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Michel Enock Jean-Marie Schwartz
Kac Algebras and Duality of Locally Compact Groups Preface by Alain Connes
Postface by Adrian Ocneanu
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Michel Enock Jean-Marie Schwartz
CNRS, Laboratoire de Mathematiques Fondamentales Universite Pierre et Marie Curie 4 place Jussieu F-75252 Paris Cedex 05, France
The sculpture reproduced on cover and frontispiece is exhibited at the Museo de antropologia de la Universidad Veracruzana, Jalapa, E. U. de Mexico
Mathematics Subject Classification (1980): 22 D 25, 22 D 35, 43 A 30, 43 A 65
ISBN 3-540-54745-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54745-2 Springer-Verlag New York Berlin Heidelberg
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41/3140-5 4 3 2 10 - Printed on acid-free paper
To Professor Jacques Dixmier "What's the matter?" Macbeth (11,2)
The question is the story itself, and whether or not it means something is not for the story to tell. Paul Auster (City of glass)
Preface
This book deals with the theory of Kac algebras and their duality, elaborated independently by M. Enock and J.-M. Schwartz, and by G.I. Kac and L.I. Vajnermann in the seventies. The subject has now reached a state of maturity which fully justifies the publication of this book. Also, in recent times, the topic of "quantum groups" has become very fashionable and attracted
the attention of more and more mathematicians and theoretical physicists. One is still missing a good characterization of quantum groups among Hopf algebras, similar to the characterization of Lie groups among locally compact groups. It is thus extremely valuable to develop the general theory, as this book does, with emphasis on the analytical aspects of the subject instead of the purely algebraic ones. The original motivation of M. Enock and J.-M. Schwartz can be formulated as follows: while in the Pontrjagin duality theory of locally compact abelian groups a perfect symmetry
exists between a group and its dual, this is no longer true in the various duality theorems of T. Tannaka, M.G. Krein,
W.F. Stinespring ... dealing with non abelian locally compact groups. The aim is then, in the line proposed by G.I. Kac in 1961 and M. Takesaki in 1972, to find a good category of Hopf algebras, containing the category of locally compact groups and fulfilling a perfect duality. It is natural to look for this category as a category of Hopf-von Neumann algebras since, first, by a known result of A. Weil, a locally compact group G is fully specified by the underlying abstract group with a measure class (the class of the Haax measure), and, second, by a result of M. Takesaki, locally compact abelian groups correspond exactly to co-involutive Hopf-von Neumann algebras which are both commutative and cocommutative. A co-involutive Hopf-von Neumann algebra is given by a morphism 1' : M --p M 0 M of a von Neumann algebra M
VIII
Preface
in its tensor square M ® M and a co-involution n which together turn the predual M* into an involutive Banach algebra. A Kac algebra is a co-involutive Hopf-von Neumann algebra with a Haar weight, i.e. a semi-finite faithful normal weight on
M which is left-invariant in a suitable way. In this book, the theory of Kac algebras and their duality is brought to a quite mature state, relying a lot on the modular theory of weights developped also in the seventies. The resulting category of Kac algebras fully answers the original duality problem, but is not yet sufficiently non-unimodular to include quantum groups. This of course opens a very interesting direction of research, undertaken recently by S. Baaj and G. Skandalis.
Paris
Alain Connes
Table of Contents
Introduction
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Chapter 1. Co-Involutive Hopf-Von Neumann Algebras 1.1
1.2 1.3 1.4
1.5 1.6
Von Neumann Algebras . . . . . . and Locally Compact Groups Co-Involutive Hopf - Von Neumann Algebras . Positive Definite Elements in a Co-Involutive . . Hopf-Von Neumann Algebra . . . . Kronecker Product of Representations . . . Representations with Generator . . . . . Fourier-Stieltjes Algebra . . . . . . . .
Chapter 2. Kac Algebras
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. . An Overview of Weight Theory . 2.2 Definitions . . . . . . . . . . . 2.3 Towards the Fourier Representation . 2.4 The Fundamental Operator W . . . 2.5 Haar Weights Are Left-Invariant . . . 2.6 The Fundamental Operator W Is Unitary 2.7 Unicity of the Haar Weight . . . . .
2.1
Chapter 3. Representations of a Kac Algebra; Dual Kac Algebra . . . . . . . . . . 3.1
3.2 3.3 3.4
3.5 3.6 3.7
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8 13
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19 23 30 36
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44
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45 55 58 60 66
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71
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76
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83
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84 89 92 97 101 104 111
The Generator of a Representation . . . . The Essential Property of the Representation A The Dual Co-Involutive Hopf- Von Neumann Algebra . . . . Eymard Algebra . . . . . . . . . Construction of the Dual Weight . . . Connection Relations and Consequences The Dual Kac Algebra . . . . . . .
1
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Table of Contents
X
Chapter 4. Duality Theorems for Kac Algebras and Locally Compact Groups . . . . . .
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Duality of Kac Algebras . . . . . . . . . 4.2 Takesaki's Theorem on Symmetric Kac Algebras 4.3 Eymard's Duality Theorem . . . . . . . for Locally Compact Groups 4.4 The Kac Algebra Ks (G) . . . . . . . . . 4.5 Characterisation of the Representations . . . . . . . . and Wendel's Theorem . 4.6 Heisenberg's Pairing Operator . . . . . . . 4.1
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5.1
5.2 5.3 5.4 5.5 5.6
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6.1
Unimodular Kac Algebras
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6.2 Compact Type Kac Algebras 6.3 Discrete Type Kac Algebras
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161
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162 166 172 174 176 184
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192
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Krein's Duality Theorem . . . . . . . . . 6.5 Characterisation of Compact Type Kac Algebras 6.6 Finite Dimensional Kac Algebras . . . . . .
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Bibliography
Index
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193 197 208 213 219 232
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6.4
Pos tf ace
136 140
144 152 158
Kac Algebra Morphisms . . . . . . . . H-Morphisms of Kac Algebras . . . . . . Strict H-Morphisms . . . . . . . . . . Preliminaries About Jordan Homomorphisms Isometries of the Preduals of Kac Algebras Isometries of Fourier-Stieltjes Algebras . .
