Kac Algebras and Duality of Locally Compact Groups

Michel Enock Jean-Marie Schwartz

. s·p·r inger-Verlag

Anthropomorphic carving representing Duality (Totonac culture)

Michel Enock Jean-Marie Schwartz

Kac Algebras and Duality of Locally Compact Groups Preface by Alain Cannes 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 antropo/ogia de Ia Universidad Veracruzana, Jalapa, E. U. de Mexico

Mathematics Subject Classification (1980): 22025,22035, 43A30, 43A65

iSBN 3-540-54745-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-54745-2 Springer-Verlag NewYork Berlin Heidelberg

Library of Congress Cataloging-in-Publication Data available This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law. of September 9, 1965, in its current version, and a permission for use must always be: obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. ©Springer-Verlag Berlin Heidelberg 1992 Printed in the United States of America (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,K), a Haar weight is a faithful, semi-finite, normal weight on M+, which is left-invariant with respect tor, i.e. such that:

(i ® cp)F(x) = cp(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 K. The quadruple (M' K, cp) is then called a Kac algebra. On L 00 (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 (£.£.£).Thus, L 00 (G), equipped with the convenient operations, is an abelian Kac algebra; we shall denote it Ka(G). Left-invariantness allows us to define the fundamental operator W, which is an isometry belonging toM® C(Hr.p), satisfying the so-called pentagonal relation (£.4.4):

r,

(1 ® W)(u ® 1)(1 ® W)(u ® 1)(W ® 1)

= (W ® 1)(1 ® W)

With the other axioms, we get that W is unitary (£.6.£), "implements" the co-product r, i.e. that we have, for all X in M (£.6.9): F(x)

= W(1 ® x)W*

Moreover, the mapping: ..\: w--+ (w ® i)(W*)

from M* to C(Hr.p) is a non-degenerate representation of the involutive Banach algebra M* (2.6.1 and 2.6.9); 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 Aa of a locally compact group G; more precisely, for the Kac algebra lKa(G), the fundamental operator is the bounded continuous function s --+ Aa( s )*, when considered as an element of L 00 (G) 0 .C(L 2 (G)), and the Fourier representation is nothing but the left regular representation of L 1 (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 H'P such that, for all x in~ (2.1.6):

(x*,w)

= (a(w) IA'P(x))

If the weight cp is left-invariant, the set I'P of such elements is a left ideal of M* (2.4.5), and, if cp is a Haar weight, we have, for all w1 in M* and w2 in

I"' (2.4.6): a(w1

* w2) =

A(w1)a(w2)

In the case of lKa( G), we recover L 1 ( G)nL 2 (G), which is a left ideal of L 1 ( G), and the formula:

h *h = Aa(h)h which links the convolution and the left regular representation of G. If lK = (M,r,,,cp) is a Kac algebra, so is JK' = (M,~r,,,cp o K), and it is possible, too, to define a Kac algebra JK' on the commutant M' of M (we suppose M is a von Neumann algebra on the Hilbert space H'P); these technical results (2.2.5) will be useful in Chap. 5.

2.1 An Overview of Weight Theory 2.1.1 Definitions and Notations. (i) A weight on a W* -algebra M is an additive mapping cp : M+ --+ [0, +oo] such that cp( Ax) = Acp( x) for all A in Jll+ 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:

rott = {x EM+, cp(x) < +oo} ~

= {x EM,

cp(x*x) < +oo}

46

2. Kac Algebras

Then ~c,o is a left ideal in M, and the involutive algebra ~~ ~ is the linear span of rott; it is denoted by rolc,o, and it is possible to extend cp to a positive linear form on rolc,o, which will be still denoted by cp. The weight cp is said to be faithful if cp( x) = 0 for x in M+ implies x = 0, semi-finite if rolc,o is ultra-weakly dense in M (finite if VJtcp = M), and normal if cp(supa xa) = supa cp(xa) for all increasing directed bounded nets {xa} in M+. If cp is normal, the set of all projections p in M such that cp(p) = 0 has a greatest element po, and q = 1- Po is called the support of cp. We may consider then cp as a faithful weight on the reduced algebra Mq. (ii) In what follows, cp will denote a faithful, semi-finite, normal w~ight on a W*-algebra M. Then, the left ideal~' equipped with the scalar product (x, y) -+ cp(y*x) (x, y E ~), is a pre-Hilbert space; let us denote Hc,o the associated Hilbert space (or H ifthere is no confusion), and Ac,o the canonical injection~-+ Hc,o. The set 21c,o = Ac,o(~ n ~~), equipped with the product T and the involution U defined by:

Ac,o(x)T Acp(y) = Acp(xy) Ac,o(x)U = Acp(x*)

(x, y E ~ n ~~) (x E ~c,o n ~~)

is an involutive algebra, dense in Hc,o, such that its involution is an antilinear preclosed mapping, and such that the left-multiplication representation of 21c,o is non-degenerate, bounded and involutive. The W* -algebra M is then isomorphic to the von Neumann on Hc,o generated by this representation ([14], th. 2.13). (iii) Generally speaking, we shall call left Hilbert algebra an involutive algebra 21, equipped with a scalar product, such that the involution is an antilinear preclosed mapping in the Hilbert space H associated, and such that the left-multiplication representation 1r of !2t is non-degenerate, bounded and involutive ([158], de£. 5.1). Let us then callS the closure of U, and F the adjoint S* of S, whose domains will be respectively denoted by vU and 1J. We shall say that in H is right bounded if the linear application from !21 to H defined by '7-+ 1r(71)e (71 E !21) is bounded; we shall then denote by 1r1(e) the element of .C(H) defined, for all7J in !21, by 7r'(e)'7 = 1r(71)e. Let us then define:

e

!21'

= {eE 'D~,

eis right bounded}

Then, 21', equipped with the product 6Te2 = 1r'(e2)6 (6,6 E !21'), and the involution e~ = Fe (e E !21') is a right Hilbert algebra (with the obvious definition). Repeating this construction, we obtain the definition of left bounded elements in H, and another left Hilbert algebra !2111 , containing !2t as an involutive subalgebra, the closed operator S being again the closure

2.1 An Overview of Weight Theory

47

of the involution of !!11 • A left Hilbert algebra is called achieved if !2t = !1!11 • Moreover, for any left Hilbert algebra !X, the left (resp. right) Hilbert algebra !!'' (resp. !X') is an achieved left (resp. right) Hilbert algebra, and the left Hilbert algebra !Xrp constructed in (ii) is achieved. Conversely, for any left Hilbert algebra !X, the left-multiplication representation 1r generates a von Neumann algebra M on the Hilbert space H, and the formula, for x in M+:

cp(x)

= { 11e11 2 +oo

if there exists e in !!11 such that elsewhere

X=

1r(e)

defines a faithful, semi-finite normal weight on M, with the left Hilbert algebra !2trp isomorphic to !!11 ([14], th. 2.11). For any in v# (resp. v"), it is possible ([14], def. 2.1) to define a left (resp. right) multiplication bye, which will be a closed operator on H, affiliated to M (resp. M'). (iv) Starting from !2trp, the polar decomposition S = Jrp.tJ}/ 2 gives rise to the antilinear isomorphism Jrp (or J if there is no confusion) from Hrp to Hrp, and the modular operator L1rp (or L1 if there is no confusion). If we consider the elements of Mas operators on Hrp, we have then JMJ = M' (and M is then in a standard position in H, in the sense of 1.1.1 (iii), thanks to the closed convex cone Prp generated by {xJrpArp(x); X E mrp}), an~ JxJ =:= x* for any x in the centre of M. Moreover, for all t in R, we have L1 at M ..1-at = M, this last formula leading to the definition of the modular automorphi8m group u'f by: ui(x) = L1itxL1-it (x E M,t E R)

e

Moreover, we have:

cp = cp o u'f and the modular automorphism group satisfies the Kubo-Martin-Schwinger condition ([14]), that is, for every x, y in ~ n m~. there exists a bounded function f on the strip {z E C, 0 ~ 1m z ~ 1}, holomorphic in its interior, · such that, for all tin R, we have:

f(t) = cp(u'f(y)x) f(t + i) = cp(xui(y)) These two properties characterize the modular automorphism group. An element x of M will be called analytic with respect to cp if the function t-+ u'f(x) has an extension to an analytic function z-+ uf(x) from C toM. We then define M'P = {x E M, u'f ( x) = x for all t in JR.}. Moreover, for any X in mrp and a in M'P' xa* belongs to ~. and we have ~( xa*) = J aJ Arp( X). For technical reasons, it is useful to know there exists a maximal subalgebra !2to of 2111 n 21', which is both a left and right Hilbert algebra, with ~ = !2t'

48

2. Kac Algebras

and !X~ = !X", and which is globaly invariant under the linear closed operators Llz for all z in C. The algebra 2lo is called the maximal modular subalgebra of !X". If 'P is a trace (i.e. ~.p(x*x) = ~.p(xx*) for all x in M), then '1tc,o = 'Jt~, Sc,o = Jc,o, Llc,o = 1, and the modular automorphism group reduces to the identity i; the algebra 'Jtcp is both a left and a right Hilbert algebra; we recover the Hilbert algebras of [24]. If trn is the canonical trace on £(H), then '1ttrH is the algebra of Hilbert-Schmidt operators on H; it is therefore complete, and we may consider HtrH as H®H, the identification AtrH being given, for all in H, by AtrH(Pe) = ® t, where Pe is the one dimensional orthogonal projection on ce. Conversely, every complete Hilbert algebra is a direct sum of algebras of that type ([24], I §8, prop. 7). For x in M'+, the formula ~.p1 (x) = ~.p(Jc,ox* Jc,o) defines a faithful semifinite normal weight on M', with 'Jlc,o~ = Jcp'Jlc,oJc,o; we identify Hc,o' with Hc,o by writing Ac,o'(Jc,oxJc,o) = Jc,oAc,o(x). If 'Pi ( i = 1, 2) is a faithful, semi-finite, normal weight on the W* -algebra Mi, then ([16]) there exists a unique faithful semi-finite normal weight 'Pl ®'P2 on M1 ® M2 such that:

e

e

('Pl ® 'P2)(xl ® x2)

= 'Pl(xl)'P2(x2) (x1 E M1, = ufl ® ur

x2 E M2)

ufl®Cf'2

If the weights are not faithful, we may define the tensor product of the reduced weights on the tensor product M1p1 ® M2p2 , which may be identified to (M1 ® M2)Pt®F.! (where Pi is the support of 'Pi), and, by composition with the reduction from M1 ®M2 to (M1 ®M2)Pt®F.!• we have a definition of the tensor product of 'Pl ® 'P2. (v) If 1.p, .,P are two faithful semi-finite normal weights on a W* -algebra M, it is possible ([16]) to define a cocycle Ut in M, such that, for all s, t in JR., and x in M, we have:

= Utuf( u uf(x) = utuf(x)ut Ut+s

8)

If 'P = 'Po uf, for all t in JR., this cocycle is a one parameter group of unitaries in MCf', which leads, thanks to Stone's theorem, to a positive self-adjoint operator h affiliated to MCf', such that Ut = hit. We have then, for all x in M+:

where he= h(1

+ eh)- 1 belongs to MCf'. Moreover: 1J(h 112 )

= {e E H;

sup(hee Ie)< +oo} E

2.1 An Overview of Weight Theory

and, for any

For any

X

49

ein "D(hl/2):

llh 112 ell 2 = sup(hee Ie)= lim(he:e Ie) e: e: in IJl~p n !Jl,p, we have: ,P(x*x)

= e-o lim lt'(h!1 2 x*xh!1 2 ) = lim IIJh!/ 2 JA~p(x)ll 2 e:-o

and so JA~p(x) belongs to "D(h 112 ), and: ,P(x*x)

= IIJhl/2JAtp(x)ll2

We shall write then ,P = lt'(h·), in the sense of ([114], th. 5.12). The operator h will be called the Radon-Nikodym derivative of ,P with respect to It'· If we have ,P ~ lj', then 0 ~ h ~ 1. If uf = uf for all tin R, then the operator his affiliated to the centre of M; as then Jh 112 J = h 112 , we get that, for any x in ~Jttpn!Jl,p, Atp(x) belongs to "D(h 1 12 ), and ,P(x*x) = llh 1 1 2 A~p{x)ll 2 . 2.1.2 Lemma. Let e; be a family of elements in M, belonging to IJl~p, analytic with respect to lj', such that, for allj, lle;ll ~ 1, and, for all zinC, Uz(e;) is weakly convergent to 1 (such a family exists, thanks to {[150], 2.16). Then, for all a in rol~p, we have:

lii;Illt'( aej) J

Proof Let x, y be in IJltp IJ'(y*xe;)

= It'(a) .

n IJl~. We have:

= (A~p(xe;) IAtp{y)) = (SA~p(ejx*) I A~p(y)) = (SejA~p(x*) I A~p(y))

= (SejSA~p(x) IAtp(y)) = (JL1 1 1 2 ejLl- 1 1 2 JA~p(x) I A~p(Y)) = (Ju~i/ 2 (ej)JA~p(x) I A~p(y))

--7 (A~p(x) I A~p(y)) = lt'(y*x) J

which, by linearity, completes the proof. 2.1.3 Lemma. Let {3 be an antiautomorphism of M (i.e. a linear mapping from M to M such that for all x, y in M, we have f3(xy) = {3(y)f3(x) and f3(x*) = f3(x)*). Let us put 9 = lt'of3; then(} is a semi-finite, faithful, normal weight on M and for all t in R, we have:

uf = tr 1 0 u"!..t 0

{3 .

50

2. Kac Algebras

'.no'

Proof. Let X and y in 'Jle n then (3( X) and f3(y) belong to IJtp n 'Jl~; therefore there exists ([14], prop. 4.4) a bounded continuous function f on the strip {z E C; 0 :::; 1m z :::; 1}, holomorphic in its interior, such that for all t in R, we have: f(t) = cp(ai(f3(y))f3(x)) f(t + i) = cp(f3(x)ai(f3(y))) It can also be written as follows:

= cp(f3(y)a~t(f3(x))) = 8((3- 1 a~tf3(x)y) f(t + i) = cp(a~t(f3(x))f3(y)) = 8(y(3- 1 a~tf3(x)) f(t)

As, on the other hand, we have, for all x in M+:

8((3- 1 a~tf3(x))

= cp(a~tf3(x)) = cp(f3(x)) = 8(x)

we see that (} satisfies the K.M.S. conditions with respect to the group of automorphisms t--+ (3- 1 a'!_tf3, then ((14] cor. 4.9) allows to conclude. 2.1.4 Lemma. Let ,P another faithful, semi-finite, normal weight on M, such that ,P is invariant under a'f for all t in R. Then A('Jtp n ~) is dense in H~.p.

Proof. Let h be the Radon-Nikodym derivative of ,P with respect to cp. Let us put h = J000 sdE8 ; en = Jt;n dEs; hn = hen. Let x be in IJtp; we have:

1/2

because en he

belongs to M

= IIJh:/ 2 JA~.p(x)ll 2

because heen is weakly convergent to hn when c goes to 0

and then xen belongs to~' and as en belongs to VJt'f', by ([114], th. 3.6), we get that xen belongs to '.Yt~.pn'.YttJ>· Moreover A~.p(xen) = JenJA~.p(x) converges

2.1 An Overview of Weight Theory

to Acp(x), because en -+ 1 when n goes to infinity. Thus, Acp(IJtcp dense in Acp(IJtcp) and therefore in Hcp.

51

n ~) is

2.1.5 Lemma. Let .,P be a semi-finite, normal weight on M, invariant by the modular automorphism group a'f. Let E be a subspace of!Jtcp, such that Acp(E) is dense in H, and that, for all x in E, we have:

cp(x*x)

= .,P(x*x)

Then, the weights

, therefore J Acp( x) belongs to 'D( h 112 ), and we have:

then, using the same arguments as ([23], lemma 23), we get that Jh 112 J = 1, therefore h = 1 and

.((i ® cp)(F(y*)(l ® x)))

(x,y E ~)

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

(t E R)

(HWiii)

2.2.2 Proposition. Let G be a locally compact group and lllla(G) = (£C:le(G), Fa, K.a) the associated abelian co-involutive Hopf-von Neumann algebra {1.2.9). Let us consider the faithful semi-finite normal weight cp 0 on L 00 (G), which is the trace arising from the Haar measure on G by:

C(Ja(i)

=!a

f(s)ds

It is a left-invariant and a Haar weight on lllla(G). Moreover, the weight cp is finite if and only if the group G is compact. Proof. We identify Hrp,. with L 2 (G), ~.. with L 2 (G) n L 00 (G), rolrp,. with L 1 (G) n L 00 (G); let fin L 00 (G)+; since fa llfll 2 (st)dt =fa llfll 2 (s)ds, it is clear that (LIW) holds. The translation of (HWii) is: lag(t)f(st)dt = lag(s- 1t)f(t)dt

which results straightforwardly from the left-invariance property of ds. And (HWiii) is trivial, cp being a trace.

56

2. Kac Algebras

2.2.3 Proposition. Let E = (M, r, /\:) be a co-involutive Hopf-von Neumann algebra, and 1.{) be a faithful semi-finite normal weight on M. If 1.{) satisfies (LIW) (resp. (HWi), (HWii), (HWiii)), then ~.p1 satisfies the same property. Proof. Let x be in M'+, and

1.{)

a left-invariant weight; then:

(i ® ~.p')r'(x) = (i ® ~.p')((J ® J)r(JxJ)(J ® J))

= J(i ®~.p)T(JxJ)J = ~.p(JxJ)l = ~.p1 (x)1 So ~.p1 is a left-invariant weight. Proving axiom (HWi), (HWii) and (HWiii) is just a straightforward calculation of the same kind, using m(G), Fa, 1\:a, l.{)a) is a Kac algebra, denoted Ka(G). For any Kac algebra K = (M, r, 1\:, ~.p ), we denote K' = (M', r', 1\:1, ~.p1 ) and K' = ( M, ~ r, 1\:, 1.p o 1\:) 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(G0 PP). Let K = ( M, r, 1\:, ~.p) be a Kac algebra, and a > 0. It is clear that ( M' r, 1\:, a~.p) is a Kac algebra, which will be denoted a:K. Let K1 = (Mt, T1, 1\:t, 1.{)1) and K2 = (M2, T2, /\:2, 1.{)2) be two Kac algebras. We shall say that K1 and K2 are isomorphic if there exists an E-isomorphism: u:

{Mt,T1,/\:1) - t (M2,r2,/\:2)

and a > 0 such that 1.{)2 o u = a~.p1. Clearly, then, u is implemented by a unitary u from H 'Pl onto H 'P2 defined for all X in m'Pl ' by:

UAp1 (x) = a- 112 Ap2 (u(x)) Therefore K and a:K are isomorphic.

