1987), 613-665. [202] S.L. Woronowicz: Tannaka-Kreen duality for compact matrix pseudogroups. Twisted SU(N) group. Invent. Math. 93(1988), 35-76. [203] S.L. Woronowicz: Differential calculus on compact matrix pseudogroups (quantum groups). Comm. Math. Phys. 122(1989), 125-170. [204] S.L. Woronowicz: Unbounded elements affiliated with C*-algebras and non-compact quantum groups. Com. Math. Phys. 136(1991), 399-432.

[205] S.L. Woronowicz: A remark on compact matrix quantum groups. RIMS Kyoto, preprint 1990. [206] S.L. Woronowicz and S. Zakrzewski: Quantum Lorentz group induced by the Gauss decomposition. Preprint.

Index

Abelian co-involutive Hopf-von Neumann algebra Achieved left Hilbert algebra . . . . . . . Affiliated . . . . . . . . . . . . . . . Analytic element . . . . . . . . . . . . Antipode . . . . . . . . . . . . . . .

Bochner's theorem C * -algebra

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Coextension of a H-morphism . . . . Co-involutive Hopf-von Neumann algebra Commutant Kac algebra . . . . . . Compact type Kac algebra . . . . . Complex Hopf algebra . . . . . . . Conditional expectation . . . . . . Connection relations . . . . . . .

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.

1.2.5 2.2.5 6.2.1 6.6.6

2.1.8 (ii) .

Discrete type Kac algebra . . . . . . . Dual co-involutive Hopf-von Neumann algebra Dual Kac algebra . . . . . . . . . Dual morphism . . . . . . . . . . . Dual weight . . . . . . . . . . . . .

Ernest algebra . . . . . . Ernest's theorem . . . . . Extended positive part of M Extension of a H-morphism . Extension of a representation Eymard algebra . . . . . Eymard's duality theorem . Eymard's theorem . . . .

1.1.1 (i) . 5.2.3

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3.6.1

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6.3.1

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5.1.2 3.5.3

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1.6.8 4.7.3 2.1.8 (i) . 5.2.3 4.6.8 . . 3.4.4 . 4.3.3 . 3.4.6 . .

256

Index

. Fourier algebra . . . . Fourier-Plancherel mapping . Fourier-Plancherel transform Fourier representation . . . Fourier-Stieltjes algebra Fourier-Stieltjes representation Fourier transform . . . . Fundamental operator . . .

3.3.4 .

3.5.3

4.3.7 2.5.3 1.6.9 1.6.9 .

4.3.7

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2.4.2

Generator

1.5.2

H-morphism . . . . . . . Haar weight . . . . . . . . Heisenberg bicharacter . . . Heisenberg's commutation relation Heisenberg pairing operator . . Heisenberg's theorem . . . . Hopf algebra . . . . . . . *-Hopf algebra . . . . . . . Hopf-von Neumann algebra . .

1.2.6 2.2.1 4.6.6 (ii)

4.6.7 4.6.2 4.6.1 6.6.6 6.6.6 1.2.1

Induced von Neumann algebra Intrinsic group . . . . . .

1.1.1 (ii)

1.2.2

Johnson's theorem

5.6.9

Kac algebra . . . . . . . . Kac-Paljutkin's theorem . . . K-morphism . . . . . . . Krein algebra . . . . . . . Krein's theorem . . . . . . Kronecker product . . . . Kubo-Martin-Schwinger condition

2.2.5 6.6.5 5.1.1

6.4.2 6.4.6 1.4.1

2.1.1 (iv)

Left Hilbert algebra Left-invariant weight Left regular representation

2.1.1 (iii) 2.2.1 1.1.5

Modular automorphism group

2.1.1 (iv)

Operator-valued weight Opposite Kac algebra

Pentagonal relation

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2.1.8 (i) 2.2.5

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2.4.4

Index

Peter-Weyl's theorem . . . Plancherel weight . . Pontrjagin's duality theorem Positive definite elements

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_

_

_

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Radon-Nykodim derivative of weights Reduced Kac algebra . . . . . . Reduced von Neumann algebra . . Represent able . . . . . . . . . Square-integrable element of M* Standard von Neumann algebra Stone's theorem . . . . . . Strict ]HI-morphism

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6.2.7 (iv) . 3.6.11 4.3.8 . .

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257

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1.3.1

2.1.1 (v) . 2.2.6 1.1.1 (ii) .

1.3.6

Sub-Kac algebra . . . . . . . . . . . . . Symmetric co-involutive Hopf-von Neumann algebra

Tannaka's theorem . . . . . . . . . Takesaki's theorem . . . . . . . . . . . . . . . . . Tatsuuma's theorem Tensor product of von Neumann algebras . . . Tensor product of weights . . . Tensor product of operator-valued weights Unimodular Kac algebra Von Neumann algebra

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6.2.8 4.2.4 . . 4.7.4 1.1.1 (ii) 2.1.1 (iv) 2.1.8 (iii)

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1. 1. 1 (i)

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W*-algebra . . . . . . . . . . . . . Walter's theorem on Fourier algebras . . . Walter's theorem on Fourier-Stieltjes algebras . . . . . Weight . . . . . . . . . Weil's theorem . . . . . . . . . . . Wendel's duality theorem . . . . . . . Wendel's theorem . . . . . . 0

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1.1.1 (i)

. 5.5.12 . 5.6.10 2.1.1 (i) 4.2.6

4.5.9 .

5.5.11