Chapter 6. Special Cases: Unimodular, Compact, Discrete and Finite-Dimensional Kac Algebras
125 130
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4.7 A Tatsuuma Type Theorem for Kac Algebra Chapter 5. The Category of Kac Algebras
124
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243
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245
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255
Introduction
In the early nineteen thirties, L.S. Pontrjagin ([121]) established his famous duality theorem for abelian locally compact groups: he showed that the set of characters of an abelian locally compact group G is an abelian group,
called the dual group of G, and noted G, which is locally compact for a suitable topology. Moreover, the bidual G^^ is canonically isomorphic and homeomorphic to G; the Fourier transform carries the convolution algebra L1(G) into the multiplication algebra L' (G"'); conversely, the Fourier inverse transform carries the convolution algebra L1(G) into the multiplication algebra L°O (G) . These two transformations are transposed of each other in the following scheme: Fourier
L1 G (
)
LOO(G)
Fourier inverse
L°O( G )
L1 (G)
Since this fundamental result, which can be considered as one of the founding theorems of abstract harmonic analysis, a series of duality theorems for nonabelian locally compact groups has been gradually set up, but these dualities were not formulated within the category of locally compact groups. In 1938, T. Tannaka ([166]) proved a duality theorem for compact groups, involving the irreducible representations of a compact group G (one should bear in mind, that, in the case of an abelian locally compact group, the characters are the irreducible representations); although no group-like structure is to be put on that class (in particular, the irreducible representations being finite-dimensional, the tensor product of two irreducible representations is still finite-dimensional, but may no longer be irreducible), Tannaka showed
that it is possible to recover the compact group G from the set of its irreducible representations. In 1941, M.G. Krein ([82]) obtained an equivalent result, and, in 1949, ([83], [84] ), he took the system of matrix algebras generated by these repre-
sentations as the dual of the compact group G, and, without invoking the group G, gave an intrinsic description ("matrix block algebras") of these objects.
2
Introduction
The next step was done in 1959 by W.F. Stinespring's duality theorem for unimodular locally compact groups in [148]. There, Stinespring emphasizes the crucial role played by the unitary operator W on L2 (G x G) defined by:
(Wf)(s,t) = f (s, st)
(s,t E G; f E L2(G x G))
and by the left regular representation AG of G on L2 (G), which are linked by:
W*(AG(s) ®1)W = AG(S) ® AG(S)
(s E G)
The operator W allows us then to define a Hopf structure on the von Neumann algebra £(G) generated by the representation AG. Stinespring showed how it is possible to recover the group G from this Hopf structure. One must notice that this unitary W defines, too, a coproduct on L°O(G) by the formula:
(W(1® f )W*)(s, t) = f(st)
(s, t E G; f E L°O(G))
Then, in 1961, G.I. Kac ([66], [70]) introduced the notion of "ring-group", closely related to Hopf algebras; he gave a satisfactory abstract framework to Stinespring's results, and was again in a situation similar to Pontrjagin's theorem, where the initial objects and their duals are both of the same kind. These "ring-groups" generalize the algebra L' (G), for a unimodular locally compact group G, as well as the von Neumann algebra ,C(G), these two objects, for a given G, being dual to each other. In 1964, P. Eymard ([46] ), at last, gave a duality theorem valid for every locally compact group, even non-unimodular: taking an arbitrary pair f , g in the Hilbert space L2 (G), the convolution product f * gb (where gb (s) is equal to (_1) for all s in G) belongs to LOO (G), and the functions of that type form an involutive algebra A(G), called the Fourier algebra of G. With a suitable norm, it becomes an involutive Banach algebra, whose dual is the von Neumann algebra ,C(G). Moreover, the spectrum of A(G) is equal to G. In order to explain the importance of that theorem, let us assume again that G is abelian. Then, the Fourier-Plancherel transform carries respectively L2(G) onto L2(G), the left regular representation of L1(G) onto the Fourier tranform, and the von Neumann algebra £(G) onto the von Neumann algebra L°O(G), acting by multiplication on L2(G). So, the predual A(G) is isomorphic to the convolution algebra L1(G), whose spectrum is, by definition, the bidual G^^. By dealing with this particular case, Eymard's theorem allows us to recover Pontrjagin's duality theorem. In the general case, Eymard's theorem brings us to write down the following scheme, which is a generalization of the abelian case: AG
Gelfand transform
Introduction
3
In 1967, N. Tatsuuma ([167]. [168]) gave another duality theorem for arbitrary locally compact groups, recovering G, not as Eymard did, only from the left regular representation of G, but, in the spirit of Tannaka, from all the unitary representations of G. In [44], J. Ernest showed how Tatsuuma's theorem was closely related to the Hopf structure of the algebra ,C(G) introduced by Stinespring, and to the Hopf structure of the enveloping algebra W * (G) that he introduced in [43] and [44].
In 1968, M. Takesaki, using both Kac's work and these recent duality theorems, gave a characterization of group algebras: every co-involutive com-
mutative von Neumann algebra with a left invariant measure corresponds to a locally compact group ([157] ). This crucial result, making the link between the abstract constructions of Kac and the duality theorems for locally compact groups, is actually the keystone of the theory. Afterwards, Takesaki endeavoured, in 1972 ([159]), to complete Kac's programme for non-unimodular groups, i.e. to establish a general duality theory which would work inside a wider category than the one of locally compact groups, and would restrict itself to the Eymard-Tatsuuma-Ernest theorem for locally compact groups. Due to the incompleteness of non-commutative integration theory at that moment, he did not reach a perfect symmetry. This was done in 1973, independently by L.I. Vainermann and G.I. Kac ([170], [179], [180] ), and the authors ([34], [35], [36]) who named "Kac algebras" this wider category, because of Kac's 1961 basic work in that direction. The duality obtained may be sketched in the following scheme:
M* - M M4-M* where M and M are von Neumann algebras, their preduals M* and M* being involutive Banach algebras, and A and A non-degenerate faithful representations. Many improvements were made, by E. Kirchberg ([78], [79] ), J. De Canniere ([18] ), the authors ([135], [136], [38], [40] ), J. De Canniere and the authors ([21], [22]). In particular, was done in [22] the generalization of M.E. Walter's work, who had noticed ([193], [194]) in 1970, that the Banach algebra A(G) characterizes the group G (although A(G) is always abelian, even when G has not the same property), exactly like L1(G) does ([198] ), and had strongly indicated these two results should be just two particular cases of a stronger one (see also Akemann-Walter [2]). As of now, the theory of duality of Kac algebras, and the links with duality of locally compact groups, seem to have reached a state of maturity, which
is described in the present book. Surveys of Kac algebras theory may be found in [45], [20], [139] and [175]. For a historical view of abstract harmonic analysis, we refer to J.-P. Pier ([117] ).