2.2 Definitions

57

2.2.6 Proposition. Let lK = (M, r, 11:, It') be a K ac algebra and R a projection in the centre of M, such that: F(R) ~ R®R ~~:(R) = R We shall denote by IKR the quadruple (MR,rR,II:R,tt'R) 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)

FR(r(x)) = (r ® r)(F(x)) II:R(r(x)) = r(~~:(x))

(iv) the weight It' R on M R is obtained by reduction from It' on M as in

{[16], def. 9.2 ..4). Then IKR is a Kac algebra, called a reduced Kac algebra of IK, and r is a surjective l8l-morphism. Proof. It is trivial to check that (MR, FR, t~:R) is a co-involutive Hopf-von Neumann algebra and that r is an l8l-morphism. We have rot'PR = r(rot'P) which implies:

Let x, y be in S)l'P. We have: (i ® tt'R)((1 ® r(y)*)rR(r(x))) = (i ® tt'R)(r ® r)((1 ® y*)r(x)) = (i ® tt')(r ® r )((1 ® y*)r(x )) = (i ®It')(( R ® Ry*)F(Rx)) = R(i ® tt')((1 ® Ry*)F(Rx)) = R~~:(i ® tt')(F(Ry*)(1 ® Rx)) = ~~:(i ® tt')(F(Ry*)(R ® Rx)) = ~~:(i ® tt')((R ® R)F(y*)(1 ® x)) = II:R(i ® tt'R)(r ® r)((F(y*)(1 ® x)) = 11:R(1 ® tt'R)(F(r(y)*)(1 ® r(x)))

As

ruf = ufRr, the axiom (HWiii) is trivially proved.

2.2.7 Proposition. Let lK = ( M, r, ~~:,It') a K ac algebra and M a sub von Neumann algebra of M such that:

58

2. Kac Algebras

{i} r(M) c (ii) ~~:(M) =

M®M M (iii} u'f(M) = M (t

E JR.)

(iv) the restriction cp I£"1+, which will be denoted cj;, is a semi-finite weight. We shall denote lK the quadruple (M, it, cj;) where and it are respectively the restrictions of r and 11: to M; the canonical one-to-one morphism M --+ M will be denoted by j. Then, i is a Kac algebra, called a sub-Kac algebra of :K, and j is a oneto-one Ill-morphism.

r,

r

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

we have

uf = uilv·

2. 3 Towards the Fourier Representation In what follows, (M,r) is a Hopf-von Neumann algebra and cp a faithful, semi-finite, normal weight on M, satisfying (HWi). 2.3.1 Lemma. For all w in M., we have: {i} (w ® i)r(rot"') c rot"' {ii} (w ® i)F('Jk,o) C 'Jkp. Proof. It is clear that (HWi) implies F(rot'l') C roti®..' associated to the co-involutive Hopf von Neumann algebra l!ll' and the weight r.p1• Let w be in. M.; we note w' the element of M' defined, for all x in M', by:

(x,w')

= (Jx* J,w)

We easily get (w o K) 1 = w1 o K 1 , and w'(JxJ) Then, we have, for x in ~:

J>..'(w')JA,.,(x)

= w(x) for all x in M.

= J>..'(w')Acp~(JxJ) = J A,.,,((w' o K 1 ® i)T'(JxJ)) = JA,.,,(((w o K) 1® i)(J ® J)r'(x)(J ® J)) = A,.,((w o K ® i)T(x)) =

>..(W)Atp( X)

by 2.3.5

and then:

>..'(w') = J>..(w)J

2.4

The Fundamental Operator W

In what follows, (M, T) is a Hopf-von Neumann algebra and r.p a left-invariant weight with respect tor (so r.p satisfies (HWi)). 2.4.1 Lemma. For all x, y in

~'

we have:

(r.p ® r.p)((x* ® 1)T(y*y)(x ® 1)) = r.p(x*x)r.p(y*y). Proof. By definition (2.2.1), we have:

(i ® r.p)T(y*y) = r.p(y*y)1 whence, by ([59] 2.1 (3); cf. 2.1.8 (i)):

r.p(y*y)x*x = x*(i ®r.p)T(y*y)x = (i ® r.p)((x* ® 1)T(y*y)(x ® 1)) Then:

r.p(x*x)r.p(y*y) = (r.p ® r.p)((x* ® 1)T(y*y)(x ® 1)).

2.4 The Fundamental Operator W

61

2.4.2 Proposition. (i) There ezi8ts a unique i8ometrg W such that, for every

x, y in '.ntp, we have:

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

= W(1 ® x)

(iii) W belongs toM® .C(HIP) Then, W is called the fundamental operator associated to (M, r, cp). Proof. By polarization and linearity it follows from 2.4.1 that, for all

Xt,

x2,

Yl , Y2 in '.ntp we have:

which can be also written as: (~®tp(F(yt)(xt ® 1))

I Atp®tp(F(y2)(x2 ® 1)))

= (~(xt) ® Atp(Yt) I AIP(x2) ® Atp(Y2)) As Atp('.ntp) ® Atp('.ntp) is dense in HIP® HIP, there exists an isometry W in .C(HIP ®HIP) such that, for all x, yin '.ntp, we have:

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

F(x)W(Atp(Y) ® Atp(z))

= F(x)Atp®tp(F(z)(y ® 1))

by (i)

= AIP®IP(r(zz)(y ® 1))

= W(Atp(Y) ® ~(xz)) = W(1 ® x)(Atp(y) ® ~(z))

by (i)

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

(JaJ ® JbJ)W(Atp(x) ® Atp(y))

= F(y)(x ® 1)(J~(a) ® JAIP(b)) = F(y)(JaJAIP(x) ® J~(b)) = F(y)(JaJ ® 1)(Atp(x) ® JAIP(b))

This equality still holds, by continuity, when AlP( x) converges to any vector e in HIP and when a strongly converges to 1. Therefore, we have:

(1 ® JbJ)w(e ® ~(y)) = r(y)(e ® J~(b))

62

2. Kac AlgebrBB

which is valid for ally in IJtcp and e in Hcp. Let x' be in M' and let us replace by x' in the above formula:

e

e

(1 ® JbJ)W(x' ® l)(e ® Acp(y))

= r(y)(x' ® l)(e ® JAcp(b)) = (x' ® 1)r(y)(e ® JAcp(b) = (x' ® 1)(1 ® JbJ)W(e ® Acp(Y))

which still holds when b strongly converges to 1, then we get:

W(x' ® 1)(e ® Acp(y)) = (x' ® 1)W(e ® Acp(Y)) By linearity, density and continuity we can conclude that W and x' ® 1 commute, therefore W belongs toM® C(Hcp)· 2.4.3 Lemma. We have, for all x, y

w((i ® cp)(r(y*)(1 ® x)))

in~

and w in M.:

= (Acp(x) I(w ® i)(W)Acp(y)).

Proof. Let a be in 'Jtcp; we have: wA.(w o t>,)Ac,o(x)

by 1.2.5

0

0

Since Ac,o('Jlc,o n 'Jl~) is a core for Sc,o, we get:

which, by (i), implies:

J>.(w)* J

= >.(w o "')

which completes the proof. 2.5.6 Corollary. For all t in R, we have:

ruf

= (i ® uf)r .

Proof. Let x be in 'Jlcp, w in M*;we have:

Ac,o((w o"' ® i)(Fuf(x)))

= .A(w)Ac,o(uf(x))

by 2.3.5

it

= >.(w)LlcpAc,o(w) •t = Ll~>.(w)Ac,o(w)

= Ll~Ac,o((w

by 2.5.5(i)

o"' ® i)(F(x)))

by 2.3.5

= Ac,o(uf((w o "'® i)(F(x))) = Ac,o((w o"' ® i)((i ® uf)(F(x))) which implies:

(w o "'® i)(Fuf(x))

= (w o "'® i)((i ® uf)r(x)))

therefore, for all x in 'Jlcp, we get:

ruf(x)

= (i ® uf)r(x)

which completes the proof, by density and continuity. 2.5.7 Corollary. Let x in M, analytic with respect to 'P· Then, for all w in M*, the element (w ® i)F(x) is analytic with respect to((', and, for all z in C, we have:

ur((w ® i)F(x))

= (w ® i)r(ur(x))

.

70

2. Kac Algebras

Proof The function z - t (w ® i)F(uf(x)) defined on Cis analytic, and, by 2.5.5, it extends to C the function defined on R by t - t ui((w ® i)F(x )).

r, ~t) be a co-involutive Hopf-von Neumann algebra, r.p a faithful, semi-finite, normal weight on M, satisfying {HWi) and {HWii). Then r.p is left-invariant.

2.5.8 Theorem. Let llll = ( M,

Proof Let w be in Mf, y in ~ and x; a family of elements satisfying the hypothesis of 2.1.2. We can write down:

r.p(((w ® i)F(y))(x;)) = r.p((w ® i)(F(y)(1 ® x;)) = (w ® r.p)(F(y)(1 ® x;)) = (w o ~t ® r.p)((1 ® y)F(x;)) = r.p(w o ~t ® i)((1 ® y)F(x;)) = r.p(y(w o ~t ® i)(r(x;)))

by (HWii)

It follows from 2.3.1 (i) that (w ® i)F(y) belongs to ~, and from both 2.3.1 (ii) and 2.5.7 that (woK.® i)F(x;) satisfies the hypothesis of 2.1.2. By passing to the limit, we get, for all win Mf andy in~:

r.p((w ® i)F(y))

= r.p(y)w(1)

with w being fixed, these two expressions will define two faithful semi-finite, normal weights on M which coincide on ~. The modular group of the second one is and we have:

ur

(w ® r.p)F(ut(y))

= (w ® r.p)(i ® uf)F(y) = (w ® r.p)F(y)

by 2.5.6

Therefore, the first one is invariant by that group. It follows from ([114], prop. 5.9), that they are equal. Then we have, for ally in M+ and win Mf:

w(( i ® r.p )F(y))

= w( r.p(y)1)

By linearity it is true for all win M., which completes the proof. 2.5.9 Example. The Fourier representation >..' of llll' satisfies:

(w 1 ) = ..\(w),

..\1

Vw EM*

where w'(x) = w(Jx* J), for all x in M'.

Proof. In 2.3.7, we have got ..\1 (w 1 ) = J..\(w)J; using 2.5.5(ii), 2.5.3 and 1.2.5, we get the result.

2.6 The Fundamental Operator W Is Unitary

71

2. 6 The Fundamental Operator W Is Unitary We keep on with the same hypothesis as in 2.5. The weight t.p being leftinvariant, thanks to 2.5.6, it is possible to use all the result of 2.4, in particular the construction of the fundamental operator W (2.4.2), the links between W, .X, and the ideal Icp (2.4.6). Let us recall that M is the von Neumann algebra generated by the Fourier representation .X (2.5.3). 2.6.1 Proposition. The fundamental operator W, the Fourier representation .X and the ideal Icp are linked by: {i) For all w in M., we have:

.X(w) (ii) For all

n in M.,

= (w o 11: ® i)(W) = (w ® i)(W*) we have:

.x.(n) = (i ® n)(W*)

= ~~:(i ® n)(W)

(iii) For a, {3, "'(, 6 in Hcp, we have the connection formula:

(W(a ® {3) 11 ® 6)

= (f31.X(w-y,a)6)

{iv) The fundamental operator W belongs to M ® (v) For all w in M. and w1 in Icp, we have:

.X(w)a(w')

M

= a(w * w') .

Proof. We have already proved (i) in 2.4.6(i) and (v); then (ii) and (iii) are straightforward corollaries of (i), (iv) is a consequence of (i) and 2.4.2(iii), and (v) is a rewriting of 2.4.6 (iv). 2.6.2 Corollary. The fundamental operator W is unitary. More precisely, for any antilinear isometric involution .7 of Hcp implementing 11: (i.e. such that ~~:(x) = .7x* .7 for all x in M), we have:

W* = (.7 ® J)W(.7 ® J). Proof. Let us recall that, for all a, {3 in Hcp, W:Ja,:T/3 = w~,/3 (1.2.8). So:

(w ® i)((.7 ® J)W(.7 ® J)) = J(w0 ® i)(W)J = J.X(w 0 o ~~:)J = J.X(w o ~~:)* J = .X(w) = (w ® i)(W*) and the result is proved, by linearity and density.

by 2.6.1 (i) by 2.5.3 by 2.5.5 (ii) by 2.6.1 (i)

72

2. Kac Algebras

2.6.3 Corollary. {i) The Fourier repre8entation A uW*u i8 it8 generator in the 8en8e of 1.5.2. {ii) For all x in M, we have:

&8

non-degenerate, and

F(x) = W(1 ® x)W* .

Proof. The assertion (i) is clear by 2.6.1 (i), 2.6.2 and 1.5.3 and (ii) is clear by 2.4.2 (ii), thanks to W being unitary. 2.6.4 Corollary. Let A be a von Neumann algebra. For all X in A® M, we have: (i ® F)(X) = (1 ® W)(1 ® u)(X ® 1)(1 ® u)(1 ® W*) .

Proof. Let a be in A, x in M. We have: ( i ® F)( a ® X)

= a ® T( X) = a® W(1 ® x )W* by 2.6.3 (ii) = (1 ® W)(a ® 1 ® x)(1 ® W*) = (1 ® W)(1 ® u)(a ® x ® 1)(1 ® u)(1 ® W*)

which completes the proof, by linearity, continuity and density. 2.6.5 Proposition ([79]). Let A be a von Neumann algebra, V a unitary of A ® M 8uch that: (i ® F)(V) = (V ® 1)(1 ® u)(V ® 1)(1 ® u)

Then, for every w in M, we have:

(i ® w o ~~:)(V) = (i ®w)(V*) The mapping r : M* -+ A defined by r(w) = (i ® w )(V) i8 a non-degenerate repre8entation of M, and V i8 the generator of r. Proof. By 2.6.4, we have:

(1 ® W)(1 ® u)(V ® 1)(1 ® u)(1 ® W*) = (V ® 1)(1 ® u)(V ® 1)(1 ® u) which can also be written: (V ® 1)(1 ®u)(1 ® W*)(1 ®u)(V* ® 1) = (1 ®u)(1 ® W*)(V ® 1)(1 ®u) (*)

2.6 The Fundamental Operator W Is Unitary

73

Let us consider the representation of M* in A® .C(H) defined by:

= V(1 ® .X(w))V*

J-L(w)

Therefore, for all w in M't 1 , we have, using 2.6.1 (i):

J-L(w)

= (i ® i ® w)((V ® 1)(1 ® uW*u)(V* ® 1)) =

For all

(i ® i ® w)((1 ® u)(1 ® W*)(V ® 1)(1 ® u))

by ( *)

fh in At 1, il2 in .C(H)t1 , and w in M., we have:

(J-L(w), ilt ® .G2) = (.Gt ® il2 ® w)((1 ® u)(V* ® 1)(1 ® W)(1 ® u)) = (.Gt ®w ® .G2)((V* ® 1)(1 ® W)) = (.Gt ®w)((i ® i ® .G2)((V* ® 1)(1 ® W))) = (ilt ®w)(V*(1 ® (i ® .G2)(W))) = w((.Gt ® i)(V*(1 ® (i ® il2)(W))) = w((ilt ® i)(V*)(i ® .G2)(W)) Therefore, by 1.5.1 (ii), we have: ~t((.Gt ® i)(V*)(i ® il2)(W)) = ((.Gt ® i)(V*)(i ® .G2)(W))*

which can be written as follows:

since

nl and .a2 are positive.

H we apply 1.5.1 (ii) again, we get:

or:

(i ® il2)((~t(il1 ® i)(V) ® 1)W)

= (i ® il2)(((il1 ® i)(V*) ® 1)W)

which, by linearity, will still holds for all (~t(.Gt ®

n2 in .C(H)., and therefore implies:

i)(V) ® 1)W = ((ilt ® i)(V*) ® 1)W

As W is unitary, we have: ~t(.Gt ®

i)(V) = (.Gt ® i)(V*)

74

2. Kac Algebras

and by 1.5.1 (i) and (ii), r is a representation and we have, for all win M.:

(i ®w o ~~:)(V) = (i ®w)(V*) As V is unitary, r is non-degenerate (1.5.3).

2.6.6 Corollaries. (i) Let u be in the intrinsic group of (M, r, ~~:). Then, we have: ~~:(u) = u* and the application w -+ w( u) is a one-dimensional representation of M., which has u as generator. (ii) If u, v are two elements of the intrinsic group of (M, r, ~~:) the Kronecker product of the two one-dimensional representations w -+ w( u) and w -+ w( v) is w -+ w( uv). So, the Kronecker product, restricted to the intrinsic group, is the usual product. (iii) If J1. is a representation of M., we have, for all w in M., with the notations of 1.1.1 (ii): (JJ. x u)(w) = JJ.(u · w) (u X JJ.)(w) = JJ.(w · u).

Proof. By 1.2.3, u is a unitary and satisfies F( u) = u ® u. We can then apply 2.6.5, with A = C, and (i) is proved; then (ii) is just a corollary of 1.5.5; (iii) is an application of 1.4.3, with the representations J1. and u.

2.6. 7 Proposition. Let R be in the centre of M, such that:

F(R)

~

~~:(R)

Then, we have: (i) W(R®R) = (R®R)W (ii) F(R)(R ® 1) = F(R)(1 ® R)

R®R = R

= R ® R.

Proof. By hypothesis, we have, using 2.6.3 (ii):

R®R

= (R ® R)F(R) = (R ® R)W(1 ® R)W*

Using 2.6.2, we get:

R®R

= (R ® R)(.:T ® J)W*(.:T ® J)(1 ® R)(.:T ® J)W(.:T ® J)

2.6 The Fundamental Operator W Is Unitary

75

and, using the fact that J RJ = R (because R belongs to the centre of M) and that :JR:J = R (because ~~:( R) = R), we have: R®R= (R®R)W*(1 ®R)W Taking adjoints, we get: R®R= W*(1 ®R)W(R®R) and, as W is unitary: W(R®R) = (1 ®R)W(R®R) = (R®R)W(1 ®R) = (R®R)W

by 2.6.1 (iv) by(*)

So, (i) is proved. We have then: F(R)(R ® 1)

= W(1 ® R)W*(R® 1) = W(R®R)W* =R®R

by 2.6.3 (ii) by 2.6.1 (iv) by (i)

Applying this result to (M, r;F, ~~:,

, (9.6. 7). We prove that this weight is a Haar weight (9. 7.4 ), and we have so defined a dual Kac algebra :i = (M, f, K, rji), the fundamental operator W of which is 0'W* u, and the Fourier representation 5. of which is given by (where ,\* : M. -+ M is obtained by predualizing -\). As ,\ is non-degenerate, 5. is faithful. On ..C( G), the weight so constructed is equal to the Plancherel weight cp8 studied by Haagerup in [58]. So, 'K8 (G) = (..C(G),rs, "-s,cps) is another example of a Kac algebra (9.7.5), 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 'f= 0 and r(x) = x ® x), are unitaries, verify "-(X) = x*, and, with the weak topology of M, form a locally compact group, called the intrinsic group of the Kac algebra (9.6.10). See also (1.2.2), (1.2.9) and (2.6.6).

,,x.