4
Introduction
In the theory of Kac algebras, the crucial idea is the role played by the coproduct on L°O(G); it is well known that, for a locally compact semi-group,
the product (s, t) -- st is completely given by all the functions on G x G defined by 1'(f) : (s, t) -> f (st), for all f in a suitable space of functions on G (which separates the points of G, as, for example, C, ,(G) or L°O (G) for a suitable measure on G); if G is a group, the inverse s -+ s-1 will be given by all the functions ic(f) : s -+ f (s-1), for all f in C0(G) or L°O (G). Moreover,
it is known, thanks to a A. Weil's result ([197]), that the topology of the locally compact group G is completely given by its Haar measure. These simple remarks show how, on L°O(G), the coproduct, the co-involution, and the Haar measure give back the group G, both algebraically and topologically. Many important results have been obtained about actions of Kac algebras on von Neumann algebras and crossed-products (i.e. non-commutative dynamical systems) ([33], [37], [39], [103], [61], [106]) and are to be linked to the numerous works which have been made about duality and crossed products ([161], [156], [151], [102], [58], [88], [89], [188], [132], [190], [104], [153], [187], [77], [134], [139], [90], [60] ). In [37], the authors have proved a dual-
ity theorem for Kac algebras crossed-products, which generalized Takesaki's theorem ([161]) about crossed-products by group actions. In [107], [108], [60], [61], [106] is developped, by A. Ocneanu and others, a Galois theory for inclusions of von Neumann algebras, in which Kac algebras play a crucial role; are obtained intrinsic characterizations of crossed-products by a Kac algebra in term of position of the initial algebra in the crossed-product. More precisely ([106] ), if Mp C M1 is an inclusion of factors, with a faithful semi-finite normal operator valued weight from M1 to M0, then M1 is the crossed product of Mp by an outer action of some Kac algebra, eventually twisted by a cocycle, if and only if the relative commutant Mo n M1 is C and Mo n M3 is a factor (where Mp C M1 C M2 C M3 C ... is the canonical tower of von Neumann algebras constructed from the initial inclusion); these results open a wide range of opportunities for further study of Kac algebras. Other directions have been studied: amenability of Kac algebras ([112], [192], [41], [115], [116])); G*-Kac algebras ([111], [96], [187], [4], [42]), and actions of G*-Kac algebras ([187], [3], [4]). Precise examples of Kac algebras which are neither abelian, nor symmetric (i.e. which are neither groups, nor group duals) have been found and studied in [73], [74], [75], [19], [97], [4]. To exhibit and classify these examples, even in the finite-dimensional case, is a difficult problem, which is far from being solved. We refer to [4] for recent researches on that question. This theory, which remained in a kind of shade for some years, is nowadays
valuable to more and more mathematicians, first of all those who want to give an even more general framework for duality theory, for instance Yu. M. Berezanskii ([6], [7], [8], [9]), L.I. Vainerman ([182], [171], [172], [173], [174], [175], [181], [176], [178]), S.L. Woronowicz ([200], [201], [202], [203], [120], [204], [205], [206]), A. Ocneanu ([107], [108], [109]), S. Baaj and G. Skandalis
Introduction
5
([3], [4]), M.E. Walter ([195], [196]) or S. Doplicher and J. Roberts ([26], [27]) (we apologize to the others when we may have ommitted them). In the early eighties, many theoretical physicists from the Leningrad school
introduced the notion of quantum group, as it appeared in the quantum inverse scatterring transform method ([144], [140], [141], [143], [165], [85], [87], [137] ). A quantum group is generally considered to be mean a Hopf algebra obtained from a Lie group by deforming the envelopping algebra of its Lie algebra, in a way to obtain a non-commutative algebra. Historically, the first example of such a construction is to be found in [73], where Kac and Paljutkin gave a non-trivial infinite dimensional example of a Kac algebra. These ideas were developped in ([86], [142], [30], [63], [64] ), and the link was made with Lie bialgebras ([10], [5], [28], [29], [51]), mostly by V.G. Drinfel'd. This theory has become very fashionable, thanks to V.G. Drinfel'd's lecture at the ICM-86 ([31] ), and his Fields medal at the ICM-90, and many mathematicians are nowadays involved in quantum groups, from Yang-Baxter equation ([52], [53], [54], [55] ), Poisson-Lie groups ([91], [138] ), studying representations of quantum groups ([92], [93], [100], [101], [128], [80], [81]), or
making the link with a geometrical point of view ([99], [129], [169], [164], [123], [17] ). The non-commutative function algebras on quantum groups are studied in ([183], [184], [185], [145], [186], [146]). Other recent works are [32], [98], [122], [147], [119]. Constructing such objects in a C*-algebra setting has been done, mostly by S.L. Woronowicz ([200], [201], [202], [203], [204], [205] ); related works has been made by M.A. Rieffel ([125], [126] ), A. Van Daele ([189] ), P. Podles ([118] ), P. Podles and S.L. Woronowicz ([120]), S.L. Woronowicz and S. Zakrzewski ([206] ), I. Szymczak and S. Zakrzewski ([155] ). M. Rosso has shown ([127], [130]) the link between Drinfel'd's work and Woronowicz's "pseudo-groups" (see also J. Grabowski [50] ). For a survey of quantum group theory, we refer to [31], [191], [13], [131]. Let us now describe briefly the link between Kac algebras and Woronow-
icz's point of view. Roughly speaking, in Woronowicz's "compact matrix pseudo-groups" ([201], [205] ), the co-involution ic satisfies weaker axioms:
in Kac algebras, it is involutive (i.e. rc(x*) = rc(x )* for all x in M) and is an involution (i.e. rc2(x) = x for all x in M); in Woronowicz's algebras, ic is defined only on a dense sub-involutive algebra A and satisfies r.(,C(x )* )* = x
for all x in A. Woronowicz then obtains a left-invariant state, i.e. satisfying (i ®Sp)-P = co, which is not a trace. On the contrary, in Kac algebra theory, whenever the Haar weight is finite (we shall say then that the Kac algebra is "of compact type"), it is a trace, invariant by the co-involution (we shall say then that the Kac algebra is "unimodular" ). For the links (and the differences) between compact type Kac algebras and Woronowicz's compact matrix pseudo-groups, see [177].