3.1 The Generator of a Representation 3.1.1 Lemma. With the definition of 1.9.6, we have:

Proof. Let X be in p R(JK) n 1)1~. It follows from 1.3.6 that there exists a Hilbert space 1-£, a non-degenerate representation p. of M. on 1i and a vector in 1-£, such that, for all w in M.:

e

(x,w} = (p.(w)e I e)

3.1 The Generator of a Representation

85

As the space I,.,ni~ is norm dense in M* by 2.4.6{iii), the algebra Jl(I,.,n~) is dense in Jl( M* ). Thanks to Kaplansky's theorem, it exists a sequence {wn}neN of elements of Irp such that IIJ.l{wn)ll :::; 1 and that Jl(wn) strongly converges to 1. Let us consider the linear forms on M* defined by:

4in(w) = (x,w~*W*Wn)

(wE M*)

We have, by hypothesis on x:

4in(w 0 * w) = (x, (w * wnt * (w * Wn)) 2:: 0 Therefore 4in is positive definite. Moreover, since x belongs to by 2.4.5, w~ * w * Wn belongs to I,.,, we get:

4in(w)

Let us put J.?n and therefore:

=

= (a(w~ * w * Wn) IA,.,(x*)) = (.A(w~ * w)a(wn) I A,.,(x*)) = (.A(w)a(wn) I.A(wn)A,.,(x*)) = (.A(w), {.?a(wn),A(wn)A.x1-1 ® 1

and so we deduce that the projector P>.x1-1 may be written 1 ® Q, where Q is in Aw Moreover, we get: 1 ® VV* = (u ® 1)(1 ® uWu)(u ® 1)UU*(u ® 1)(1 ® uW*u)(u ® 1) = (u ® 1)(1 ® uWu)(u ® 1)(P>.xp ® 1)(u ® 1)(1 ® uW*u)(u ® 1) = (u ® 1)(1 ® uWu)(u ® 1)(1 ® Q ® 1)(u ® 1)(1 ® uW*u)(u ® 1) =1®Q®1 And so we get that V*V = VV* = Q ® 1 and the theorem is proved. 3.1.5 Corollary. (i) Let JJl and JJ2 two non•degenerate representations of M.; then JJl X JJ2 is non-degenerate. (ii) The triple (W*(K), c;s11'X11'• s;r ), with the definitions of 1.6.5 and 1.6.6, is a co-involutive H opf-von Neumann algebra; it is symmetric if K. is abelian, and abelian if ][{ is symmetric. Proof. The assertion (i) is a direct corollary of 1.5.5 and 3.1.4, as degenerate and (ii) is a trivial application of 1.6.7 and 3.1.4.

11"

is non-

3.2 The Essential Property of the Representation A

89

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

,(Ax J.L)(w)

= V(10 A(w))V*

.

Proof. It results from 1.5.5 that the generator U of A X J.L is equal to:

(, 0 i)(i 0 aW*a)(10 V) =(a 01)(10 a)(10 W*)(10 a)(a 01)(10 V) =(a 01)(10 a)(1 0 W*)(V 01)(10 a)(a 01) Therefore, we have, for all win M*: '(A

X

J.L)(w)V = (, 0 i)(i 0 i 0w)((a 01)(10 a)(10 W*)(V 01)(10 a)( a 01))V = (i 0 i 0 w)((10 a)(1 0 W*)(V 01)(10 a))V = (i 0 i 0 w)((10 a)(1 0 W*)(V 01)(10 a)(V 01)) = (i 0 i 0w)((10 a)(10 W*)(i 0 F)(V)(10 a)) by 1.5.1(i) = (i 0 i 0 w)((10 a)(10 W*)(1 0 W)(10 a)(V 01) (1 0 a)(10 W*)(10 a)) by 2.6.4 = (i 0 i 0 w)((V 01)(1 0 aW*a)) = V(10 (i 0 w)(aW*a)) = V(10 A(w)) by 2.6.1 (i)

which completes the proof. 3.2.2 Theorem and Definitions. Let J.L be a non-degenerate representation of M* with a generator V. Then: (i) The mapping which sends any element x of M to V(10x)V* is a oneto-one normal morphism from M to AJS 0 M which shall be denoted by ,:YJS; (ii) The representations A and A x J.L are equivalent; moreover, we have, for all w in M*: (iii) We have: "fJS 8 A

= 'SAXJS

(iv) The mapping ,:YA is a coproduct on M; it shall be denoted by i' and by transposition it induces a product* on M*. For all x in M, we have:

F(x)

= aW*a(1 ® x)aWa

90

3. Representations of a Kac Algebra; Dual Kac Algebra

(v) For all u in the intrin,ic group of K., and w in M., we have:

.Yu(.\(w)) = .\(u ·w). Proof. As Vis unitary, it is enough to check in which space belongs V(1®x )V* for any x in M. It results from 3.2.1 that for all w in M., we have: .Y,(.X(w))

= ~(.\ x JL(w)) E A,® 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 JL!. JL2 be two non-degenerate repre.5entatiom of M., with, re8pectively, generator, V1, V2; then: (i) For all x in M, and t in Hom(JLl, JL2), we have:

{ii) For any morphi8m ~ : A, 1

-+

A,2 , .5uch that ~ o JLl = JL2, we have:

Proof Let w be in M*. We have: 1)~(.\

by 3.2.2 (ii) by 1.4.5(i) by 3.2.2 (ii)

= (~®i)(VI(1 ®x)Vt) = V2(1 ® x)V2*

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

(t ® 1)1, 1 (.\(w)) = (t ®

X JLl)(w) = ~(.\ x JL2)(w)(t ® 1) = '1,2 (.\(w))(t ® 1)

by continuity and density, we get (i). Let x be in M; we have: (~®i)7, 1 (x)

=.Y,2(x) which completes the proof. 3.2.4 Lemma. Let JL a non-degenerate repre8entation of M., in (A,) •. We have:

w in M.

and

n

3.2 The Essential Property of the Representation

~

91

Proof. Let w be in M •. We have: {-X.((fl ® w) o -)'p),w}

= {1-p(.\(w)), n ® w} = {.\ x JL(w),w ® fl} = {-X.(w)J.L•(fl),w}

by 3.2.2 (ii) by 1.4.3

which completes the proof.

3.2.5 Lemma. Let Ill and 1-'2 be two non-degenerate representations of M •. We have:

Proof. Let w be in M., fl1 in (Ap 1 )., fl2 in (Ap 2 )* and win

M•. We have:

{-)'pl Xp2(.\(w)), fl1 @ fl2@ w}

={.\X Ill X JL2(w),w ® fl1 ® fl2}

by 3.2.2 (ii)

= {.X.(w )(J.Ll)•( fll)(J.L2)•(fl2), w}

by 1.4.3

= {.X.((fll ® w) o -)'p1 )(J.L2).(fl2),w}

by 3.2.4

by 1.4.3 ={(.\X J.L2)(w), (fll ® w) O-)'p1 ® fl2} = (,(.\X J.L2)(w), (fl2 ® fl1 ® w) O (i ® -)'p1 )} = {-)'p 2 (-X(w)), (fl2 ® n1 ® w) o (i ® -)'p 1 )} by 3.2.2 (ii) = ((, ® i)(i ® -)'p1 )-)'p 2 (.\(w)), n1 ® n2 ®w}

which completes the proof by linearity, density and continuity.

3.2.6 Proposition. Let 1-' be a non-degenerate representation of M •. We have, with the notations of 3.2.2:

(i'p ® i)i' = (i ® r).y~'

.

Proof. We have:

(.y~' ® i)r =

( .y~' ® i)i'~

by 3.2.2 (iv)

= i'.y,.o~

by 3.2.3 (ii)

= .Y,~xp = (' ® i)i'~xp

by 3.2.2 (ii)

= (i ® 1'~)-)'p

= (i ®T)i'p

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, >.a the left regular representation of G, fJa a unitary representation of G. The representation fJa ®>.a is then quasi- equivalent to >.a. More precisely we get, for any sinG: V(1 ® >.a(s))V* = fJa(s) ® >.a(s) where Vis the unitary in C('Hp)®L 00 (G) defined by the continuous bounded functions-+ fJa(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 fJ associated to fJa (1.5.10), that M is then the von Neumann algebra£( G) generated by the left regular representation>. of L 1 (G) (2.5.4). So the morphism ,:Yp satisfies: ,:Yp(x)

= V(1 ® x)V*

,:Yp(>.(J)) =~().X fJ){f)

(x E C(G)) (! E L 1(G))

From this last relation, we deduce, using 1.4.7, for the unitary representations of G, associated to :Yo>. and~>. X fJ respectively: ,:Yp(>.a(s))

= fJa(s) ® >.a(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 >.a. There exists a unique normal injective morphism F8 from C( G) to C( G) ® C( G) such that, for all s in G, we have: Fa(>.a(s)) = >.a(s) ® >.a(s) and F8 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 C(Hr.p) onto itself defined by x -+ Jx* J for all x in C(Hr.p) is an involutive anti-automorphism of C(Hr.p)· The restriction of this mapping to M is an involutive antiautomorphism of M in the sense of 1.2.5. It shall be denoted it. Moreover, the involutions "' and it are linked by the following relations: it(>.(w)) =>.(woK.) ,(>..(w)) = >..(w o it)

(wE M.)

(wE M.).

3.3 The Dual Co-Involutive Hopf-Von Neumann Algebra

Proof. For all x in M, let us put i\:(x)

= Jx* J.

93

Let w be in M •. We have:

11:(-\(w)) = J,\(w)* J = -\(w o ~~:)

by 2.5.5{ii)

which altogether provides the first equality and ensures, by continuity, that for all x in M, i\:(x) belongs toM; the involutive character of i\: is trivial. For all win M., we have: (~~:(-\.(w)),w}

= (-\.(w),w o ~~:}

= (-\(w o ~~:),w} = (il:(-\(w)),w}

by the first equality

= (-\(w),w oil:}

= (-\.(w o il:),w} which completes the proof. 3.3.2 Theorem. The triple ( M, i', i\:) is a co-involutive H opf-von Neumann algebra. It will be called the dual co-involutive Hopf-von Neumann algebra of

:K. Proof. Let w be in M*, WI. w2 in M*. We have:

(Fil:-\{w),w1 ® w2} = (F-\(w o ~~:),w1 ® w2} = ((,\ X -\)(w o ~~:),w2 ® w1} = (-\.(w2)-\.(wi),w o ~~:)

by 3.3.1 by 3.2.2 (iv) by 1.4.3

= (~~:-\.(wl)~~:-\*(w2),w} = (-\.(wl o il:)-\.(w2 o il:),w}

by 3.3.1 = ((-\ x -\)(w), (w1 ® w2) o {il: ® il:)} by 1.4.3 = (~(il: ® il:)r-\(w), (w1 ® w2)} by 3.2.2 (iv) Therefore we have:

Fi\:,\(w) = ~{il: ® ii:)F-\(w) and we can complete the proof by continuity. 3.3.3 Proposition. The mappings.\ is an E-morphism from (W*(.K), ~s1rx1r, s;r) to ( M' i\:). If .K is abelian, ( M' i'' i\:) is symmetric, and if K. is symmetric, (M, i', il:) is abelian. ·

r'

Proof. Let w be in M*. We have: i's.\1r(w)

= :Y.\s.\1r(w) = ~s.\x.\7r(w) ~(,\ x ,\)(w)

= = ~(s.\ ® s.\)(1r x 1r)(w)

= (s.\ ® s.\)~s1rx1r1r(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: And: KS~1r(w)

= KA(w) = .\(w o ~~:) = s~1r(w o ~~:) = s~1i"(w)

by 1.6.1 (ii) by 3.3.1 by 1.6.1 (ii) by definition of 1i" (1.6.6)

= S~S,r1r(w) and we get: KS~

= s~s,r

At last, since.\ is non-degenerate, we have s~(1) = 1, by 1.6.1 (iii); ass~ is one-to-one, it completes the proof, together with 3.1.5 (ii). 3.3.4 Definition and Notations. Let us denote Cl (IK) the C* -algebra generated by the Fourier representation .\ of lK. By 1.6.1 (i), we may identify its dual (Cl(IK))* with a closed subspace B~(IK) of B(IK). Mo~e precisely, to e~ element (}of (Cl(JK))*, one associates the element (} o .\of B(K) (where.\ denotes the restriction of s~ to C*(IK); cf. 1.6.1 (i) and (ii)). By 1.6.2 (ii), the mapping (s~)· is an isometry from M. into B(IK), the image of which is contained in B~(IK). By transposing 3.3.3, (s~). is then an isometric Banach algebra morphism, the image of which will be denoted by A(IK) and called the Fourier algebra associated to lK. Every element of A(IK) vanishes over Kers~; conversely, let(} be in B(K), such that (x, 0) = 0 for all x in Ker s~. We can define a linear mapping won M by writing: (s~(z),w)

= {z,O)

(z E W*(JK))

In fact, w appears as the composition of the restriction of(} to the reduced algebra W* (K)supp 8 ~, with the canonical isomorphism between W* (K)supp 8 ~ and M. Therefore w is ultraweakly continuous and belongs to M., and we have(}= (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 11:.\* is a non-degenerate faithful representation of M. in M. Iu generator is W and we have:

{i) {ii)

(i ® i')(W) = (W ® 1)(1 ® u)(W ® 1)(1 ® u) (i ®w o K)(W) = (i ®w)(W*) (wE M.).

Proof. By 1.6.1 (ii) we have ~~:.\* = 11:1r*(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

win

M*,

95

we have:

therefore: KA*(w)

= (i 0

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, .C(G) be the von Neumann algebra generated by the left regular representation Aa. There exists a unique normal morphism Fa from .C(G) to .C(G) 0 .C(G), and a unique normal antiautomorphism Ka in .C( G) such that, for all s in G: Fa(Aa(s)) = Aa(s) 0 Aa(s) Ka(Aa(s))

= Aa(s- 1 )

Then, (.C( G), Fa, Ka) i.<J a .<Jymmetric co-involutive Hopf-von Neumann algebra; we shall denote it JH[a( G). It is the dual co-involutive Hopf-von Neumann algebra associated to the Kac algebra Ka(G). Moreover, the morphism s_x from the envelopping W* -algebra W* (G) to .C( G) .<Juch that we have, for all sinG: s_x('7ra(s)) = Aa(s) i.<J an JH[-morphi.<Jm from the Erne.<Jt algebra of G (cf. 1.6.8} to JH[a(G). Proof. The existence of K8 is the only non-trivial result; applying 3.3.1, we see there exists a co-involution P;, on (.C(G),F8 ), defined in 3.2.8, such that, for all fin L 1 (G):

K(A(j)) = A(j o Ka) From 1.1.3, we have, for any sinG, fin L 1 (G):

and so, we have:

We then get, for any sinG:

96

3. Representations of a Kac Algebra; Dual Kac Algebra

3.3.7 Theorem. Let][{= (M, r, K,cp) be a Kac algebra, :K' the commutant Kac

algebra. Then the dual co-involutive Hop/ von Neumann algebra (M'~, r'~, ,'~) is equal to (M, c;F, K.). Proof. Usin~ 2.5.9, we see that the Fourier representation >..' generates M; so M'~ = M. By 2.4.8, the fundamental operator W' associated to K:' is W' = ( J ® J) W( J ® J). The coproduct r'~ is, then, using 3.2.2 (iv ), such that: r'~(x) =

(J ® J)uW*u(J ® J)(1 ® x)(J ® J)uWu(J ® J) = (K. ® K.)FK(x) by 3.3.1 and 3.2.2 (iv) = c;F(x) by 1.2.5 applied to (M,r,K.)

As, by the identification of HIP with HIP'' the associated antilinear isomorphism JIP and JIP, are equal, we see, by 3.3.1, that ,_,~ = K, and the theorem is proved.

= (M' r, K, cp) be a K ac algebra, and i. = (M' i'' 'k, cp) be a Kac subalgebra in the sense of 2.2.7. Let us denote j the canonical imbedding from M into M, which is an 18!-morphism. There is then a canonical surjective 18!-morphism r from (M,F,K.) to (M~,t~,'K~) such that r(>..~w)) = X(w o j), for all w in M., where X is the Fourier representation of:K.

3.3.8 Proposition. Let lK

Proof. Let us call I the isometry from Hrp to HIP defined, for all element x in 1Jtrp = ~ n M by: (i) IArp(x) = Atp(j(x)) As j is an 18!-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 w-y,a o j = WJ•-y,I•a· Let x in !Jtrp, w in M •. We have:

IX(w o j)Arp(x)

= IArp((w o j o 'k ® i)i'(x))

by 2.3.5

= IArp((w o K o j ® i)i'(x)) = AIP((w o K o i))F(j(x)))

by (i)

= >..(w)AIP(j(x)) = >..(w)IAq,(x) therefore:

IX(w oj) Let us put, for X in M, r(x) because I is an isometry.

= >..(w)I

(ii)

= rxi. We have r(l) = 1 and r(>..(w)) = X(woj),

3.4 Eymard Algebra.

Let now

97

wbe in (M~) •. We have: {j(X.(w)),w) = {X.(w)),woj)

= {X(w o i), w) = {r(A(w)),w) = {A.r.(w),w) then:

A.r.(w)

= jX.(w)

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

3.4 Eymard Algebra 3.4.1 Lemma. The .set B_x(K) i.s a .self-adjoint part of B(K). Proof. For all x in M and w in M*, we have:

by 1.2.5 and, by 3.3.1:

IIA(w 0 o ~~:)II= IIK(A(w))*ll = I!A(w)ll By using 1.6.2 (i) we then see that thanks to 1.3.4, we get the result.

'~~"•(B.x(K))

is stable by involution, and,

3.4.2 Proposition. (i) Any norm-one po.sitive element of B.x(K) i.s the limit, for the u(B(K), C*(K)) topology, of norm-one po.sitive element8 of A(:K). {ii} The .space B_x(K) i8 the u(B(K), C*(K))-do.sure of A(K). {iii) The .space B_x(K) {re.sp. 1r*(B_x(:K))) i.s compo.sed of the element8 of the form (sp).(il) {re.sp. 1-'•(il)}, where 1-' i.s a repre.sentation of M. weakly contained in A, in the .sen.se of [47), and n an element of (Ap)•· Proof. Any element of B_x(K) vanishes on Ker X (cf. 1.6.1 (ii)). By ([25], 3.4.2(i)) any norm-one positive element of B_x(K) is thus the limit, for the u( B(K), C* (K)) topology, of elements of the form ( s .x)• ( il), where n is normone positive in M.; at last 3.3.4 allows to deduce (i). It results from (i), by linearity that B_x(K) is contained in the closure, for the cr(B(:K), C*(K)) topology, of A(K); on the other hand, since B_x(K) is the annihilator of Ker A, it is u(B(K), C*(K)) closed, which completes the proof of (ii).

98

3. Representations of a Kac Algebra; Dual Kac Algebra

Let I" be a representation of M* weakly contained in>. and n in (Ap);t". By ([25], 3.4.4), (sph(il) is the u(B(lK),C*(JK)) limit of elements of the form (sA)*(w) where w belongs to M;t", i.e. of positive elements of A(JK). By using (ii), we can conclude that (sp)*(il) belongs to BA(JK); by linearity this conclusion still holds for any n. To prove the converse, let us note that those elements of the form (sp)*(il) with I" weakly contained in >. and n is in (Ap)* compose a vector space. Indeed if I" (resp. IL') is weakly contained in >. and n (resp. il') belongs to (Aph (resp. (Ap' )*), it is easy to check that:

(sp)*(il)

+ (sp' )*(il') =

(spEilp' )*(il

EB il')

and that I" EB I"' is weakly contained in .>.. By linearity, it is therefore enough to consider x_ in BA(JK)+. Let y be the positive linear form on C~(JK) such that x =yo>. (cf. 3.3.4). The Gelfand-Naimark-Segal construction allows to associate toy a triple ('H, I")· Then I" o >.is a representation of M* and we have: ((I" o >.)*(n~),w) = (l"(>.(w))e I0

e,

= (y,>.(w)) =(yo .\,1r(w)) = (x, 1r(w)) = {1r*(x),w)

Therefore, we have: and:

x

= (spoA)*(il~)

since it is clear that I" o >. is weakly contained in

.>., it completes the proof.

3.4.3 Lemma. The product of B(JK) is u(B(JK), C*(JK)) separately continuous on the bounded part8. Proof. Let 8i be a bounded family of elements of B(JK) converging to an element 8 in B(JK). For all 81 in B(JK) and w in M*, we have:

{1r(w), 8i * 8')

= {1r*(8i * 8'),w) = (7r*(81 )7r*(8i),w) = {1r*(8i),w · 1r*(81 ))

= {1r(w · 7r*(81)),8i) which converges to:

thanks to the same computation.

by 1.6.9

3.4 Eymard Algebra

99

Since the ll9i * 9'11 are bounded by 119'11 sup ll9ill which is finite, by the density of 1r(M.) in C*(JK), we get that 9i * 91 converges to 9 * 91 for O"(B(JK), C*(JK)). The left multiplication is dealt with in the same way. 3.4.4 Theorem. The sets A(JK) and B,x(lK) and are norm-closed self-adjoint ideals of B(JK). Specifically, B,x(lK) is an involutive Banach algebra which we shall call the Eymard algebra associated to K. Proof. Let

wbe in M., 9 in B(JK) and win M •. We have: by 1.6.9 and 1.6.1 (ii) (1r.(6 * (s.x).(w)),w) = (-X.(w)7r.(9),w) by 1.4.3 = ((.\ x 1r).(w ® 9),w) = ((.\ x 1r)(w),w ® 9) by 3.2.2 (ii) = (,Y,...\(w),9®w) = (.\(w),(9®w)o.y,..) = (-X.((9®w)o,Y,..),w)

From what we obtain that:

By 3.3.4, it follows that A(JK) is a left ideal of B(JK). As A(JK) is self-adjoint and norm-closed, the first part of the theorem is secured. Now, let 9 be in B,x(JK)+l. By 3.4.2 (i), 9 can be O"(B(JK), C*(JK)) approximated by norm-one positive elements of A(JK). Applying the first part of this proof, as well as 3.4.3 and 3.4.2 (ii), we find that, for all 91 in B(JK), 91 * 9 belongs to B,x(lK). By linearity, we can conclude that B,x(lK) is a B(JK)-left ideal; since, by 3.4.1 and 1.6.1 (i), B,x(JK) is norm-closed and self-adjoint, the proof is completed. 3.4.5 Proposition. The restriction to B,x(JK) of the Fourier-Stieltjes representation of B(JK) is the transposed of the mapping iU from M. to Cl (JK) (once B,x(lK) is identified with the dual ofCl(JK)). Proof. Let 9 be in (Cl(JK))* and w in M •. Then by 3.3.4, 9 o Xbelongs to B,x(K) and we have: (~~:7r.(9o.X),w)

= (7r(wo~~:),9o.X) = (.\(w o ~~:),9) = (iU(w))),9)

which completes the proof.

by 3.3.1

100

3. Representations of a Kac Algebra; Dual Kac Algebra

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 e, 11 in H, and a unitary representation fiG of G on H, weakly contained in the left regular representation, such that, for all t in G:

f(t)

= (Pa(t)e l11)

(ii) We have: sup{la lf(t)h(t)dtl, hE L 1 (G), IIA(h)ll :::; 1} < +oo

The space of such functions is noted B>. (G); it is a closed ideal of the FourierStieltjes algebra of G, and will be called the Eymard algebra of G. Moreover, iff is in B>.(G), its B(G)-norm is equal to: sup{

!a

lf(t)h(t)dtl, hE L 1 (G), IIA(h)ll ::=; 1}

The space B>.(G) can be identified with the dual of C~(G) (the C*-algebra generated by the left regular representation of L 1 ( G)), the duality being given, if f(t) = (pa(t)e 111), and h in CHG), by:

u. h) = (p(h)e l11) where fL denotes again the associated representation of C~ (G) (recall that fL is weakly contained in A). Moreover, for every w in £(G)*, the set: A(G)

= {s-+ (A(s- 1 ),w)

(s E G)}

is a norm-closed ideal of B>.(G). It will be called the Fourier algebra of G, and its dual is £(G). Proof. Using 3.4.2(iii), we see that property (i) characterizes 11'*(B>.(Ka(G))); by 1.6.3 (iii), it is the same for property (ii). So, we get B>.(G) = 11'*(B>.(0Ca(G))) = l\':a11'*(B>.(0Ca(G))) and all other properties of B>.(G) come then from 1.6.3(iii) and 3.4.4. Let us now consider l\':a11'*(A(OCa(G))). Using definition 3.3.4, it is the set of all elements which may be written, for all [lin £(G)*:

In 1.3.10, we have seen that A*(il) is the functions -+ (Aa(s), il); by then

1\':aA*(il) is the functions-+ {Aa(s- 1 ), il).

3.5 Construction of the Dual Weight

IOI

So, we get A( G)= Ka'll'*(A(lKa(G))), and all properties of A( G) come from 3.4.4.

3.5 Construction of the Dual Weight In that paragraph, we consider the set 2:l = a(Irp n I~). 3.5.1 Proposition. Let w, w' be in Irp

n ~· The formulas:

a(w )T a(w') = a(w * w1 )

{i)

{ii}

a(w)U = a(w 0 )

allow us to equip 2:l with a structure of left Hilbert algebra, dense in Hrp. Let us denote by 7r the left multiplication of 2:l. We have, for all w in Irp n I~:

?r(a(w)) = .X(w)

{iii)

and the von Neumann algebra generated by 7r(2:l) is equal toM. Proof. (a) We have seen in 2.4.6 (iii) that Irp n I~ is an involutive subalgebra of M*. As a is a bijection from Irp n I~ to 2:l, we see that 2:l, equipped with

T and # is an involutive algebra. (b) Let e be in Hrp, orthogonal to 2:l; by 2.4.5, we have, for all WI,W2 in Irp: 0 = (a(w!

* W2) Ie)= (.A(wi)*a(w2) I e)

by 2.4.6 (ii)

= (a(w2) I.A(wi)e) because of the density of a(Irp) in Hrp (2.1.7 (ii)), it implies .A(wi)e = 0 for all in Irp; because of the density of Irp in M* (2.1.7 (ii)), it implies, for all win M*, .X(w)e = 0, which, in turn, because of A being non-degenerate (2.6.3(i)) implies e = 0. Therefore 2:l is dense in Hrp. (c) For all WI fixed in Irp n ~the mapping a(w)-+ a(wi)Ta(w) is continuous from 2:l to 2:l. In fact, we have:

WI

a(wi)Ta(w) = a(wi *W) = .X(wi)a(w) (d) For all WI,W2,W3 in Irp

by definition by 2.6.1 (v)

n I~, we have:

(a(wi)Ta(w2) I a(w3)) = (.A(wi)a(w2) Ia(w3)) = (a(w2) I.A(wl)a(w3)) = (a(w2) I a(wl)Ta(w3)) = (a(w2) I a(wi)UTa(w3))

by (c) by 2.5.3 by (c) by definition

102

3. Representations of a Kac Algebra; Dual Kac Algebra

(e) Let e in HI(J orthogonal to ~t~. We have, for all WI,W2 in II(J n I;:

o=

(a(wl)Ta(w2) Ie)= (.X(wl)a(w2) Ie) = (a(w2) Ia(wl)e)

by (c) by 2.5.3

By (b) it implies .\(w1)e = 0, since a(I'fJ n I;) is dense in H by (b); by continuity, it implies .\(w)e = 0, for all win M.; therefore because of.\ being non-degenerate, it implies e = 0; so, ~T~ is dense in H'fJ. (f) Let w be in Itp n I; and X in 'Jltp n 'Jltpolt• We have:

(Atp(x)la(w)j) = (Atp(x)la(w 0 )) = (x*,w 0 ) = (~~:(x),w) = (a(w) I ~(~~:(x*)))

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

because SJltpo~e = ~~:('Jt~). Therefore, the mapping j has an adjoint, the restriction of which to ~(SJltp n 'Jltpo~e) is the mapping A'fJ(x) -+ A'fJ(~~:(x*)). By 2.7.5(iv), this adjoint mapping is densely defined, therefore j 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), 11'(~) generates M, we have completed the proof. 3.5.2 Definitions. We shall denote~' the right Hilbert algebra associated to~' ~ 11 the achieved left Hilbert algebra, and ~ 0 the maximal modular 8ubalgebra of ~11 (cf. 2.1.1 (iii)). We shall still note T and j (resp. b) the product and the involution on ~ 11 (resp. ~'). We shall note S and F, the closures of j and b, with respective domains denoted to vi and vb. In particular' we have, for all X in SJltp n SJltpo~~::

He in Htp is left bounded with respect multiplication" by ([14), def. 2.1).

e

to~'

we shall still note 7r(e) the "left

3.5.3 Definitions. We shall note~ the faithful, semi-finite normal weight on canonically associated to ~ ([14), th. 2.11), and call ~ the dual weight a88ociated to IK. For all w in I'fJ n by 3.5.1 (c), .\(w) belongs to 11'(~) and therefore to ~ n sn~. Moreover, for WI and W2 in Itp n I;, we have:

M

:r;,

3.5 Construction of the Dual Weight

I03

To the weight cj; we associate the Hilbert space Hq, and the canonical oneto-one mapping Ac:p : 'Jtc:p -+ Hq,. We shall note !i the left Hilbert algebra associated to cj;, i.e. Ac:p('Jtcp n 'Jt~), which is isomorphic to ~11 (2.1.1 (iii)). More precisely, the mapping which, to every in ~", associates the vector:

e

can be uniquely prolonged into a unitary operator from Hcp to Hc:p still denoted by :F. It will be called the Fourier-Plancherel mapping and will allow us to identify Hc:p and Hcp, and, through this identification of Hcp, we have ~" = !i. Using the definition 2.1.6 (ii), we shall note a instead of aq,. 3.5.4 Proposition. For all win Icp, a(w) is left-bounded with respect to~' and we have: (i) 11-(a(w)) = A(w) {ii) a(w) = Ac:p(A(w)) {iii) for all a,"( in 2l1, 7To.b is left-bounded with respect to~ and we have:

Proof. Let

WI

be in Icp, W2 in Icp n I;, e in ~'. We have:

A(w2)1r'(e)a(wi) = 1r'(e)A(w2)a(wi) by 2.6.1 (v) = 1r'(e)a(w2 * wi) = 11-( a(W2 * WI ))e because W2 * WI belongs to Icp n /; = A(w2 * wi)e by 3.5.1 (c) = A(w2)A(wi)e As 1 is in the closure of A(lcp n ~)

= 11-(~), we have:

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 nat,/31 M for some vectors a, fJ in Hcp. This element shall be written wat,/3· 3.5.8 Corollary. {i) The algebra C(G) is in a standard position in L 2(G). {ii) The predual C(G). is equal to the set {n1,9 IC(G), j,g E £2(G)}.

104

3. Representations of a Ka.c Algebra; Dual Ka.c Algebra

(iii) For all fin L 2 (G), let us put j(s) = f(s- 1 )- for all sin G. The set of all {! * g, f, g E £ 2 ( G)} is the Fourier algebra A( G) defined in 3.4.6. Proof. The aBsertions (i) and (ii) are just applications of 3.5.5 to IKa(G). In 3.4.6, A(G) haB been defined aB the set offunctions s-+ (>.a(s- 1 ),w}, for all w in £(G) •. But we have: (>.a(s- 1 ),wf,g} = (>.a(s- 1 )flg) = laf(st)g(t)dt = laf(t)g(s- 1t)dt

fa

= f(t)g(C 1s)dt = (! * g)(s) And so, (ii) implies (iii).

3. 6 Connection Relations and Consequences 3.6.1 Proposition (Connection relations). The operations T and by the following relations: (i) For any a, 'Y in !211 and fJ, 5 in H'P, we have:

(ii) For any a, 'Y in !211 and

fJ, 6 in !B1,

f

are linked

we have:

(iii) The set !B'T!B' is included in A'P(~) and, for any a, 'Y in H'P and we have:

fJ, 5 in !B1,

Proof. Combining 2.6.1 (iii) and 3.5.4 (iii), we get (i). Therefore, with the hypothesis of (ii), we have:

(W(a ® fJ) I'Y ® 6) = (fJ I11-'(6)('YT ab)) = ( 11-'(6)* fJ I 'YTab)

= (1r1(6b)fJ I'YTab) = (fJT5G 1'YTab) which is (ii).

3.6 Connection Relations and Consequences

105

It can also be written as follows:

It follows that:

Thus, {3T 8~ is left-bounded with respect to 21 (cf. 2.1.1 (iii)), and we can write: (W(a ® /3) I7 ® 8) = (1r(/3T8~)a I7) which, by continuity, still holds for any a, 7 in H cp· This completes the proof. 3.6.2 Lemma. {i} The set 23 1T23 1 is included in Acp('Jlcp n 'Jlcpo~t)· More precisely, for {3, 8 in 23 1 , we have:

and:

~t( 1r(f3T 8~ )*) = 1r(8T 13~)

(ii) The space Acp('Jlcp n 'Jlcpo~t) is a core for

F.

Proof. Let {3, 8 be in 231 , a, 7 in Hcp. We have:

(..\.(wcS,,B)*a h)= (..\.(wcS,,B),w-y,a)= (..\(wooy,a), WcS,,B)= (f31..\(wooy,a)8) = (W( a ® {3) I"{ ® 8)

by 2.6.1 (iii)

= (1r(/3T 8~)a I7)

by 3.6.1 (iii)

from what follows the first equality. For win M., we have, then:

(~t( 1r(/3T8b)), w) = (K(..\.(wcS,,B)*), w)

= (..\.(wcS,,B),w = (..\(w WcS,,B)-

0 )-

0 ),

= (..\(w)*,w6,,8)= (..\(w),w,B,cS) = (..\.(w,B,cS),w)

by 1.2.5

106

3. Representations of a Kac Algebra; Dual Kac Algebra

Therefore, we have:

by the first equality; as !Jtpo~~: = ~~:(~), the proof of (i) is completed. As, by ([158], p. 17), ~~t~' is a core for F, (ii) is immediate. 3.6.3 Lemma. Let x bt;. in M. • (i} For any a in TJ, xa belong& to 'Db, and we have:

Fxa = ~~:(x)* Fa (ii} For any /3 in

vi,

x/3 belong.s to

vi,

and we have:

Sxf3=~~:(x)*Sf3.

Proof. Let a be in Acp(IJ'tp n IJ'tpo~~:), w in lcp

(xa Ia(w)i)

n I~, x in M. We have:

= (xa Ia(w = (A;p 1 (xa)*,w 0 ))

0 )-

by 3.5.1 by 2.1.6 (ii)

= (~~:A;p 1 (xa),w)

= (~~:(xA;p 1 (a)),w)

= ((~~:(x*)~~:(A;p 1 (a)*))*,w) = ((~~:(x*)A;p 1 (Fa))*,w) = (A;p 1 (~~:(x*)ra)*,w)

by 3.5.2(i)

= (a(w) l~~:(x*)F~a)

by 2.1.6 (ii)

Therefore xa belongs to 1Jb and:

Fxa = ~~:(x*)Fa As Acp(mcp n mcpo~~:) is a core for Let a be in

vb' /3 in vi

and

X

F by 3.6.2 (ii), we have proved (i). in M. We have:

(x/31 ra) = (/31 x* F~a) = (/31 r~~:(x)a)

= (~~:(x)a I S~/3) =(a l~~:(x*)S~/3) which completes the proof.

by (i)

3.6 Connection Relations and Consequences

3.6.4 Proposition. The modular operator Proof. Let a be in

~0

xLia =xi'Sa

as ~ 0 is a core for

Li =

107

L1rf> is affiliated to M 1 •

and x in M. We have:

= P,.(x*)Sa = FSxa = Lixa

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

Li, we have xLi C Llx, which completes the proof.

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

rat= (i ® af)i'. Proof. For all x in

M,

and t in R, we have:

i'af(x) = aW*(af(x) ® l)Wa = aW*(LiitxLi-it ® l)Wa = aW*(Liit ® l)(x ®!)(,&-it® l)Wa

by 3.2.2(iv)

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

i'af(x)

= a(Liit ® l)W*(x ® l)W(..&-it ® l)a = (1 ® jit)aW*(x ® l)Wa(l ®,&-it) = (1 ®Liit)r(x)(I 0Li-it)

by 3.2.2(iv)

= (i ® af)F(x) which completes the proof. 3.6.6 Corollary. For any x in M, we have:

(i) (ii) Proof. Let a be in

"'(x*) = JxJ (l®J)W(J®J)=W*. ~o

and x in M. We have:

xla = xLi112 sa = ..& 112 xSa = jl/2§,.(x*)a

= J"'(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 ..& i, affiliated to the centre of M; moreover, it i.'J the Radon-Nikodym derivative of the weight cp with re8pect to the weight cp o K-, in the .'Jen.'Je of [114] (cf. 2.1.1 (v)).

Proof. By 2. 7.6 (i), we have arK. = u'f for all t in lll. The theorem 5.4 of [114] gives then the existence of a unique positive self-adjoint operator h, affiliated to the centre of M, such that cp o K. = cp(h·) (cf. 2.1.1 (v)). Let x be in '.ltp n '.ltpoK. j then Acp( X) belongs to V( h1/ 2 ), and we have:

llh 112Acp(x)ll 2 = cp o K-(x*x) = 11Acp(K-(x*))ll 2 = IIFAcp(x)ll 2

by 3.5.2(i)

= IIi,&-1/2 Acp(x)ll2 = 11..&-1/2 Acp(x)il2 Now, by 3.6.2 (ii), we see that Acp('Jlcp n 'JlcpoK.) is a core for F, thus also for ..&- 112; on the other hand, 3.6.4 implies that h112 and ..&- 1/ 2 commute. Using the same arguments as in ([23] lemma 23), we can conclude that h1/2 = ..& - 1/2, and so h = ..& - 1. The operator ..& is therefore affiliated to the centre of M and cp o K-(Ll·) = cp, which completes the proof. 3.6.8 Corollary. For all t in lll, we have: ~

(p

K.O't =

(p

~

0' -tK.

.

Proof. By 3.6.7, for all tin lll, ,&it belongs to the centre of M. We then have for all x in M:

~uf(x) = Juf(x)J = J..&itx,&-it J = ,&-itJxJ..&it

by ([14], 4.10) by 3.3.1

= ,&-it~(x)..&it

= u~t~(x) which completes the proof. 3.6.9 Lemma. Let for all w in M*:

(i) {ii) {iii}

X

in M,

X

=/:-

o,

8Uch that r(x)

.X(w)x = x.X(K-(x) · w) x.X(w) = .X(x · w)x .X(K.(x) · w) = K-(x).X(w)x .

= X® X.