6
Introduction
More generally, if a coproduct r on a von Neumann algebra M is given by a unitary W by the formula:
F(X) - w(l 0 X)yy* as it is the case for L' (G), then, the co-associativity of r leads to a pentagonal relation for W, which occurs both in Kac algebras and in pseudo-groups. This relation is the starting point of S. Baaj and G. Skandalis ([4]), where such unitaries are called "multiplicative". They define supplementary conditions to get "Kac systems", where duality theorems are proved. Both Kac algebras and Woronowicz's pseudo-groups are Kac systems, and, within Kac systems, pseudo-groups are characterized. So, the close relationship between Kac algebras, Woronowicz's work and quantum groups appears more and more often. We do hope our work will be useful for researches in that direction. We would like to express our profound gratitude to Jacques Dixmier, who oriented us to this subject and who always provided us with true help and
support. Special thanks are due to Alain Connes, who kindly accepted to write the preface of this book. We are indebted to Masamichi Takesaki, whose
work is the founding stone of the present one, and who gave us permanent encouragement. Thanks also to Adrian Ocneanu, who contributed most to bring back Kac algebras into bright light, and wrote a very encouraging postface. We are thankful to many other colleagues for fruitful exchanges, specially to Francois Combes for so many substantial conversations, and to the C.N.R.S. to which we both belong and which made this research possible. Last but not least, we are obliged to Ms. C. Auchet, who processed the typescript.
Chapter 1 Co-Involutive Hopf-Von Neumann Algebras
This chapter is devoted to the structure of co-involutive Hopf-von Neumann algebras, which has been introduced by Ernest ([44] ), and mostly studied by Kirchberg ([79] ), and de Canniere and the authors ([21] ). The paradigm, from which the whole theory comes, is the algebra L°°(G) of all the (classes
of) essentially bounded measurable (with respect to a left Haar measure) complex valued functions on a locally compact group G, equipped with a coproduct and a co-involution, which are nothing but the duals of the usual product and involution of the involutive Banach algebra L1(G) of all (classes of) integrable (with respect to a left Haar measure) complex valued functions on G (let us recall that L1(G) is the predual of L°°(G)). Other examples will be found later: the von Neumann algebra of G (1.6.8 ),
the von Neumann algebra generated by the left regular representation of G (3.3.6 ), the dual of the involutive Banach algebra M1(G) of bounded measures on G (4.4.1). A co-involutive Hopf-von Neumann algebra (1.2.5) is a von Neumann algebra M, equipped with a coproduct, i.e. a normal injective unital morphism r from M to the von Neumann tensor product M 0 M, which is co-associative, i.e. which satisfies:
(r®i)r=(i®r)r
and a co-involution x satisfying:
Fx=s(K0K)r where c(a 0 b) = b ® a for any elements a, b of M. The predual M* is then, in a natural way, an involutive Banach algebra, and so we may study the theory of continuous positive forms on it (called positive definite elements in M (1.3.1)) and the theory of representations of M*. In the case of L' (G), we recover the usual positive definite functions on G (1.3.11) and the unitary representations of G (1.1.6 ). Convenient morphisms (H-morphisms) for co-involutive Hopf-von Neumann algebras are defined and allow us to define a category (1.2.6).
8
1. Co-Involutive Hopf-Von Neumann Algebras
A crucial result is the construction of the Kronecker product of two representations, due to Kirchberg (1.4.2 ), which, in the case of L°O (G), is the tensor product of unitary representations of G (1.4.7) and may be considered as an important and useful tool for the general theory as well. The Kronecker product enables us to put, on the enveloping von Neumann algebra of M*, almost a structure of co-involutive Hopf-von Neumann algebra (eventually, we only miss the coproduct being non-degenerate; (1.6.5) and (1.6.6)). So, its predual (which is, too, the dual of the enveloping C*-algebra of M* ), is therefore an involutive Banach algebra, called the Fourier-Stieltjes algebra (1.6.9 ); it has a faithful representation in M, called its FourierStieltjes representation. In the case of L°O(G), we get B(G), the FourierStieltjes algebra of G, studied by Eymard ([46] ), and the canonical imbedding of B(G) into L°O(G). It is well known that non-degenerate representations of L' (G) are given by unitary representations of G. A similar situation, for a general co-involutive Hopf-von Neumann algebra, is given by the "representations with gener-
ator", which, after Kirchberg, are studied in 1.5. The particular case of finite-dimensional representations is especially studied, and will be useful in Chap. 6.