Then, we have,

3.6 Connection Relations and Consequences

109

Proof. Let y be in l)tp. We have:

.X(w)xA'f'(y)

= .X(w)~(xy) = A'f'((w o "'® i)r(xy)) = A'f'((w o"' ® i)((x ® x)r(y))) = xA'f'((w o "') · x ® i)r(y)) = xA'f'(((K(x) · w) o "'® i)r(y))

= x.X(,.,(x) · w)A'f'(y)

by 2.3.5

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* .X(w 0 ) = .X((K(x) · w) 0 )x* = .X(x* · w0 )x*

by 1.2.5

and, changing w to w0 , x to x* (which satisfies the same hypothesis), we get (ii). Let us now assume that w is in I'f', and let w1 be another element of I'f'. We have:

A(K(x) · w)ia(w1)

= 1r'(ia(w1 ))a(K(x) · w) = i.X(w')iK(x)a(w) = i.X(w')x* ia(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 1 and x = ix* .X(K(x*) · w')ia(w) by 3.6.6(i) = K(x)i-X(K(x*) · w')ia(w) 1 and 3.5.1 (c) by 3.5.4(i) = K(x)1r'(ia(K(x*) · w ))a(w) by 3.5.4(i) = K(x).X(w)ia(K(x*) · w1) by 2.1.7(iii) = K(x).X(w)iK(x*)a(w1) 1 by 3.6.6(i) = K(x).X(w)xia(w )

By continuity, we get:

.X(K(x) · w) = K(x).X(w)x for all win I'f', 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

andr(x)=x®x. Proof. Let x be a character on M., that is, x belongs toM, and is such that X =F 0 and r(x) =X® x. As K(x) satisfies the same hypothesis, we have, for

110

3. Representations of a Kac Algebra; Dual Kac Algebra

all win M.: x~~:(x ).>.(w)

= x.>.(~~:(x) · w)~~:(x) = .>.(w)x~~:(x)

by 3.6.9 (ii) applied to ~~:(x) by 3.6.9(i)

So, by continuity, x~~:(x) belongs toM', and, then, by 2.7.2(i), it is equal to a scalar a. But then, for all w in M., we have:

.>.(w)x =

x.>.(~~:(x)

· w)

by 3.6.9(i) by 3.6.9 (iii)

= x~~:(x).>.(w)x = a.>.(w)x By continuity, we get x =ax, and, as x #- 0, we have a= 1. So x~~:(x) = 1, xis invertible, and the theorem is proved.

3.6.11 Proposition. Let G be a locally compact group, IKa(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),T8 ,~~: 8 ) (cf. 3.3.6}, and the dual weight (C,Oar on .C( G) is the Plancherel weight C,Os studied in [58], associated to the left Hilbert algebra JC( G) of continuous functions on G with compact support.

Proof. By definition 3.5.2, the weight (c.oar is associated to the left Hilbert algebra a(I'P n I~), that is, by 2.4.7(ii) and 1.1.2(ii), the set:

{! E L 1 (G) n L 2 (G); the functions-+ r(s)

= /(s- 1 )L1a(s- 1 )

belongs to L 1 (G) n L 2 (G)} equipped (by 3.5.1 and 1.1.2 (ii)) with the usual convolution product and the involution °. We have JC(G) C a(I'P n ~),and the operations on JC(G) being the restrictions of those on a(I'P n ~).Both generate the same von Neumann algebra£( G). Thanks to 3.6. 7 and 2.2.2, it appears that the modular operator associated to ( C,Oar is the Radon-Nikodym derivative of the left Haar measure with respect to the right Haar measure, that is the modular function L1a. It is also the modular operator associated to c,o 8 • Therefore, using ([114], prop. 5.9), we have:

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 such that I =F 0 and Fa(/)= I® I (i.e. l(st) = l(s)l(t), a.e.)

I in L 00 (G)

3.7 The Dual Kac Algebra

Let now g in /C( G) such that (g, f)

(g,J)f(s)

=

111

-:f. 0; we have:

(fa g(t)f(t)dt) f(s) =fa f(st)g(t)dt =fa f(t)g(s- t)dt 1

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 ~; then F(y )( x ® 1) belongs to ~rp®rp and we have:

Arp®,p(F(y)(x ® 1)) = uW*u(A,p(x) ® A,p(y)) . Proof. Let /31, P2 and 6 in 23'; let w be in IV' and x in ~· We have:

((1 ® 1i-1(6))uW*u(A,p(x) ® a(w)) I/31 ® /32) = (uW* u(A,p( x) ® a(w)) I /31 ® /32 T6b)

= (W*(a(w) ® A,p(x)) I /32 T6b ® P1) = (.X(Da(w),,B2 TcS&)A,p(x) IP1) = (.X(w · 1r(/32 T6b)*)Arp(x) I/31) = (.X(w · .x.(wo,p2 ))A,p(x) IP1) = (.X(w . .x.(wo,p2)),wAq,(z),.Bl) = (.X.(wAq,(z),,Bl),w . .x.(wo,.B2)) = (.X.(wo,,B2).X.(wAq,(z),pJ, w) = (.X.(wAq,(z),p *w0,p2 ),w) = (.X(w),wAq,(z),.B1 *wo,p2)

by 2.6.1 (iii) by 2.1.7(iv) by 3.6.2

1

by 3.3.5

= (i'(.X(w))(A,p(x) ® 6) IP1 ® P2)

So, by linearity and density, we have:

(1 ® 1i-1(6))uW*u(A,p(x) ® a(w))

= F(.X(w))(A,p(x) ® 6)

Let 61 be in 231 • We have:

(71-'(61) ® 1i-1(6))uW*u(A,p(x) ® a(w))

= (71-1(61) ® 1)(F(.X(w))(Arp(x) ® 6) = F(.X(w))(1i- (61)Arp(x) ® 6) = F(.X(w))(x ® 1)(61 ® 6) 1

112

3. Representations of a Kac Algebra; Dual Kac Algebra

Then, we can deduce that 17W* u( A.(w) =(woK. 0 i)(O'W*O')

= (i 0

w o K.)(W*)

Using 2.6.1 (ii) and 3.3.1 we get:

Using 3.3.5, we see that ). is a faithful non-degenerate representation; so, by 2.4.6 (iv), the weight cp satisfies (HWii); it satisfies (HWiii) by 3.5.3, and then (M, i', K., cp) is a Kac algebra. 3. 7.4 Definition. The Kac algebra Kac algebra of K

lK

= ( M, i', K., cp) will be called the dual

3.7.5 Theorem. Let G be a locally compact group. Then the quadruple (.C(G), F8 , K- 8 , cp 8 ), where .C( G) is the von Neumann algebra generated by the left regular representation >..a of G, F8 , K- 8 and cp 8 have been defined respectively in 3.2.8, 3.3.6 and 3.6.11, is a symmetric Kac algebra, denoted lK8 ( G), and we have:

Ks( G) = !Ka(

or

Its fundamental operator is equal to the functions--+ >..a(s) (s E G), considered as an element of .C(G) 0 L 0 '>(G). Its Fourier representation >.(w) is defined as being, for all win .C(G)., the functions--+ (>..a(s- 1 ),w} (s E G); the Fourier representation). oflK8 (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 (.C(G),F8 ,K- 8 ,cp8 ) is the dual Kac algebra of Ka(G). Therefore, we get, by 2.4.7(i) that O"W*O" is the function s --+ >..a(s), when considered as an element of .C(G) 0 L 00 (G). By 3.7.3, it is the fundamental operator of lK8 ( G). By 2.6.1 (i), applied to OC8 ( G), we get then the Fourier representation of K 8 (G).

3. 7 The Dual Kac Algebra

115

3.7.6 Proposition. Let K1 = (Mt.Tt,~Ct,cpl) and OC2 = (M2,T2,"2,cp2) be two Kac algebras, U a unitary HIP 1 -+ HIP 2 which implement& an llli8omorphi&m from OC1 to OC2 (cf. 2.7.9 and 2.2.5). Then U implement& an lll-i8omorphi&m from lK1 to lK2, too. Proof. Let u an lEI-isomorphism from M1 to M2, such that 'P2 o u (cf. 2.7.9) and let U be defined by (cf. 2.2.5):

= acp1

U ~~ (x) = a- 1 / 2 ~ 2 (u(x)) Then, for any w in I'P 2 , w o u belongs to IIP 1 , and we have:

So, we get, using 2.6.1 (v), for all w 1 in M2*:

U>.t(w 1 o u)at(w o u)

= Ua1(w 1 o u * w o u) = Uat((w 1 * w) o u) = a 1l 2a2(w1 * w) = a 1l 2>.2(w1 )a2(w) = >.2(w')Ua1(w o u)

from which we get that the application

u(x)

u defined, for

x in M2, by:

= U*xU

is a von Neumann isomorphism from M2 to

Mt

such that, for all w' in Mt.:

From that, by predualizing, we get u., a Banach space isomorphism from Mt. to M2*' such that, for all w1 in Mt.:

and, as >.2* is injective (3.3.5), we see that u. is multiplicative and involutive, and, so, u is an JBl-isomorphism. Moreover, we have then, for all win l'P2 :

.2(w)* >.2(w))

= tPt(>.t(w o u)* >.t(w o u)) = llat(w o u)ll 2

by 3.5.2 (ii)

= al!a2(w)1! 2 = acp2(>.2(w)* >.2(w))

by 3.5.2 (ii)

from that, we deduce, by 2.7.9, that

tPl o u = acp2.

116

3. Representations of a Kac Algebra; Dual Kac Algebra

3.7.7 Proposition. Let][{ be a Kac algebra, W its fundamental operator, the modular operator associated to the dual weight tP· Then: (i) We have: W(1 ® Ll)W* = Li ® Li

Li

{ii) For all t in R, .Jit belongs to the intrinsic group of K {iii) For all t in R, w in M*, we have:

(iv) For all x in M+, we have: (cp ® i)(F(x)) = cp(x).J-I {v) Let x be in 'Jl,., n '.ncpo~~: and "' in '.ncp n '.ncpo~~: and we have:

~o. Then ( i

® w 11 )F( x) belongs to

Proof. The fundamental operator of the dual Kac algebra i is uW*u (3.7.3); so (i) comes from 2.7.6(vii) applied to OC. From (i), we get W(1 ~~ .Jity~r:. =~ .Jit ~~lit; so (ii) comes from 3.2.2 (iv).. By 3.6.6(i) we have ~~:(Llzt) = JLldJ = Ll-zt; so, 3.6.9(iii) applied to Ll-zt yields (iii). Let w be in M't. We put w1 = w(Ll- 1 ·) in the sense of ([114], prop. 4.2). By 3.6.7 and ([16], 1.1.2(b)), we have: cp ® w = (cp o 11: ® w 1 )((Ll ® Ll)·)

Therefore, for all x in M+, by ([114], prop. 4.2), we have: ( cp ® w )(F(x ))

= e-+0 lim( cp o 11: ® w

1

)((Ll ® Ll)eF(x ))

where: We have: W(1 ® (1

and so:

+ c:Ll))W* = 1 ® 1 + c:W(1 ® Ll)W* = 1 ® 1 + c:(Li ® Ll)

by (i)

3.7 The Dual Kac Algebra

and:

(Li ® Li)e

117

= (Li ® Ll)W(1 ® (1 + e..&)- 1)W* = W(1 ® LnW*W(1 ® (1 + eLl)- 1)W* = W(1 ® L1A(1

+ e..&)- 1)W*

= r(Lie)

where: therefore, we get:

(c,o ® w )(F(x ))

= e-->0 lim( c,o o K ® w')r(Liex) = lim c,o o K(Liex)w 1(1)

by 2.2.4

= c,o(x)w(..&- 1 )

by 3.6.7

e-->0

which is (iv). We assume 111711 = 1. Then (i ® w'l) is a conditional expectation and we have:

(i ®w'l)(F(x*))(i ®w'l)(F(x)):::; (i ®w'l)(F(x*x)) which implies: c,o(( i ® w'l)(F(x*))( i ® w'l)(F(x))) :::; (c,o ® w'l)(F(x*x))

= IIL1A-1/217112c,o(x*x) therefore ( i ® W'l )( F( X)) belongs to

K((i ®w'l)F(x))*

~·

by (iv)

Furthermore, we have:

= (i ®w'l o K)(K ® K)F(x*) = (w'l o K ® i)F(K(x*))

which, by hypothesis, and by 2.3.1 (ii), belongs to 'Jlcp; therefore (i®w'l)(F(x )) belongs to have:

'.ncpo~~:, and by 3.5.2(i), ~((i ®w'l)F(x)) belongs to 1)~, and we F~((i ® w'l)(F(x)))

= Acp((w'l o K® i)F(K(x*)) = ,\(w'l)Acp(K(x*)) = -\(w'l)FAcp(x)

Therefore, we have: ~((i ® w'l)r(x)) = F-\(w'l)FAcp(x)

= j J.-1/2 ,\(w'7)J.1/2 j Acp(x)

by 2.3.5 by 3.5.2(i)

118

3. Representations of a Kac Algebra; Dual Kac Algebra

ein !Bo, we have:

On the other hand, for any

(.\(w.21-t/4,)e Ie)= (w.21-1/4'1 ® we)(W*)

by 2.6.1 (i)

= (W*(..::i- 1/ 4 71 ®e) 1.&- 1/ 4 71 ®e) = (w*(..::i- 11271 ®e) 111 ®e)

by 3.6.7

= (w*(..::i-1/271 ® ..:1-1/2 j1/2e) 171 ®e) = {{1 ® ..:1-ll2)w*(71 ® Li1t2e) 111 ®e) by (i) = (w*(71 ® Li 1t 2e) 111 ® .&- 1t 2e) = (.\(w,)..::i1/2e I j-1/2e) by 2.6.1 {i) Therefore we have the proof of (v).

.1- 1/ 2.\(w,)..::i1/ 2 C

.\(w.21-t/4'1) which allows to complete

3. 7.8 Theorem. Let K = (M, r, tt, r.p) be a K ac algebra, IK' the commutant Kac algebra. The dual Kac algebra K'A is equal to K'. Proof. We have seen in 3.3. 7 that the dual co-involutive Hopf-von Neumann algebra (M'A,r'A,tt'A) is equal to {M,d',K.). Let us now compute the dual weight cp'A; let us recall {2.3.7) that, if w belongs to M., if we define w' in M. by w1(x) = w(Jx* J), we have .\1(w1 ) = .\(w). Let us now suppose that w' belongs to lr.pt. We have, then, for any X in

m:,:

w(JxJ)

= w(Jx* J)= w'(x)= (a (w I Ar.pt(x*))1

1)

= (a 1(w 1 ) I JAr.p(Jx* J))-

= (Ja'(w') IAr.p(Jx* J)) sow belongs to lr.p, and a(w) = Ja1(w1 ). For w1, w1 in lr.p', we have: c,01(.\1(w')* .\1(w 1 ))

= (a 1(w 1) Ia1(w 1)) = (Ja(w2) I Ja(w1))

by 3.5.2(ii) applied to c,O'

= (a(wt) I a(w2))

= c,O(.\(w2)* .\(w1))

by 3.5.2 {ii) applied to c,O = c,O(.\(w2 o tt).\(w1 ott)*) by 1.2.5 and 2.5.3 = c,O(K.(.\(w2))K.(.\(w1)*)) by 3.3.1 = c,O o K.(.\(w1)* .\(w2))

= c,O o K.(.\1(w')* .\1(w1)) Using 2.7.7, we get c,O' = c,0 o K., which ends the proof.

3.7 The Dual Kac Algebra

119

3.7.9 Proposition. Let][{= (M, r, ~t, cp) be a Kac algebra and i: = CM, i', it, cp) be a K ac subalgebra in the sense of 2.2. 7. Then: (i) there is a unique faithful normal conditional expectation E from M to M such that cp o E = cp. Moreover, E satisfies:

i' o E

= (E ® E)r

itE

= E~t

and the projection P defined by P Acp( x)

= Acp( Ex)

belongs to the centre of

M· '(ii)

-

the canonical surjective E-morphism r from M toM~, defined in 3.3.8, has P as support, and identifies i:~ to Kp (cf. 2.2.6).

Proof. As M is o-f-invariant and cp!M semi-finite, by [160], there is a unique normal conditional expectation E such that cp o E = cp (cf. 2.1.8(ii)). As in 3.3.8, let j be the canonical imbedding M -+ M, and I the isometry defined, for all X in ~~ = ~ n M by: IA~(x)

= Acp(j(x))

Let P =II*. We know, from [160] that, for ally in

~cp:

A~(Ey) =I* Acp(y)

Acp( Ey)

= P Acp(Y)

and, moreover, that, for all x in M, Ex is the unique element of M such that (Ex)P = PxP. Let w be in Icp; it is easy to see that w o j belongs to I~, and that: a(w 0 j)

= I*a(w)

We have then, using 3.3.8: cp~(r(A(w)* A(w))) = cp~(.\(w o j)*.\(w o j)) = l!a(w o j)ll 2

by 3.5.2 (ii) applied to cp

= I!I*a(w)ll2 = I!Pa(w)l! 2

Let R be the support of r. Using 2.7.8, we know there ifl a> 0 such that: cp~(r(A(w)* A(w)))

= acp(RA(w)* A(w)) = ai!Ra(w)l! 2

by 3.5.2 (ii)

120

3. Representations of a Kac Algebra; Dual Kac Algebra

So, we have 11Pa(w)ll 2 = a11Ra(w)ll 2 for all win IV'; so we get P = R (and so P belongs to the centre of M), a= 1, cpA= c(;p, which ends the proof of (ii). Now, E ®Eisa faithful normal conditional expectation from M ® M to M ® M such that, for all X in ~IP®IP:

Atp®IP((E ®E)( X))= (P ® P)Atp®tp(X) So, for x,y

in~.

we shall have:

AIP®IP((E ® E)(F(y)(x ® 1))) = (P ® P)A~P®IP(F(y)(x ® 1)) = (P ® P)W(AIP(x) ® AIP(y)) = (P ® 1)W(1 ® P)(AIP(x) ® AIP(y)) because P belongs to Z ( M) = (P ® 1)W(A1P(x) ® Atp(Ey)) = (P ® 1)A1P®1P(F(Ey)(x ® 1)) As E ® i is also a faithful normal conditional expectation from M ® M to

M ® M such that, for all X in ~IP®IP: AIP®IP((E ® i)(X))

= (P ® 1)A~P®1P(X)

We have then:

AIP®IP((E ® E)(F(y)(x ® 1)) = Atp®IP((E ® i)(F(Ey)(x ® 1))) and, therefore:

(E ® E)(F(y)(x ® 1)) = (E ® i)(F(Ey)(x ® 1)) By continuity, we get, for all y in

(E ® E)(F(y))

~V':

= (E ® i)(F(Ey)) = F(Ey)

and, by continuity again, we have:

(E®E)F=FE Let now x be in M; we have: ~t(Ex)P = }(Ex)JP

= }(Ex)PJ

by 3.6.6(i) because P belongs to Z(M)

= JPxPJ

=PJxJP = P~t(X)P = E~t(X)P

because P belongs to Z(M) by 3.6.6(i)

3.7 The Dual Kac Algebra.

121

We have then K(Ex) = EK(x), which ends the proof of (i) and of the proposition.