1.1 Von Neumann Algebras and Locally Compact Groups 1.1.1 Notations and Definitions. We refer to the usual text books about operator algebras ([105], [12], [24], [25], [133], [124], [113], [152], [162]); here follow some basic facts about C*-algebras and W*-algebras:
(i) A C*-algebra A is a complex Banach algebra, equipped with an antilinear involution x -+ x* such that (xy)* = y*x* and llx*xil = Iix112 for all x, y in A. Let H be a Hilbert space; we shall denote by NC(H) the involutive Banach algebra of all bounded linear operators on H equipped with the norm:
lxii =
6 E H, 11611 <_ 1}
the involution x --p x* being defined by, for all 6, ri in H:
(x*Ir7) =(6Ixri) Then £(H) and all its norm-closed involutive subalgebras are C*-algebras; conversely, by Gel'fand-Naimark's theorem, for every C*-algebra, there exists a Hilbert space H such that A is isomorphic to a C*-subalgebra of ,C(H). A W* -algebra M is a C*-algebra which is the dual of some Banach space; then, the above Banach space is unique, called the predual of M, and denoted
1.1 Von Neumann Algebras and Locally Compact Groups
9
M. The algebra £(H) is a W*-algebra, its predual ,C(H)* being the space of all trace-class operators, that is the set of elements of £(H) such that Iti = (t*t)1/2 satisfies trH(Itl) < +oo (where trH is the canonical trace on £(H)), equipped with the norm t --+ trH(Iti), whose dual is then the set of all the linear forms t -+ trH(xt), for all x in NC(H), and may be identified, as a Banach space, with NC(H). The topology cr(L (H), ,C(H)*) is called the ultraweak topology. A von Neumann algebra M on H is an involutive subalgebra
M of £(H), ultra-weakly closed, containing 1. It is easy to show that it is then a W*-algebra, the predual of M being then the Banach space of all the ultra-weakly continuous linear functionals on M. Conversely, by Sakai's theorem, for every W *-algebra, there exists a Hilbert space H such that M is isomorphic to a von Neumann algebra on H. (ii) In all what follows, M will denote a von Neumann algebra on a Hilbert space H, and M* its predual, Z(M) its centre, M+ (resp. M*) the positive
part of M (resp. M* ), M+1 (resp. M* 1) the subset of M+ (resp. M* ) composed of the elements of norm one. For all w in M* and x in M, to emphasize the fact that M is the dual of M*, we shall often write (x, w) instead of w(x). If y is in M and w in M*, we define w, y w and w y in M*, for x in M, by:
w(x) = w(x*)-
(x,y.) = (xy,w) (w . y)(x) = w(yx)
If , r,i are in H, we shall denote by Sl,., the linear form on £(H) defined by:
(x,Qe,) = (xe 10 If M is a von Neumann algebra on H, we shall still
and Q instead of fI . write fl 1,7 instead of fl c,7 I M, if there is no confusion. We have fly,,7 = f217 For any subset S of £(H), we define the commutant S' by:
S'={x, xE,C(H), xy=yxforallyinS} which is a von Neumann algebra on H. Then, by von Neumann's theorem, S" is the von Neumann algebra generated by S, that is the intersection of all von Neumann algebras on H containing S. If T is a closed operator on H, we say that T is affiliated to M if uTu* = T for all unitaries in M'. So, thanks to von Neumann's theorem, if T is bounded and affiliated to M, it belongs to M. If P is a projection in H, belonging to M, the reduced von Neumann algebra Mp is the set of all x in M such that Px = xP = x, and is a von Neumann algebra on the Hilbert space PH; its commutant will be the induced von Neumann algebra (M')p, whose elements are all the restrictions to PH of elements of M'. If Mi (i = 1, 2) is a von Neumann algebra on a Hilbert space Hi, the von Neumann tensor product M1 0 M2 is the von Neumann algebra on the tensor
10
1. Co-Involutive Hopf-Von Neumann Algebras
product Hilbert space H1 ® H2 generated by the algebraic tensor product M1 O M2. The algebraic tensor product (Ml) * ® (M2) * is then norm dense in the predual (Ml 0 M2)*. Moreover, by Tomita's theorem, we have:
(M1®M2)' =MGDM2 (iii) Let M be a von Neumann algebra on H; M is said to be standard in H ([57]) if there exists an antilinear bijective isometry J : H --+ H and an autopolar convex cone 2 in H such that:
JMJ=M' J = C for all C in P
xJxJ2CPforallxinM We shall say, eventually, that the triple (H, J, 2) is standard for M, or that (H, J) is standard for M. For example, the von Neumann algebra ,C(H), identified to C(H) ® CH, is standard in H ® H (where H is the conjugate
Hilbert space of H), thanks to the antilinear isometry J from H ® H to H ® H defined by J(6 (6,77 E H) and the closed convex cone P ri GD generated by {6 ®1; 6 E H}. In an obvious way, the von Neumann algebra ®iEIL(Hi) is standard in ®iEI(Hi ® Hi). Thanks to Haagerup's result, every von Neumann algebra can be represented on a Hilbert space in which it is standard, and, then, two technically very useful events occur: first, for all w in M*, there exist 6,77 in H such that w = 9k17 M. We shall write then instead of fflC,7 1 M, if there is no confusion. Secondly, every automorphism a of M is actually implemented by a unique
unitary u of ,C(H) (that is a(x) = uxu* for all x in M) such that uJ = Ju and uP C P. Then u is called the standard implementation of a. 1.1.2 Notations and Definitions. In all the chapters, G will denote a locally compact group; we refer to the usual text books about locally compact groups ([197], [11], [62], [12], [25]). We shall denote by Cb(G) (resp. C0(G), resp. K(G)) the algebra of bounded continous complex valued functions on G (resp. vanishing at the infinity, resp. with compact support). Let ds be a left Haar measure on G. We shall write L2(G) for the Hilbert space of all (classes of) square-integrable measurable (with respect to ds) complex-valued functions on G. Let LOO(G) denote the (abelian) W*-algebra of all (classes of) essentially bounded measurable (with respect to ds) complex-valued functions
on G. Let us recall that the predual of L' (G) is isomorphic to the Banach space L1(G) of all (classes of) ds-integrable functions over G. The essential results we shall need are recalled thereafter: (i) The von Neumann tensor product L' (G) ® L°° (G) may and will be identified with L' (G x G). For every f in L°° (G), we define a two variables function I'a (f): (s,t E G) ra(.f)(s, t) _ f(st)
1.1 Von Neumann Algebras and Locally Compact Groups
11
Thanks to the above identification, fa (f) belongs to LOO (G) ®L°° (G). Along the same way, we define rya (f) by the equality: wa(f)(s) = AS-1)
(sec)
Evidently, Fa and 1a are linear, multiplicative and preserve the adjoint. (ii) The Banach space L1(G) is a Banach involutive algebra, when equipped with the convolution product * and the involution ° respectively defined by,
f * g(t) = Ms)
JG
f (s)g(s-lt)ds
(f, g E Ll(G), s, t E G)
= aG(s-l)f(s-i)
(f E Ll(G),
S E G)
where AG is the modular function on G. It is known that L1(G) has a bounded approximate unit, and has a unit if and only if G is discrete; the unit of L1(G) is then the characteristic function of {e}, where e is the unit of G. (iii) Let M1(G) be the Banach space of bounded measures on G, which is dual to the Banach space C0(G). Then, M1(G) is a Banach involutive algebra, when equipped with the convolution product * and involution ° defined by, for m, n in M1(G), f in Co(G), where s, t belong to G:
m.n(f)= MOM =
J
f(st)dm(s)dn(t)
(JGj51m5i
It is known that L1(G) may be considered as a closed subspace of M1(G), a function f of L1(G) being identified with the bounded measure f (s)ds. Under that identification, L1(G) is a two-sided ideal of M1(G), and is equal to M1(G) if and only if G is discrete. It is known that L1(G) and M1(G) are abelian if and only if G is abelian. 1.1.3 Proposition. The morphisms 1'a and 'a defined in 1.1.2 verify, for any
fin L' (G), h, k in L1(G): (l'a(f), h ® k) =
fG
f(t)(hrk)(t)dt
(Xa(j)jh)- = IG f (s)h'(s)ds
So, Fa and rCa are normal.
12
1. Co-Involutive Hopf-Von Neumann Algebras
Proof. Let f be an element of L°O(G), h and k be two elements of Ll(G). We have:
IG f(t)(h * k)(t)dt =
(G h(s)k(slt)dsl dt
G
JGxG
/
f (t)h(s)k(s-lt)dsdt f(st)h(s)h(t)dsdt
JGxG GxG
Ta(f)(s) t)(h ®k)(s, t)dsdt
_ (I'a(f),h®k) Since the algebraic tensor product L'(G) 0 L1(G) is weakly dense in the predual of L°O (G) 0 L°° (G), this equation ensures the normality of Ta . With the same assumptions, we have also: f(s),Ac(s-')h(s-')ds
IG
= IG f (s)(s)ds = (a(f), n)
which brings the normality of 'a and completes the proof. 1.1.4 Definition. Let µG be a continuous unitary representation of G on some Hilbert space ?-l. Then it is well known ([25] 13.3) that the equality:
(f E Ll(G))
P(f) = fG f (s),UG(s)ds
defines a non-degenerate representation of L' (G) on 'H; we establish by this way a one-to-one correspondance between continuous unitary representations of G on ?-C and non-degenerate representations of L1 (G) on '. More generally, it is true as well that there is a one-to-one correspondance between continuous unitary representations of G and representations of M1 (G), whose restriction to Ll(G) is non-degenerate, given by: N(m) = fG
(m E Ml(G))
1.1.5 Definition. For all s in G, we define a unitary operator AG(s) on L2(G), by, for any f in L2 (G), t in G:
(AG(s)f)(t) = As-
it)
It is well known that AG is a continuous unitary representation of G, called the left regular representation of G.
1.2 Co-Involutive Hopf-Von Neumann Algebras
13
The corresponding representation of M1(G), called the left regular representation of M1(G), is given, for all m in M1(G), f in L2 (G), t in G, by:
(ac(+n)J)(t) =
G f (s-lt)dm
The corresponding non-degenerate representation of L' (G), called the left regular representation of L1(G), is given, for all f in L1(G), g in L2(G), tin G, by:
(A(f)g)(t) = JG f(s)g(st)ds We shall write £(G) for the von Neumann algebra generated by the representation AG of L'(G). If f is in L'(G), and g in L1(G) f1 L2(G), we have: AG(f)g = f * 9
1.2 Co-Involutive Hopf- Von Neumann Algebras 1.2.1 Definition. A couple (M, r) is called a Hopf-von Neumann algebra if: (i) M is a W*-algebra; (ii) I' is an injective normal morphism from M to the W*-algebra MOM, such that I'(1) = 1®1, and which has the co-associativity property, i.e. which makes the following diagram commute:
M
MOM
MOM
M®M®M r is then called a co-associative co-product. If (M, I') is a Hopf-von Neumann algebra, so is (M, cI') where c(a ® b) is equal to b 0 a for any elements a, b of M. The co-product I' will be called symmetric if SI' = F. For every x in M and every w, w' in M*, let us write: (x, w * w') = (I'(x), w ®w') This formula defines a product * on M* . As equipped, M* becomes a Banach algebra; it is abelian if and only if I' is symmetric.
14
1. Co-Involutive Hopf-Von Neumann Algebras
We have then, for all w, w1 in M* :
(w*w')-=w*w/ 1.2.2 Definition ([44]). Let (M, I') be a Hopf-von Neumann algebra. The set of all invertible elements x of M such that:
I'(x) = x ® x
is clearly a subgroup of the group of invertible elements in M, called the intrinsic group of (M, I'). 1.2.3 Proposition ([44]). The elements of the intrinsic group of a Hopf-von Neumann algebra are unitaries. Proof. If x is an element of the intrinsic group, so is x*x. Therefore, we have:
II='=II = I= IIx*x 0 ='=II = II='=IIZ which implies IIx*xII = 1. Since (x*x)_l belongs also to the intrinsic group,
we have J11 = 1 as well; x*x being a positive operator, the spectral theorem then implies x*x = 1. The same arguments work for xx* = 1, which completes the proof.
1.2.4 Remark. Since the weak, ultraweak, strong and ultrastrong topologies
coincide on the unitary group of a von Neumann algebra (cf. [24] ), the intrinsic group of a Hopf-von Neumann algebra is a topological group for this topology.
1.2.5 Definition. A triple H = (M, I', n) is called a co-involutive Hopf-von Neumann algebra if: (i) (M, I') is a Hopf-von Neumann algebra (ii) tc is an involutive anti-automorphism of M, i.e. a linear mapping from
M to M such that, for every x, y in M: tc(xy) tc(x*)
tc(Y)K(X)
tc(x)*
K(K(X)) = x
ic is then called a co-involution.