3. 7.10 Proposition. Let ][{ = ( M' r, K, 'P) be a K ac algebra, R be a projection of the centre of M such that F(R) ~ R Q9 R, K(R) = R and JKR be the reduced Kac algebra in the sense of 2.2.6. There is a canonical one-to-one llll-morphism j from (JKRt to lK which identifies (KRt with a Kac subalgebra ofOC (cf. 2.2.7). More precisely, ifr denotes the reduction x - t XR of M on MR, we shall have, for all w in (MR)*: j(>..R(w))

= .\(w or)

where AR denotes the Fourier representation of JKR. Proof. Let us call I the projection R, considered as an element of .C(H'f', H'f'R). We have, then:

I* I= R II*= ln'PR IA~.p(x)

= A'f'R(r(x))

r(x) = Ixl*

(x E l)'tp) (x EM)

Moreover, if w is in I'f'R• it is easy to check that w or is in I'f' and that:

a(w or)= I*a(w) The reduction r is an llll-morphism, thus the mapping w - t wor from ( M R)* to M* is multiplicative and involutive; therefore, the set { .\( w or), w E (M R)*} is an involutive subalgebra of M; let us call N its weak closure. Let f3 in H 'f' such that {/31 xc5) = 0 for all x in N and c5 in H 'f'· We have, for all a, 1 in H'f'R• all c5 in H'f': 0=

{/31 A(WJ*-y,I•a)c5)

= (W(ra ® /3)

r1 ® c5) =((I ®l)W(ra Q9 /3) 11 Q9 c5) 1

which implies, for all a in H'f'R:

(I ®l)W(ra Q9 /3)

=o

(R ®l)W(I*a Q9 {3)

=0

or:

by 2.6.1 (iii)

122

3. Representations of a Ka.c Algebra; Dual Ka.c Algebra

as R is in the centre of M, W belongs toM® M and Rr = r, it implies:

W(ra®/3)

=o

As w is unitary, it gives r a ® /3 = 0, for all a in H cp R' which implies /3 = 0. Then, N is a non-degenerate algebra on Hcp; it is a von Neumann subalgebra of M. Let y be in '.>lcp, win (MR)•· We have:

D.(w o r)Acp(Y) = IAcp((w oro K ® i)T(y)) = AcpR((w o K ® i)(r ® r)T(y)) = AcpR((w o K ® r)T(ry)) = AR(w)AcpR(r(y)) = AR(w)IAcp(Y)

by 2.3.5

therefore:

D.(w or)= AR(w)I and:

RA(w or)=

r AR(w)I

By passing to the adjoints this equality yields that R belongs to N'. So R belongs to the centre of M, and, for all in!!~, we have:

e,"'

Now, let z be in

M such that zR = 0. For all e.., in!!~, we shall have: zA(wRe,R'I) = z1r(ReT R"'b)

= zR1r(eT.,b)

by 3.5.4 (iii) by the above remarks

which implies, by continuity, for all e,"' in Hcp:

zA(wRe,R'I)

=0

or, also, for all -y,a in HcpR:

that is, for all w in (MR)•:

zA(wor)=O and zN = 0, which ensures z = 0 by the above results on N. So, the reduction N -+ N R is an isomorphism. Let us call 9 the inverse isomorphism, and for x

3. 7 The Dual Ka.c Algebra

123

in ( M RY, let us put j ( x) = ~(I* xi). It is clearly a one-to-one homomorphism from (MRY toM such that j(l) = 1. Moreover, we have, for all win (MR)*:

j(>.R(w))

=~(I* >.R(w)I)

= ~(R>.(w or))= >.(w or)

The range of j is therefore equal to N. Now, let wbe in M*. We have:

and then r >.* = >. R*i*. From what it is straightforward to prove that j* is involutive and multiplicative and therefore that j is an lBI-morphism. For all t in JR, we have:

afp.(w or)) = >.(w oro L ~it) by 3.7.7 (iii) = >.(w o Lr(~it) or) which belongs toN therefore N is af-invariant. Let win I'PR" We have:

cp(j(>.R(w)* >.R(w)) = cp(>.(w o r)*>.(w or)) = lla(w o r)ll2 = III*a(w)ll 2 = lla(w)ll 2 = .R(w)* ).R(w)) Therefore j(>.R(I'PR) C SJl.p, which implies that cp IN is a semi-finite weight. Finally, we see that N is a Kac subalgebra of M, j is an lBI-isomorphism from (MRr toN and, by 2.7.9 and the above calculation cp o j = ®cp and the operator which sends AtJ>®®.a(s) U>.(w)U* = Qw.

Proof. Thanks to 4.2.2 it is clear that U as defined in the proposition is unitary. Moreover, we have, for all s, tin G and win Icp:

(>.a(s)Qw')(t)

= Qw1(s- 1t) = (Q(s · w 1 ))(t)

by 4.2.1 {ii)

Therefore:

>.a(s)Qw'

= Q(s · w') = Ua(s · w1) = Usa(w 1)

by definition of U by 2.1.7(iii)

which gives {i). Let w in M., w1 in Icp. We have:

U >.(w )U*Qw1 = U >.(w )a(w1 ) = Ua(w * w1 ) = Q(w*w') = QwQw1

by 2.6.1 (v)

4.2 Takesaki's Theorem on Symmetric Kac Algebras

133

Since Qlcp is dense in L 2 (G), we have:

U>.(w)U* = Qw which completes the proof. 4.2.4 Takesaki's Theorem. Let lK = (M, r, ,.,,.(w)U*

= Qw

and we can deduce, thanks to 1.4.5 (ii), that we have:

(U ® U)F>.(w)(U* ® U*) = (U ® U)c;(>. x >.)(w)(U* ® U*) = c;Q x Q(w) Let J,h,h in L 1 (G(IK)). We have:

(Q.J,w) = (Qw,J) = faf(s)Qw(s)ds = fa(s*,w)f(s)ds which implies:

and:

(c;Q

X

Q(wl),h ®h)= (Q.hQ.h,w) = {

laxG

=

f

lax a

h(t)h(s)(t*s*,w)dsdt Qw(st)fl(s)h(t)dsdt

= (Fa9w, h

®h)

Thus we have, for all w in M.:

(U ® U)F(>.(w))(U* ® U*) and, by density, for all x in

= Fa(9w)

M:

(U ® U)F(x)(U* ® U*)

= Ta(UxU*)

134

4. Duality Theorems for Kac Algebras and Locally Compact Groups

Moreover, we have: Uk(.A(w))U* = U.A(w o K)U*

by 3.3.1

=Q(woK.)

= Ka(Qw) by density, for all x in

by 4.2.1(i)

M, we get: Uk(x)U* = Ka(UxU*)

Finally, by 4.2.2, we have, for all w in lr.p: ..a,(v) on G', and (U ® I)W*(U* ®I) is the identity function on G1, considered as an element of L 00 (G') ® L 00 (G,mt. So, for all win {L00 {G,mn., we clearly see that U(i ® w)(W)U* is the function v - (v,w) on G' and, as {i ® w)(W) is the functions- (~t(s),w) on G, we infer that the isomorphism f - U* JU from £ 00 ( G') to £ 00 ( G, m) is just the composition by I'· Using f = XG'-p(G)• we see then that G1 -~t(G) is of Haar measure 0. So, if v belongs to G1 -~t(G), v~t(G) ~ G1 -~t(G) is of Haar measure 0, and so is I'( G), which is impossible; so I' is surjective, and the theorem is proved.

136

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 OC8 ( G). 4.3.1 Lemma ([44)). The left regular representation >..a is a one-to-one homeomorphism from G to the intrinsic group of OC8 ( G). Proof. Lets be in G. Then, by 3.3.6, >..a(s) belongs to G(OCs(G)); the mapping >..a is also clearly one-to-one. Now let {>..a(s 11 )} 11 denote a net converging to the identity 1 = >..a(e), where e is the unit of the group G. We wish to show that s 11 converges to e.

Proceeding by way of contradiction, we suppose it does not. Then there exists a subnet, say Sn of s 11 and a compact neighbourhood V of e such that sn rf. V, for all n. Choose a compact neighbourhood U of e such that uu-l c v. Then, for each Sn, we have (xu denotes the characteristic function of U and 11 the Haar left measure on G):

(>..a(sn)xu Ixu)= faxsnuxud/1 = 1-L(snU nU) = 0 Thus:

l((>..a(sn) -1)xu I xu)l =(xu I xu) = 11(U) > 0

Thus >..a(sn) does not converge weakly to 1, which contradicts our first assumption and completes the proof. 4.3.2 Theorem. The left regular representation >..a is a bicontinuoul! bijection from G onto the intrinllic group of OC 8 ( G). Proof. Let denote G0 the intrinsic group G(OCs(G)). By 3.7.5, the Fourier representation >.. of OC8 (G) is the mapping, defined, for all w in .C( Gh, by:

s-+ (>..a(s)*,w) which belongs to L 00 ( G). By 4.2.4 applied to ][{8 ( G), there exists a unitary U from L 2 (G) on L 2 (Go) such that U>..(w)U* be equal to Qw, which is the mapping defined, for all t in G 0 , by:

t-+(t*,w) which belongs to L=(G0 ). Thus, we have:

U*QwU

= Qw o >.a

4.3 Eymard's Duality Theorem for Locally Compact Groups

137

and, by continuity, for all fin L=(G0 ): U*fU

= fo>.a

By selecting f = Xao-.Xa(a)• we see that f = 0, and Go- >.a( G) is a zeromeasure set. Let so EGo and So rf. >.a( G); we have so>.a(G) C Go- >.a( G); therefore s 0 >.a( G) is a zero-measure set as well as >.a( G) by left-invariance, but this is impossible by(*). Therefore >.a(G) = G 0 , which completes the proof. 4.3.3 Corollary (Eymard's Theorem [46]). Let G be a locally compact group. Let us recall ( cf. 9.4. 6 and 9. 5. 6{ that the Fo'll:..rier al~ebra A( G) has been defined as the set {f*g, f,g E L (G)} (where f(s) = f(s- 1 ) for all sinG, fin L 2(G)), equipped with the norm:

llall

=sup

{l.la

a(s)f(s)dsl, f E L 2(G),

l!>.a(f)ll

:51}

Then, the spectrum of A( G) is G; so every character on A( G) is involutive. Proof. By 4.3.2, the spectrum of£( G). is equal to >.a( G). By 3.4.6 and 3.5.6, there is an isomorphism between C(G)* and A(G), which, to each Wj,g in C(G)., associates the function f * g, linked by:

So, we see that the spectrum of A(G) is the set {s- 1 ,s E G}, that is G. 4.3.4 Corollary. Let G1, G2 two locally compact groups, u an JH[.morphism from lHl8 (G1) to lHl 8 (G2); then there exists a continuous group homomorphism a from G1 to G2 such that u( >.a1 ( s)) = >.a2 (a( s)), for all s in G1. The image a(Gl) 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 JH[8 (G1) into the intrinsic group of JH[8 ( 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); T 8 (x) = x®x} is a closed subset of the unit-ball of £(G1), and therefore, is compact for weak topology. Its image par u, that is:

by 3.6.10 and 4.3.2, is then also compact for the weak topology of £(G 2 ). So {>.a 2 (a(s)), s E GI} is locally compact, and by 4.3.1, a(GI) is a locally compact subgroup of G2, and so is a closed subgroup of G 2.

138

4. Duality Theorems for Ka.c Algebras and Locally Compact Groups

For every win .C(G2)., we have: gw(a(s)) = (.\a2 (a(s)*),w)

by 4.2.1 by de:fini tion of a

= (u.\a1 (s)* ,w)

= (.\a1 (s)*,w o u) by 4.2.1

= g(w o u)(s)

And: gw o a= g(w o u)

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 group8j then, the following a88ertion8 are equivalent: (i) There exi8t8 a bicontinuoU8 i8omorphi8m u : G1 -+ G2 (ii) There exi8t8 an JIJ.-i8omorphi8m from l!ll0 (G2) onto l!lla(Gl)· (iii) There exi8t8 an l!ll-i8omorphi8m from l!ll8 ( G1) onto l!ll8 ( G2). Proof. We have (i) => (ii) because the application f -+ f o u is an l!llisomorphism from l!ll0 (G2) onto l!ll0 (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, K) be a 8ub co-involutive Hopf-von Neumann algebra of l!ll8 ( G), 8uch that there exi8t8 a Haar weight

..a(s))

= (E®E)rs(>..s(s)) = (E®E)(>..a(s)®>..a(s)) = E>..a(s)®E>..a(s)

So, by 3.6.10, either E>..a(s) = 0, or E>..a(s) is in the intrinsic group of lK, and is therefore unitary, which implies E>..a(s) = >..a(s). So, the subset {s E G; >..a( s) ~ M} is therefore equal to the subset {s E G; E>..a(s) = 0} and is closed. Then G' = {s E G; >..a(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 L 1 (G), and the Gelfand repre8entation of L 1 (G) is given by the Fourier transform:

{ii) There exists a Haar mea8ure dx on

G,

and an isomorphism U from

L 2 (G) to L 2 (G) defined, for all fin L 1 (G) n L 2 (G), by:

called the Fourier-Plancherel transform, such that the mapping x -+ U xU* is an isomorphism from the dual Kac algebra lK8 (G) to Ka(G), such that, for all f in L 1 (G):

U>..a(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 L 1 ( G), and the Gelfand transform, taking a coherent definition with 4.2.1, will be given by:

for all fin L 1 (G),

x in G, which is (i).

140

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 L 2 (G) to L 2 (G), defined by Uf = j for all fin L 1 (G) n L 2 (G), such that the mapping x-+ UxU* is an isomorphism of the dual Kac algebra lK8 (G) to Ka(G). By 4.2.3(ii), we get U>..a(f)U* = j, for all fin L 1 (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 GM is isomorphic to G. Proof. The isomorphism defined in 4.3. 7 (ii) sends the intrinsic group of K 8 ( G) onto the intrinsic group of Ka(G), that is onto GAA. For all fin L 1(G), we have, using 4.3. 7 (ii ): kf(s)(x,s- 1 )ds

= ](x) = kf(s)U>..a(s)U*ds

from which we can deduce that U>..a(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 GAA is isomorphic to the group G.

4.4 The Kac Algebra lKs(G) It is now possible to describe the various objects associated to the Kac algebra lK8 (G) by the general theory.

4.4.1 Proposition. {i) The enveloping C* -algebra C*(Ks( G)) is the algebra Co(G) of continuous functions on G, vanishing at infinity; the canonical representation of C(G). into C*(JK8 (G)) is then the Gelfand transform Qw(s) = (>..a(s)*,w) for all win C(G)., s in G. {ii) The Fourier-Stieltjes algebra B(lK8 ( G)) is the algebra M 1 ( G) of bounded measures on G, and its Fourier-Stieltjes representation is the left regular representation of M 1 (G); an element x of C( 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 s·uch that: x

=fa >..a(s)dm(s)

(iii) The enveloping W*-algebra W*(JK 8 (G)) is the dual M 1 (G)* of M 1 (G). This Banach space, which is a W*-algebra, being equal to the bidual ofC0 (G),

4.4 The Kac Algebra K,(a)

141

has then a structure of co-involutive Hopf-von Neumann algebra, given by:

(F(8), m1 0 m2)

= (8, m1 * m2)

(~~:(8),m) = (O,m 0 ) -

(8 E M 1 (G)*, mt. m2 E M 1(G)) (8 E M 1(G)*, mE M 1(G))

where* is the multiplication of M 1 (G), and 0 its involution, and where 0 is defined by (O,m) = (8,m)-, with m(f) = (J jdm)- for all fin C0 (G). (iv) The canonical imbedding (s.x)• from L 1 (G) to B(lK8 (G)) = M 1 (G) is the usual imbedding from L 1 (G) to M 1 (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 C(G). is its Gelfand transform. So (i) results from 4.3.3. As B(lKs( G)) is the dual of C*(lK8 ( G)), we deduce, from (i), that B(lK8 (G)) is equal, as a Banach space, to M 1 (G). Let us compute its Fourier-Stieltjes representation K 8 7r*j if m is in M 1 ( G), j,g in L 2 (G), we shall have: (~~: 8 7r*(m),wf,g)

= (1r(wf,g o ~~:),m) =

Ia

=

la(>•a(s),wf,g}dm(s)

And so, we have: Ks7r•(m) =

Ia

(i(wf,g o

~~:)dm by 3.7.5 and (i)

.Aa(s)dm(s)

We can deduce from it that the multiplication and the involution of B(lK8 ( G)) are the usual ones on M 1 (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 1 (G), and m = (s_x).(f). As ~(f) is fa .Aa(s)f(s)ds, and, by 1.6.1 (ii) and 3.7.3, equal to Ks7r•(s_x).(f) which is, by (ii), equal to fa .Aa(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 p. of C(G). is given by a spectral measure Pf.J on G, with values in 1if.J, as defined, for example in {[105], IV, §17.4), such that, for all w in C(G).: p.(w) =

fa(>. a(s)*,w)dPf.J(s)

{ii} Let p. be a non-degenerate representation of£( G)., Pf.J its associated spectral measure on gf.J, il an element of (Af.J)*; then ~~: 8 p..(il) is the image by the left regular representation of the bounded measure d(Pf.J ( s ), il) on G.

142

4. Duality Theorems for Kac Algebras and Locally Compact Groups

(iii} Let f-1.1, f-1.2 be two non-degenerate representations of C( Gh, P1411 P142 their associated spectral measures. The spectral measure associated to the Kronecker product f-1.1 X f-1.2 is the convolution product of P141 and P142 defined by:

Proof. By 4.4.1 {i) and 1.6.1 {i), there is a representation jl. of C0 (G) on 1£14 such that jl. o g = J.l.i by {[105], IV, §17.4), the representation p. is given by a spectral measure on G with values in 1£14 , which gives {i). We have then:

{Kaf..l.•{il),w}

= {J.I.(w o ~~: 8 ), il} = fa{>·a(s)*,wo~~:)d{P14 (s),n} =

fa {-Xa(s),w}d{P14 (s), il}

and therefore:

Kaf..l.•(n) =fa -Xa(s)d{P14 (s), il} which gives {ii). Let now il1 be in {A141 )., il2 in {A142 h; we have:

{(f-1.1

X

J.1.2)(w), il1 ® il2} = {J.I.t.(il1)J.I.2•(n2),w}

by 1.4.3

= {~~:af..1.2•(il2)~~:af..1.h(il1),w o ~~:a}

= fa{-Xa(s),wo~~:a}d({P142 ,il}*{P1411 il})(s) =

by{ii)

fa{.Xa(s)*,w}d({P142 ,il} * {P141 ,il}){s)

therefore, the measure d{P#Jl X#J2' n1 ® .02} is the convolution product of the measures d{P142 , il2} and d{P141 , il1}; which gives (iii). 4.4.3 Theorem. Let P be a spectral measure on G with values in 'H. Then, there exists a unitary U in C( G) ® C('H) such that, for all w in C( G)., 1J in 'H:

e,

We shall write:

U=

la

-Xa(s)* ® dP(s) .

4.4 The Kac Algebra K.(G)

143

Proof. Let i' be the non-degenerate representation of Co( G) associated to the spectral measure P, i.e. such that, for any fin C0 (G):

~LU) =

fa f( s )dP( s)

Let us put v = i' o {}; then v is a non-degenerate representation of .C( G). such that, for any win .C(G)., we have:

v(w)

=fa (Aa(s)*,w)dP(s)

By 3.1.4, there exists a unitary U, in £(1-l) ® .C( G) which is the generator of v, and is such that, for all~. 71 in 1-l, win .C(G).:

SoU= uU,u satisfies the theorem. 4.4.4 Proposition. Let G be an abelian locally compact group, G its dual group, in the sense of4.3.7. For any min M 1 (G), let us define the Fourier transform ofm by:

(x E G) Then: (i) For any m in M 1 (G), we have:

U

(fa Aa(s)dm(s)) U* = m

(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 i' of G is given by a spectral measure Pp on G, with values in 1-lp, such that, for all s in G, we have:

Proof. For any fin L 1 (G) (which is an ideal of M 1 (G)), (i) has been proved in 4.3. 7 (ii). So, using the non-degeneracy of the representations, (i) is proved for any min M 1 (G). As L 1 (G) has a bounded approximate unit, we see, using 4.4.1 (ii), that every positive definite element in L 00 (G) is of the form U(J0 Aa(s)dm(s))U*, with min M 1 (G)+. So (ii) is proved, using (i) and 1.3.11.