1.2 Co-Involutive Hopf-Von Neumann Algebras
15
(iii) the co-product and the co-involution are such that the following diagram commute: IC
M
M
r
r
M®M
M®M
x®r,
s
M ® M
It is easy to check that (ii) implies i (1) = 1. If x belongs to the intrinsic group of (M, F), it is clear it is the same for rc(x). Let H = (M, r, rc) be a co-involutive Hopf-von Neumann algebra; then ]His = (M, sr, x) is also a co-involutive Hopf-von Neumann algebra, called the symmetric of H. When M is commutative (resp. F is symmetric), H is said to be abelian (resp. symmetric). For every x in M and w in M*, let us define w° = Co o rc, that is: (x,w°) =
(n(x)*, w) -
this formula defines an involution ° on the Banach algebra M* (1.2.1); as equipped, M* becomes an involutive Banach algebra, and we have, for all w in M* : CO
=Lao orc=(wor.)o
If x, y are in M, w in M*, we have, with the notations of 1.1.1 (ii): (Y' (K(x) - W)O) = ('C(Y*), K(X) - W) - = (/C(xy*), W) -
= (yx*,wO) = (y,x*
. w°)
and so:
(ic(x) - w)° = x* wo
1.2.6 Definition. Let H1 = (M1, r1, ri1) and H2 = (M2, I'2, K2) be two coinvolutive Hopf-von Neumann algebras; a morphism from H1 to H2 is a normal morphism u from M1 to M2 such that: U(1) = 1
I'ZU = (u 0 u)I'1 !2u = ulc1
Such morphisms will be called H-morphisms. The class of all co-involutive Hopf-von Neumann algebras, equipped with H-morphisms, is a category.
16
1. Co-Involutive Hopf-Von Neumann Algebras
1.2.7 Proposition. With the notations of 1.2.6, let Pu be the greatest projector
of the ideal Keru, Ru = 1 - Pu be the support of u; then Pu and Ru belong to the centre of M1. We have:
r'1(Ru)?Nu0 Ru
R.
rt, 1(Ru) = Ru
(+al
Proof. We have:
(u0u)r1(Ru) =rlu(Ru) which proves (i). We have: UKJ(PP) = ic2u(Pu) = 0 so:
K1(Pu) : PU As x1 is involutive, we get (ii).
1.2.8 Notations. Let H be a Hilbert space such that M is standard in H (cf. 1.1.1 (iii)); the co-involution is is then implemented by an anti-linear isometric
involution of H, noted 3; that is, for every x in M, we have: 1C(x) = 3x*3
Moreover, for any x in M, we have:
(x,wJJ,) = (x3 13i)
= (I3xJ) _ (k(x)ll I e)
-
(re (X), LO
_ (K(x),w in o K)
by 1.2.5
_ (x,w,7) which implies:
1.2.9 Example. Let G be a locally compact group. It follows from 1.1.2 and 1.1.3 that the triple (L°°(G), ra, ICa) is an abelian co-involutive Hopf-von Neumann algebra. We shall denote it Ha(G). Furthermore we have seen that the involutive Banach algebra structure induced on the predual LOO(G)*, which is isomorphic to L'(G), is the usual one.
1.2 Co-Involutive Hopf-Von Neumann Algebras
17
Let G°pp stand for the opposite group of G. As the left Haar measure on G°pp is nothing but the right Haar measure AG(s)ds on G, the algebras L°O(G°pp) and L°°(G) are identical, and it is immediate that Ha(G°pp) is equal to Ha (G) is symmetric if and only if G is abelian. 1.2.10 Proposition. Let H = (M, I', i) be a co-involutive Hopf von Neumann algebra. Let H be a Hilbert space, and J be an anti-linear bijective isometry
on H such that (H, J) is standard for M (1.1.1 (iii)); then M' = JMJ; let us now define, for all x in M':
r'(x) = (J 0 J)I'(JxJ)(J ® J) K'(x) = Ji(JxJ)J then:
(i) H' = (M', .r', i') is a co-involutive Hopf von Neumann algebra. (ii) the application u : x -+ Jx(x)*J from M to M' is an H-isomorphism from Hs to H'. Proof. A straightforward calculation left to the reader gives (i); further on, we have:
.P'U(X) = (J 0 J)F(tc(x*))(J 0 J)
= t(J 0 Jkk 0
)r(x)*(j 0 J)
= (U 0 U)4;.F(x)
and:
K'u(x) = JiC(Ju(x)J)J = J/C(IC(x*))J
= Jx*J = u(ic(x)) which proves (ii).
1.2.11 Lemma. Let I be a set, and, for all i in I, let di be in N, H, be a Hilbert space of dimension di, {}1<j
in N. such that El 1 mi) j,kl dkl = didj and a unitary Ui j from Hi 0 Hj to ®1(Cmi,i,kl 0 Hkl) such that, for all x = ®kEIxk in D, we have: r(x)(pi 0 pj) = Ui,j (®1(1G Cmi,I kl) 0 xki ))Ui*,j
18
1. Co-Involutive Hopf-Von Neumann Algebras
(ii) for all i in I, there exist it in I such that n(,C(Hi)) = C(H i), and di = dii. Moreover, there exists a unique unitary Vi from Hi to Hii such that, for all x in £(H1), we have: c(x) = VixtVi*
where xt is the element of £(Hi) whose matrix with respect to the basis {}1<j
Pi = d 1 E e j,k ®l(ek,j ) j,k
belongs to ,C(Hi) 0 ic(,C(Hi)) and is the projection on the one-dimensional subspace of Hi 0 Hii generated by the vector >2j ® V V .