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4. Duality Theorems for Kac Algebras and Locally Compact Groups

Let us consider the non-degenerate representation of £ 1 (G) obtained from J.L by 1.1.4. By 4.3.7(ii), L 1 (G) is isomorphic to .C(G)., and, by 4.4.2(i) and 4.3.7(i), there is a spectral measure PIJ on G, with values in 1{/J, such that, for all fin L 1 (G), we have: J.LU)

=

Ja /(x)dP~'(x)

from which we get (iii).

4. 5 Characterisation of the Representations and Wendel's Theorem Let lK = (M,F,K,cp) be a Kac algebra. Let A be a von Neumann algebra. In this paragraph, f3 will denote a normal, one-to-one morphism from if to A 0 if such that

(/3 0 i)i' = (i 0 i')/3 f3(1M) = 1A ®1M By 3.2.6, for any non-degenerate representation J.L of M., the algebra AIJ and the morphism :YIJ fulfill these conditions. 4.5.1 Proposition. We have, for all x in if+ and t in JR.:

(i)

(i 0 cp)f3(x) = cp(x)1A

(ii)

(i 0 af)/3 = f3af .

Proof. Let x in if+. We have:

(i 0 cp)f3(x) ®1M= (i 0 i 0 cp)(i 0 F)f3(x)

by 3.7.2

= (i 0

i 0 w)(/3 0 i)F( X) = f3((i 0 cp)i'(x))

by hypothesis

= cp(x)f3(1M)

by 3.7.2 by hypothesis

= cp(x)(lA ®1M)

which brings (i). Lett in JR.. We have:

((i 0 af)/3 0 i)i'

= (i 0

af 0 i)(/3 0 i)i'

4.5 Characterisation of the Representations and Wendel's Theorem

= (i ® uf ® i)(i ® F)f3

145

by hypothesis

= (i ® i ® uf)(i ® F)f3 by 2.7.6 (ii) applied to

= (i ® i ® uf)(f3 ® i)i'

K

by hypothesis

= (f3 ® i)( i ® uf)i'

= (f3 ® i)( uf ® i)i'

by 2. 7.6 (ii) applied to

K

From what follows that (i ® uf)f3 ® i and f3uf ® i coincide on F(M); as it is obvious that they coincide also on C ® M, thanks to 2. 7.6 (iv) applied to K, they will coincide on M ® M, which completes the proof. 4.5.2 Proposition. Let '1/J be a faithful, semi-finite, normal weight on A (to simplify, we shall suppose A C C(Ht!J)). Then: (i} for all x in 'Jtv,, y in IJlcp the operator f3(y )(x ® 1) belongs to 1)11/J®cp, and there is an isometry U in H.,p ® H such that:

U(At!J(x) ® Acp(Y))

= At/J®cp(f3(y)(x ® 1))

(ii) U belongs to A® M. (iii} for all z in M, we have:

f3(z)U

= U(1 ® z)

.

Proof. We have: ('1/J ® ®((,B 0 i)(F(y1))(x 0 Y2 01)) = Ati>®cf>®cp((i 0 F)(,B(yl))(x 0 Y2 0 1))

by hypothesis

= At/>®cp®cp((i 0

F)(,B(yi)(x 0 1))(x 0 Y2 0 1)) = (1 0 a)(1 0 W*)At/>®cp(,B(yi)(x 0 1)) 0 A.p(Y2)) by 2.4.9 applied to cj; and 3. 7.3 = (1 0 a)(1 0 W*)(U 01)(Att>(x) 0 A.p(YI) 0 A.p(Y2))

by (i)

This isometry does therefore coincide with (1 0 a)(1 0 W*)(U 0 1)(1 0 a) on those vectors of the form Att>(x) 0 A.p(Y2) 0 A.p(YI), which, by linearity, density and continuity, completes the proof of (ii). 4.5.4 Proposition. With the notation.! of 4.5.2, the i8ometry U 8ati8jie8:

(i 0 F)(U)

= (10 a)(U 01)(10 a)(U 01).

Proof. By 2.6.4, we have:

(i 0 c;F)(U)

= (1 0

a)(1 0 W)(10 a)(U 01)(10 a)(10 W*)(10 a)

Let x in !Jltt>, Y!. Y2 in !Jtcp. We have: (10 W)(10 a)(U 01)(10 a)(10 W*)(Att>(x) 0 Acp(YI) 0 A.p(Y2))

= (10 W)(10 a)(U 01)(Att>(x) 0

A.p0.p(F(y1)(y2 01)))

by 2.4.2(i) applied to][{ and 3.7.3

= (1 0

W)(1 0 a)Att>®(((,B 0 i)(f'(yi)(Y2 01))(x 01 01)) by 4.5.3(i)

4.5 Characterisation of the Representations and Wendel's Theorem

= = = =

149

(1 ® W)(1 ® a)A.p®ip®,:P((/3 ® i)i'(yi)(f3(Y2)(x ® 1) ® 1)) (U ® 1)(1 ® a)(A.p®,;p(f3(Y2)(x ® 1)) ® Aq,(Yl)) by 4.5.3 (ii) (U ® 1)(1 ® a)(U(A.p(x) ® Aq,(y2)) ® Aq,(YI)) by 4.5.2(i) (U ® 1)(1 ® a)(U ® 1)(1 ® a)(A.p(x) ® Aq,(Yl) ® Acp(Y2))

therefore, we have: (1 ® W)(1 ® a)(U ® 1)(1 ® a)(1 ® W*) = (U ® 1)(1 ® a)(U ® 1)(1 ®a) which completes the proof. 4.5.5 Proposition. With the notation8 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 1-'· Moreover, we have:

A = A"' and {3 =

7"' .

Proof. Let P the projection UU*. We have: (i ® F)(P) = (U ® 1)(i ® ~)(P ® 1)(U* ® 1) $ UU*®1 =P®1

by 4.5.4

Applying 2.7.3(i) to IK', we get the existence of a projection Q in A such that P=Q®1 Let z in

M. We have: f3(z)(Q ® 1) = {3(z)UU* = U(1 ® z)U* = UU*U(1 ® z)U* = (Q ® 1)U(1 ® z)U* = (Q ® 1){3( z )( Q ® 1)

by 4.5.2(iii)

by the same calculation

Passing to the adjoint operators, we get:

{3(z)(Q ® 1) = (Q ® 1){3(z) Now let x in 'Jl.p, y in 'Jlcp. We have:

U(Q ® 1)(A.p(x) ® Aq,(y)) = U(A.p(Qx) ® Aq,(y)) = A.p®,;p(f3(y)(Qx ® 1)) = A.p®,;p((Q ® 1){3(y)(x ® 1)) = (Q ® 1)A,p®,;p(f3(y)(x ® 1)) = (Q ® 1)U(A,p(x) ® Aq,(y))

by 4.5.2(i) by the result above by 4.5.2(i)

150

4. Duality Theorems for Kac Algebras and Locally Compact Groups

By linearity and density, we get: UP= U(Q®1)

and:

= (Q®1)U = PU = UU*U = U

uu• = P = u•u P = u•u = 1

Therefore U is unitary. By 2.6.5, U is the generator of a non-degenerate representation of M.; let us note it p.. Then, we have, for z in M: {1(z)

= U(1 ® z)U*

by 4.5.2 (iii) by 3.2.2(i)

=.Y,(z) which completes the proof.

r' ;.,,

= (M' r, tt, cp) be a K ac algebra, i = (M' cj;) the dual Kac algebra, A a von Neumann algebra, {1 an injective normal morphism from M to A® M. Then, the following assertions are equivalent: (i) We have:

4.5.6 Theorem. Let ][{

({1 ® i)i' = (i ® r){1

and {1(1)

=1

{ii) There ezists a non-degenerate representation p. from M. to A such that: {1 0). = ~). X p. Proof. That is clear from 3.2.6 and 4.5.5. 4.5. 7 Corollary. Let G be a locally compact group, A a von Neumann algebra, {1 an injective normal morphism from C( G) to A® C( G). Then, the following assertions are equivalent: {i) We have, for all s in G:

(i ® ra)f1(>.a(s)) = f1(>.a(s)) ® >.a(s) {1(1)

=1

{ii) There ezists a unitary representation P.G of G, such that A is the weak closure of P.a(L 1 (G)), and, for all s in G:

{1(>.a(s))

= P.a(s) ® >.a(s).

Proof. It is just an application of 4.5.6 to Ka(G). 4.5.8 Corollary. Let K be a Kac algebra. Then an element u of C(H) belongs to the intrinsic group G(K) if and only if it is the canonical implementation of an automorphism p of M such that (!1 ® i)i' = rp ([18]).

4.5 Characterisation of the Representations and Wendel's Theorem

151

Proof. Let fJ an automorphism of M such that (fJ®i)i' = i'{J. By 4.5.6, there is a unitary u which is the generator of a one-dimensional representation of M. (so, by 2.6.6 (i), u belongs to G(K)), such that fJ = .:Yu· Moreover, by 4.5.2, we shall have: uAcp(Y) = Acp(fJ(y))

Therefore u is the canonical implementation of fJ. Conversely, let u in G(K). By 2.6.6 (i), u is a one-dimensional representation of M., and then .:Yu is an automorphism of M satisfying:

Using the first part of this proof, we see that the canonical implementation of ~u is an element v of G(K), which satisfies ~u = ~v· Then uv* belongs toM' n M, which means, by 2.7.2, that there exist a complex a such that u = av. As u and v are unitaries, we have lal = 1. As F(u) = u ® u, and r(v) = v ® v, we get a= a 2 , so a= 1 and u = v.

4.5.9 Wendel's Theorem ([199]). Let G be a locally compact group; every automorphism of L 00 ( G) which commutes with the right translations is a left translation. Proof. Let fJ be an automorphism of L 00 ( G) commuting with right translations; we have FafJ = (fJ ® i)Fa, from which, applying 4.5.8 to K 8 (G), we get that there is u in G(K8 (G)) such that fJ(f) = ufu* for all f in L 00 ( G). So, by Eymard's theorem (4.3.2), there exists s in G such that fJ(f) = >..a(s)f>..a(s)*, and so fJ is then the left translation by s- 1 .

4.5.10 Theorem ([163]). Let G be a locally compact group. {i) Let ( M, r, K) be a sub co-involutive H opf-von Neumann algebra of lila (G), such that there ezists a H aar weight r.p on ( M, r, K). Then, there ezists a normal subgroup H of G such that M is the subalgebra of the functions in L 00 (G) invariant by H. Then (M,F,K) is isomorphic to H:Ia(G/H). (ii) Let ][{ be a sub Kac algebra of Ka(G); then there ezists a normal compact subgroup K ofG such that][{ is isomorphic to Ka(G/K). (iii} Let Ks(G)p be a reduced Kac algebra of K 8 (G); then, there ezists a normal compact subgroup K of G such that K 8 (G)p is isomorphic to Ka(G/K). Proof. (i) Let][{= (M,r,,,r.p), K = (M,i',K.,rp) the dual Kac algebra. By 4.2.5(ii), K is isomorphic to Ka(G(K)). As (M,F,K) is a sub co-involutive Hopf-von Neumann algebra of (L 00 (G),Fa,Ka), it is easy to see that M is globally invariant under the automorphisms implemented by >..a(s), for all s in G. Then z -+ >..a(s)z>..a(s)* (z E M) is an automorphism fJ 8 of M satisfying:

152

4. Duality Theorems for Kac Algebras and Locally Compact Groups

Ff3s = ({38 ® i)F We have then a continuous morphism from G to the group of automorphisms = ({3 ® i)F, which, by 4.5.9, is isomorphic to the intrinsic group of i. Let H be the kernel of this morphism; we then get a continuous one-to-one morphism u : GI H -+ G(K). By definition of H, M is included in the subalgebra {! E L00 (G); Ad>..a(s)f = f, '. ® S>.)(Q ® Q) = S>.(Q) ® S>.(Q) ~

S>.(Q) ® 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 ~ supps>,, which implies: Q

+ (1- supps>,) ~ 1

and:

s"II"X"II"(Q) + s7rx7r(1- supps>.) ~ 1 ® 1 and then:

Q ® Q + (1- supps>,) ® (1- supps>.)

~

1®1

And: ((1- Q)®supps>.)(Q®Q + (1- supps>,)®(1- supps>,))((1- Q)®supps>,) ~ (1- Q)®supps>. which leads to: (1- Q) ® supps>, = 0 which is impossible, Q being different from 1, and supps>. different from 0. Therefore we must have S>.(Q) = 0, which is Q ~ 1-supps>. and it completes the proof of (i). Let now Q be in 'h· As £1 is a Jordan isomorphism, i1(Q) is a projector of W*(OC2). Moreover it is not equal to 1, because i1(Q) = 1 would obviously be equivalent to Q = 1. Now, let 6 and 61 be two positive elements of B(OC2). We have: {s7r2X7r2(il(Q)),6 ® 6') = {s71"2x71"2(i(Q)i(1)*),6 ® 6} = {s7r2 x7r2 (i(Q))(i(1)* ® £(1)*), 6 ® 61} = {s7r2 x7r2 (i(Q)),i(1)* · 6 ® £(1)* · 6') = {.i(Q), (£(1)*. 6) * (£(1)*. 61 )} = {Q, T((£(1)* · 6 * (£(1)* · 6'))) = (Q, T(£(1)* · 6) * T(£(1)* · 61 ))

by 5.4.1 by 5.6.1 (i)

by hypothesis

5.6 Isometries of Fourier-Stieltjes Algebras

187

= (s1r2 x1r2 (Q), T(£(1)* ·B)® T(£(1)* · B')) ::; (Q ® Q, T(£(1)* ·B)® T(£(1)* · B'))

by asswnption

= (Q, T(£(1)* · B))(Q, T(£(1)* · B'))

== (Q, (il).(B))(Q, (£1).(8'))

by 5.6.1 (iii)

= (il(Q),B)(il(Q),B')

= (il(Q)®il(Q),B®B') So, we get:

and therefore el(Q) belongs to Q2. Therefore we get i1(Q1) C Q2. As £1 is bijective, we could prove i1 1(Q2) C Ql the same way, which completes the proof of (ii). 5.6.3 Proposition. With the above notations, we have:

(i) (ii)

(iii)

£1(1-supps_x 1 ) = 1-supps_x2 i(Ker s _x 1 ) = Ker s _x 2

T(A(K2)) = A(JK1) .

Proof. Let us apply 5.6.2 to prove (i), considering that £1 preserves the order. The ideal Ker s _x 1 is generated by the projection 1 - supp s _x 1 • Let x in W*(JKI)· As £1 is a Jordan isomorphism (5.4.2), we have: i1(x(1- supps_x1 )

= !C£1(x)i1(1- supps_x1 ) = i1(x)(1- supps_x 2 )

+ £1(1- supps_x1 )i1(x)) by (i)

As £1 is bijective, we get £1 (Ker s _x 1 ) = Ker s _x 2 • And £(1) being unitary and Kers_x 2 a bilateral ideal, it completes the proof of (ii). Let Bin B(JK2); by (ii), T(B) vanishes over Kers_x 2 if and only if B vanishes over i(Ker s _x 1 ) = Ker s _x 2 , which gives the result, thanks to 3.3.4. 5.6.4 Notations. The restriction of T to A(lK2) satisfies the hypothesis of 5.5.10. There is thus an element u of the intrinsic group of OC2 and an llllisomorphism ~from lK2 to lK1 or lKl such that, for all Bin A(JK2), we have:

Let us determine u more accurately; the mapping ( s _x 1 ) ; 1T( s _x 2 ). is an isometric linear bijection from (M2)• to (MI)• which shall be denoted by T.

188

5. The Category of Kac Algebras

Let l : M1 -+ M2 its transposed. By 5.5.9, we get u = l(l ); and by transposing the relation T( s .x 2 ). = (s .x1 ).T which defines T, we get that s.x 2 f = ls.x 1 , by definition off and l. Then, we have:

5.6.5 Lemma. With the above notationll, we have, for all() in B(K2):

Proof. To simplify, we shall put 'Y

=7

8 >. 2

(l( 1)). In 5.6.4, the above relation has

been proved for() in A(K2). Now let win M2•· Let us recall that (s.x 2 )*(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:

or:

and, by using 5.6.4 again:

which, by 4.6.9 (ii), can also be written:

by having .X2(w) converging to 1, we complete the proof. 5.6.6 Theorem. Let K1 and K2 be two Kac algebrall. We allllume that there ezilltll a multiplicative, illometric, linear, bijective mapping T from the FourierStieltjell algebra B(K2) on B(K1). Then, there ezilltll an ]8[-illomorphillm from K2 onto K1 or rrq. More precillely, iff lltandll for the tranllpolled ofT, we have: (i) The operator s.x 2 (f(l)) belongll to the intriruic group ofK2. (ii) There ill an 18[-illomorphillm iP from K2 onto K1 or K} (in the jirllt calle £1 ill a von Neumann algebra homomorphillm from W*(K1) to W*(K2), in the llecond calle it is an anti-homomorphism) such that, for all() in B(K2), we have:

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 IJi be a normal isomorphism from W*(:K1) onto W*(:K2) such that:

(i.e. such that IJi respects the canonical coproduct of W*(:Kl) and W*(K2)). Then, there exists an lEn-isomorphism !I from :K2 onto :K1 such that, for all w in M1., we have:

We have also:

Proof. Let us apply 5.6.6 to the transposed mapping .P. = B(:K2) -+ B(:Kl). As IJi is multiplicative, we are in the first case, furthermore, as .P(1) = 1, there is an lEn-isomorphism !I from :K2 onto :K1 such that, for all 8 in B(:K2), we have: Because !JI'i:2

= l'i:l !J, it can also be written:

Therefore, for all win Mh, we have: {1Ji('11"1(w)),8}

= {w,'ll"h'I/J•(8)} = {w,!J'II"2•(8)}

= (w o !1, '~~"2•( 8)} = {'11"2(W 0 !J), 8} which gives the first result. We can see that: s;r2 1Ji'11"1(w)

= s;r2 '11"2(w o !I)= 1i"2(w o !I)= '11"2(w o !I o l'i:2) = '~~"2(w o l'i:l o !J) = !Ji11"1(w o l'i:l)

= !Ji1i"1(w) = !Jis;r1 '~~"l(w) which gives the second result, by the ultraweak density of '11"1(M1.) in W*(:K1).

190

5. The Category of Kac Algebras

5.6.8 Corollary. Let IK1 and IK2 be two Kac-algebra,, tJi a IK-isomorphism from IK1 to IK2 (i.e. an lH!-isomorphism from W*(JK1) to W*(K2)). Then there exists an lH!-isomorphism u from (Mt. Ft.K1) to (M2, F2, 11:2) such that tJi is the extension of u. Therefore, IK-isomorphisms are lH!-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 compact groups. Let T a multiplicative, linear, bijective mapping from M 1( G1) to M 1 (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 M 1(G1) we have:

Proof. By 4.4.1 {ii), the algebra M 1(G1) is the Fourier-Stieltjes algebra associated to the Kac algebra IK8 {G1)· Let us recall that, by 3.6.12, the intrinsic group of 1Ka(G1) is composed of the characters on G1. Therefore, by using 5.6.6, we see that there is a character x' on G1 and an lH!-isomorphism ~ from IK8 {G1) to IK8 {G2) (because IK8 (G2)~ = IK8 {G2)) such that, for all p. in M 1(G1), we have: >.a2 (T P.) = ~(f3x' >.al (p.)) We easily compute that for all p. in M 1(G1) we have:

(**) On the other hand, by 4.3.5, there is a bicontinuous isomorphism a 1 from G1 to G2 such that, for all s in G1:

By integrating, we find, for all p. in M 1(G1):

(***) Going back to ( *), we have:

>.a2 (T P.) = ~( >.al (x' P.)) = >.a2 (a'(x' P.) and therefore:

Tp. = a 1 (~~:1 p.) = (~~:' o a 1 - 1 )(a1 (p.))

We finally reach the result by writing

x = x' o a'- 1 and a

=

a'- 1 .

5.6 Isometries of Fourier-Stieltjes Algebras

191

5.6.10 Corollary (Walter's Theorem [194]). Let G1 and G2 be two locally compact groups. Let T be a multiplicative, isometric, linear, bijective mapping from B(G1) to B(G2)· Then there ezists: {i) an elements in G1 (ii) a bicontinuous isomorphism a: from G2 to G1 or to ct;PP such that, for all t in G2 and f in B(G1), we have: (Tf)(t)

= f(s- 1 a:(t))

.

Proof. By 1.6.3 (iii), up to the Fourier-Stieltjes representations, we have B(Gi) = B(Ka(Gi)) (i = 1,2). Therefore, applying 5.6.6, we get the existence of an element u in G(K8 (G1)) and an E-isomorphism iP from Ea(Gl) to E 0 (G2) or Ea(G2)~ = E0 (G~PP), such that for all fin B(G1), we have:

By 4.3.2, there exists sin G1 such that u in L 00 (Gt) and almost all tin G1:

= Aa1 (s).

Then, we have for all

f

on the other hand, by 4.3.5, it exists a bicontinuous isomorphism a: from G2 to G1 or G~PP such that: iP(f) = f 0 0: (***) Going back to(*), we finally find, for all tin G1 and fin B(G1), that: (Tf)(t) = (f3>.a 1 (s)U))(a:(t))

= f(s- 1a:(t)) which completes the proof.

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

Let :K = (M,r,,,cp) be a Kac algebra, K = (M,i',k,cj;) the dual Kac algebra. We have seen that the modular operator ..1 = L1c,0 is the RadonNikodym derivative of the weight cp with respect to the weight cp o K ( 9.6. 7). So, it is just a straightforward remark to notice that cp is invariant under K if and only if cj; is a trace. Moreover, the class of Kac algebras whose Haar weight is a trace invariant under K is closed under duality ( 6.1..4). These Kac algebras are called "unimodular" because, for any locally compact group G, the Kac algebra :Ka( 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 cp a finite weight, then (M,T,K,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 (6.9.9). 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: with di = dim Hi < oo

M = ffiC(Hi) i

and the trace cp is then given by:

cp ( E¥xi)

= ~diTri(xi) I

where

Xi

belongs to C(Hi), and

Tri

is the canonical trace on C(Hi)·

6.1 Unimodular Kac Algebras

193

Moreover, we get, following Ocneanu ((109]), an existence theorem for a Haar trace in this case; let (M, r, ~~:) be a co-involutive Hopf-von Neumann algebra, such that M = $i£(Hi) with di = dim Hi < oo and some Hio equal to C; let p be the one-dimensional projector associated to Hio; if p gives a unity of the Banach algebra M*, and if F(p) satisfies a certain (quite natural) condition involving ~~:, then, there is a Haar trace t.p and ( M, r, ~~:, t.p) is a Kac algebra of discrete type (6.9.5). This result appears, then, to be, in the non-commutative case, the analog of Krel'n's matrix block algebras (6.4.5), and, so leads to Krel'n's duality theorem (6.4.6). More generally (6.5.2), we can associate to each co-involutive Hopfvon Neumann algebra a discrete type Kac algebra (or, by duality, a compact type Kac algebra); in the group case, we recover Bohr compactification of locally compact groups ( 6.5.4 ). We then also get an existence theorem of a Haar state in the compact type case (6.5.8). At last, we get, after Kac ((69]), an easy result ( 6.6.1) which strengthen the analogy with locally compact groups: Kac algebras which are both of compact and discrete type are finite dimensional (and vice versa). Then, following Kac and Paljutkin ((75]), we give an existence theorem for a Haar state on a finitedimensional co-involutive Hopf-von Neumann algebra ( 6. 6.4 ). This last result makes the link ((110]) with the algebraic Hopf algebra theory, as exposed in (1] or (154]. For other specific results about finite-dimensional Kac algebras, we refer to (74], (75], (110], (71], (72], (4].

6.1 Unimodular Kac Algebras 6.1.1 Lemma. Let ( M, r, 11:) be a co-involutive Hopf-von Neumann algebra, t.p be a faithful, semi-finite, normal trace on M (we consider elements of M as operators on Hcp)· For all x in 'Jtp, we shall consider the element Wz of M* defined for all y of M by: Wz(Y)

= t.p(xy) = t.p(yx)

Then: (i) For all XI. x2 in 'Jtp, we have:

Therefore the set {wz, x E 'Jtp} is dense in M*. (ii) For all x in IJtcp, the element Wz belongs to lcp and we have, with the notations of 2.1.6:

194

6. Special Cases

(iii} For all

X

in mcp, we have with the notations of 1.1.1 (ii):

(iv) For all w in

M.,

we have, with the notations of 1.1.1 (ii) and 2.1.6:

(v) If

.(w,.)>.(w 11 )• belongs to I,;p. We have also: W>.(w,.)>.(w 11 )• = WAV>(>.(w,.)),AV>(>.(w11 ))

by 6.1.1 (i)

= Wa(w,.),a(w 11 )

by 3.5.4 (ii)

= wAtp(z),Atp(y)

by 6.1.1 (ii)

Therefore we have: a(wA'P(z),A'P(y))

= a(W>.(w,.)>.(w11 )•) = A,;p(.X(wz).X(wy)*)

= a(wz*W~) = a(wz*W~~:(y)•)

by 6.1.1 (ii) applied to~ by 3.5.4 (ii) by 6.1.1 (v)

and it completes the proof. 6.1.6 Proposition. Let G be a locally compact group. The following assertions are equivalent: (i} The group G is unimodular. (ii} The Haar weight 'Pa ofKa(G) is Ka-invariant. (iii} The Kac algebra lKa(G) is unimodular. (iv) The Kac algebra lK8 (G) is unimodular. (v) The Haar weight cp 8 of lK8 ( G) is a trace. Proof. The equivalence of (i) and (ii) results from the definitions of the Haar weight on Ka(G) and of the unimodularity of G. The equivalence of (ii) and (iii) and of (iv) and (v) respectively are mere applications of the definition 6.1.3. Finally the equivalence of (iii) and (iv) is a corollary of 6.1.4.

6.1. 7 Proposition. Let lK = (M, r, "'• cp) be a K ac algebra. The following assertions are equivalent: (i) The weight cp ill 8trictly 8emi-finite (in the 8en8e of [15]}.

6.2 Compact Type Kac Algebras

197

(ii} There is a sub-Kac algebra of :K which is a trace Kac algebra. (iii) There is a reduced Kac algebra ofJK which is an invariant weight Kac algebra. (iv) The sub-algebra M'P is a sub-Kac algebra of :K. Proof. For every Kac algebra, it is clear from (HWiii) that l'i.(M"') is equal toM"' and from 2.7.6(ii) that F(M"') is included in M"' ® M"'. Moreover, we know, by (15), that r.p is strictly semi-finite if and only if the restriction of r.p to M'P is a semi-finite trace. Therefore, we see, by using 2.7.7, that r.p is strictly semi-finite if and only if M'P is a trace sub-Kac algebra of lK. Thus, (i) implies (iv) which implies (ii). Conversely, let us assume (ii) and denote by CM, K;, cp) the trace sub-Kac algebra of lK. We have, for any X in Mandt in R, ui(x) = ur(x) = X and therefore M is included in M'P, which implies that the restriction of r.p to M'P is semi-finite, so, r.p is strictly semi-finite. The equivalence between (ii) and (iii) immediately results from 6.1.2 and 3.7.9(ii) and 3.7.10.

r,

6.1.8 Corollary. Let G be a locally compact group. The weight r.p 8 on C( G) is strictly semi-finite if and only if there exists an open subgroup of G which is unimodular.

Proof. By 6.1.7, the weight r.p 8 will be strictly semi-finite if and only if there exists a reduced Kac algebra of :K8 (Gr (i.e. of :Ka(G) by 4.1.2) admitting an invariant weight. As :Ka(G) is abelian, it is a trace Kac algebra; then the assumption is equivalent to the existence of a reduced Kac algebra of :Ka(G) being unimodular. By 4.3.6 (ii), it is equivalent to the existence of an open subgroup H of G such that :Ka(H) is unimodular which is, in turn, by 6.1.6 equivalent to H being unimodular.

6.2 Compact Type Kac Algebras

r, "')

6.2.1 Theorem. Let (M, be a co-involutive Hopf von Neumann algebra. Let r.p be a finite faithful normal weight on M such that, for all x, y in M, we have: ( i ® r.p )((1 ® y*)r(x )) = "'( i ® r.p )(F(y*)(1 ® x ))

Then, ( M, r, "'' r.p) is a unimodular K ac algebra. Such a K ac algebra will be called of compact type. We have then: W*C()

C()*W = w(1)r.p r.po = r.p •

=

198

6. Special Cases

= 1 in the above formula,

Proof. Putting y

it comes:

(i ® cp)(r(x)) = cp(x)1 applied to

~~:( x ),

it gives:

cp o ~~:(x)1 = (i ®

cp)F(~~:(x)) =

(i ® cp),(~~: ®

~~:)F(x) =

(cp o 11: ® i)F(x)

and:

cp o ~~:(x)cp(1) = (cp o 11: ® cp)r(x) = cp(x)cp o ~~:(1) = cp(x)cp(1) therefore cp o 11: = cp and cp 0 = cp. Let tin lR, we have:

rar = (i ® af)r rar = (af ® i)r

by 2.5.6 by 2.7.5 (i) because cp 0 II:= cp

therefore, we get: ra~

= (af ® af)r

On the other hand, it results from(*) that (cp®cp)F(x) = cp(x)cp(1) for all x in M, and as F(M) is aj®r.p invariant, by 2.7.6 (iii), we have:

a~r.pr=rar we finally get ra~ = rar' and, r being injective, it implies a~ = af and then, for all t in JR, af = id. Therefore cp is a trace, the axiom (HWiii) obviously holds, and ( M' r, K, cp) is a Kac algebra, it is unimodular because cp is a ~~:-invariant trace. The formula W*C;? = w(1)cp is given by(*); using the involution and the fact that cp = cp 0 , we get C;?*W = w(1 )cp. 6.2.2 Theorem. Let G be a locally compact group. The following allllertionll are equivalent: {i) The group G ill compact. {ii) The Kac algebra Ka(G) ill of compact type. Proof. It is trivial.

r,

6.2.3 Lemma. Let ][{ = (M' K, cp) a compact type K ac algebra llUCh that = 1. Then there exilltll an illometry I from Hr.p to Hr.p ® Hr.p lluch that, for all x, y, z in M, all w in Ir.p, all e, TJ in H, we have:

cp(1) {i) (ii) {iii)

IAr.p(x) = Ar.p®r.p(F(x)) I*(Ar.p(Y) ® Ar.p(z)) = a(wy*Wz) (.A(w)e ITJ) = (Ia(w) ITJ ® Je).

6.2 Compact Type Kac Algebras

199

Proof. As, by 6.2.1 (•), we have (cp®cp)F(x) = cp(x) for all x in M, by density and polarization, we can define a unique isometry I from Hcp to Hcp ® Hcp verifying (i). Then, we get: (I*(Acp(y) ® Acp(z)) I Acp(x)) = (Acp(y) ® Acp(z) I IAcp(x)) = (Acp(Y) ® Acp( z) 1Acp®cp(r( x))) = (cp ® cp)(F(x*)(y ® z)) = (F(x*),wy ®wz) by 6.1.1 = (x*,wy*Wz} by 2.1.6 (ii) = (a(wy*Wz) IAcp(x)) which gives (ii). We have:

(A(w )Acp(Y) I Acp(z)) = (A(w )*, wA (ii) is obvious, the result is proved.

JK.,( G), i.e. to the set {Aa( 8 ),

6.3 Discrete Type Kac Algebras 6.3.1 Definition. Let K = (M,r, ,;,,cp) be a Kac algebra. It shall be said of discrete type when the algebra M* is unital. 6.3.2 Theorem. Let G be a locally compact group. The following assertions are equivalent: {i) The group G is discrete. {ii) The Kac algebra Ka(G) is a discrete type Kac algebra. Proof. It is well known that G is discrete if and only if the algebra £ 1 ( G) has a unit (1.1.3). 6.3.3 Theorem. Let]({ be a Kac algebra. The following assertions are equivalent: (i) The Kac algebra]({ is of discrete type. {ii) The Kac algebra K is of compact type.

6.3 Discrete Type Kac Algebras

209

Proof. Let us assmne (i). Because of M* being unital, so is .A(M.); let e be the unit of .A(M.). We have xe = x for all x in .A(M.); by having x strongly converging to 1, we get 1 = e, and so 1 belongs to .A(M.). And, as Icp n I~ is dense in M. and .A norm-continuous, we get that 1 belongs to the norm closure of .A( Icp n I~) and therefore to the norm closure of 'Jl

'il

"'"'"

Using once more (*), we get: r(p)(1 ® 4,;) = (Ed"f 1 i'

E~~:ce~,q) ® e~,p) c1 ® 4,;) p,q

212

6. Special Cases

and then:

r(ef:n)(1 ® 4,j)

= r(e::,r)(1 ® ei,j)

= di'(D,;;,

ci'

o K ® i ® i)((F ® i)F(p)(1 ® 1 ® ei;)) by ( ***)

=di'(D,;;,

ci'

o~~:®i®i)(F®i)(F(p)(1®eL 3·))

C.,m,~l

'-m'''

=

di,di 1

'

,

L(De~,ei' 0 K ® i)F(~~:(eL,q)) ® e~,j q

by the above computation This implies:

(i ®a.n.neeB (N.Yu. Reshetikhin, L.A. Takhtadzhan, L. D. Faddeev): KB&HTOB&HHe rpynn H anre6p JIH. Anre6pa H aaaJIH3 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, Craiova, 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 modeles a vertex de V. Jones en theorie des noeuds. Note C.R.A.S. Paris 307(1988), 207-210.

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252

[148] [149] [150] [151] [152] (153] [154] [155] (156] [157] [158] [159] [160] [161] [162] [163] [164]

[165]

[166] [167] [168] [169] [170]

[171]

Bibliography ero npHn. 24-3(1990), 92-93; translated in: Gel'fand-Naimark-Segal states and Weil group for the quantum group SU(n). Funct. Anal. appl. 24(1990), 253-255. W.F. Stinespring: Integration theorems for gauges and duality for unimodular locally compact groups. Trans. Amer. Math. Soc. 90 (1959), 15-56. E. St!yHKI{. aHaHH3 Hero UpHn. 8-1(1974), 7576; translated in: Characterization of objects dual to locally compact groups. Funct. Anal. appl. 8(1974), 66-67. JI.H. BaituepMaH (L.I. Valnerman):)) KoHe'IHble rHnepKoMnneKCHble cHcTeMbl Kana. Cl>yHKn. auanH3 H ero npHn. 17-2(1983), 68-69; translated in: The finite hypercomplex systems of Kac. Funct. Anal. appl. 17(1983), 135-136.

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254

[186]

[187] [188] [189] [190] [191] [192]

[193] [194] [195] [196] [197] [198] [199] [200] [201] [202] [203] [204] [205] [206]

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Index

Abelian co-involutive Hopf-von Neumann algebra Achieved left Hilbert algebra Affiliated . . . Analytic element Antipode . . .

. . 1.2.5 2.1.1 (iii) 1.1.1 (ii) 2.1.1 (iv) . . 6.6.3

Bochner's theorem

4.4.4 (ii)

C*-algebra Coextension of a lHI-morphism Co-involutive Hopf-von Neumann algebra Commutant Kac algebra . Compact type Kac algebra Complex Hopf algebra . Conditional expectation Connection relations

1.1.1 (i) 5.2.3 1.2.5 2.2.5 6.2.1 6.6.6 2.1.8 (ii) 3.6.1

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 lHI-morphism . Extension of a representation Eymard algebra . . . . Eymard's duality theorem Eymard's theorem

6.3.1 3.3.2 3.7.4 5.1.2 3.5.3 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 Generator

3.3.4 3.5.3 4.3.7 2.5.3 1.6.9 1.6.9

4.3.7 2.4.2 1.5.2

llll-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)

Induced von Neumann algebra Intrinsic group . .

1.1.1 (ii) 1.2.2

Johnson's theorem

4.6.2 4.6.1 6.6.6 6.6.6 1.2.1

5.6.9

Kac algebra . . . Kac-Paljutkin's theorem :K-morphism Kre'ln algebra Kre'ln's theorem Kronecker product Kubo-Martin-Schwinger condition Left Hilbert algebra . . . Left-invariant weight Left regular representation Modular automorphism group Operator-valued weight Opposite Kac algebra Pentagonal relation

4.6.7

.

2.2.5

6.6.5 5.1.1 6.4.2 6.4.6 1.4.1 2.1.1 (iv) 2.1.1 (iii) 2.2.1 . . 1.1.5 2.1.1 (iv) 2.1.8 (i) 2.2.5 2.4.4

Index

Peter-Weyl's theorem Plancherel weight Pontrjagin's duality theorem Positive definite elements Radon-Nykodim derivative of weights Reduced Kac algebra . . . . Reduced von Neumann algebra Representable . . . . . . .

257

6.2.7 (iv) . 3.6.11 4.3.8 . 1.3.1 . 2.1.1 (v) . . 2.2.6 1.1.1 {ii) 1.3.6

Square-integrable element of M. Standard von Neumann algebra Stone's theorem Strict l!ll-morphism . . . . . Sub-Kac algebra . . . . . Symmetric co-involutive Hopf-von Neumann algebra

2.1.6 1.1.1 (iii) 4.4.4 {iii) 5.3.1 2.2.7 1.2.5

'Thnnaka's theorem Takesaki's theorem Tatsuuma's theorem Tensor product of von Neumann algebras Tensor product of weights . . . . . Tensor product of operator-valued weights

6.2.8 4.2.4 4.7.4 1.1.1 {ii) 2.1.1 (iv) 2.1.8 (iii)

Unimodular Kac algebra

. 6.1.3

Von Neumann algebra

1.1.1 {i)

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

1.1.1 {i) . 5.5.12 . 5.6.10 2.1.1 (i) . 4.2.6 . 4.5.9 . 5.5.11