Proof. The support of the homomorphism x --+ r(x)(pi 0 pj) from D to ,C(Hi (9 Hj) is a finite-dimensional projection in the centre of D; therefore, there exist k1i . . . , kp in I such that this support is EP1=1 pkl . Moreover, the homomorphism xk, -+ (kl)(Pi ®pj)r(pk) from ,C(Hkl) to C(Hi ®Hj)r(pk ) is the composition of an ampliation and a spatial isomorphism; so, there exist mi, j,kl in N, and a unitary Ui, j,k, from r(pk,)(Hi (9 Hj) to Cmssilki ® Hkr such that, for any xki in ,C(Hk, ), we have:
F(xki)(pi ®Pj)r(pkl)
Uilj,kr (l(Cmi,j k,) ®xk1)Ui j,kt
and then, for all x = ®kElxk in D, with Ui,j = ®lUi,j,kl, we have:
f
(x)(pi ®pj) = ®l r(xk,)(Pi (9 pj)r(pk,) = Ui,j((Dl
®xkl ))U= j
which gives (i).
For all i in I, ic(pi) is a minimal projection in the centre of D; therefore, it is equal to some pig and we have then rc(,C(Hi)) = C(Hil); by dimension arguments, we get di = di1. As the mapping x -+ ic(xt) is an isomorphism from,C(H1) to C(Hii ), we get the existence (and unicity) of Moreover, we Vi.
have: 2
di
i
j,k=1
ei )k ®ic(ek, j)
_
e j,kel,rn
j)x(em)l )
j,k,l,m
ej,kei,m ®tz(emlekJ) j,k,l,m
®ic(em,j ) j,k,m di
= di L ej,k ®1c(ek,j) j,k=1
1.3 Positive Definite Elements in a Co-Involutive Hopf-Von Neumann Algebra
which shows that Pi is a projection. Moreover, let in Hi 0 Hi ; we have:
Pi = di
1
19
_ >1m al,?zV ® ViE;n
al,m e j,kd 0 tC(ek)j )ViVm j,k,l,m
= d; 1
«k,k
j
by (1)
k
from which we get (ii).
1.3 Positive Definite Elements in a Co-Involutive Hopf- Von Neumann Algebra In the following paragraph H = (M, T, ic) will be a co-involutive Hopf-von Neumann algebra. 1.3.1 Definition. An element x in M is told to be positive definite if it induces a positive form on the involutive Banach algebra M*, i.e. if for all w in M* the number (x, w° * w) is positive. The set of all positive definite elements will be denoted by P(H).
1.3.2 Lemma. Let H be a Hilbert space and J an anti-linear isometric involution on H. The closed convex cone c(J) generated in H ® H by the set
JJ 0 ; E H} is autopolar. Proof. Let H be the conjugate Hilbert space of H. The closed convex cone generated in H ® H by { ®6; 6 E H} is the autopolar cone associated to the von Neumann algebra C R ®,C(H) which is standard in ft ® H (cf. 1.1.1(iii)). It is clear, by hypothesis, that the mapping 3 0 1 is an isomorphism from
H ® H which sends the above mentioned autopolar cone onto c(J); that completes the proof.
1.3.3 Theorem. (i) The product of two positive definite elements is positive definite.
(ii) Let x in P(H). Then n(x) = x*, and this element still belongs to P(H). In other words, every x in P(H) is an hermitian form on M*.
Proof. We assume that M is standard on a Hilbert space H. Then, for all x in M and all w = in M* (with in H), we have, by 1.2.8: (x) wo
* w) = (r(x), wj 7n ®w X17) = (r(x)(Je ® 0 1 J77 0 7l)
20
1. Co-Involutive Hopf-Von Neumann Algebras
Thus, x belongs to P(H) if and only if that last quantity is positive for all , 77 in H. Using the above lemma, by linearity and continuity, it is equivalent to suppose that:
r(x)c(J) C C(J) As r is a morphism, (i) follows. Since the preceeding computation may be prolonged by:
(x, w° * w) = (J' ® I r(x*)(Ji7 0 ii)) it is clear, by virtue of the same reasoning as above, that x belongs to P(H) if and only if x * also does. Moreover, for x in P(H) we get: 0 < (x*,wo * W) _ (x,° * CO) (f(x),w o i ®w° o K)
_
by 1.2.5
((It® rt)r(x), w ®w°)
_ (1'(ic(x)),w° ®w)
by 1.2.5
By polarization, it implies that for all w, w' in M*, we have:
(r(x*), w ® w') = (r(It(x)),w ®w') By density, we deduce that r(x*) = r(#c(x)); r being injective, it completes the proof of (ii).
1.3.4 Definitions and Notations. Let i be a representation of M* over a Hilbert space '14; we shall note Aµ the von Neumann algebra generated by 12(M*) in £ (W µ) . If µ 1 i 122 are two representations of M*, we shall write Hom(121, µ2) the set of all intertwining operators between ,1 and 122.
We define a linear mapping y* from the predual of All to M by writing, for all 9 in (Aµ) * and w in M* : (12*(9),w) _ (12(w), 9)
It is obvious that y* is norm-continuous and that I I * I I < I I / I I . It is clear that 12 is non-degenerate if and only if p(M*) is weakly dense in Ap and if and only if µ* is injective.
1.3.5 Proposition. Let y be a representation of M. For any positive element t? of (Aµ)*, y*(Q) belongs to P(H). Proof. For all w in M*, we have: (12(w)*12(w), Sl) > 0
which completes the proof.
by 1.3.4 by hypothesis
1.3 Positive Definite Elements in a Co-Involutive Hopf-Von Neumann Algebra
21
1.3.6 Definition. Let x in P(IHI). If there exists a representation p of M* and a positive element Si in the predual of Aµ, such that x = p* (Q ), we shall
say that x is representable. The subset of P(H) made of all representable elements will be denoted by PR(]HI).
1.3.7 Example. Since I'(1) = 10 1 and i (1) = 1, it is obvious that 1 induces a one-dimensional representation on M* which shall still be noted 1; using that representation, it is trivial to show that 1 (as an element of M) belongs to PR(H). 1.3.8 Proposition. (i) Let x be in P(H); x belongs to PR(IH[) if and only if it exists a positive number p such that, for all w in M*: I(x,w)12
< p(x,w° *w)
More precisely, in that case, there exists a Hilbert space H, an involutive non-degenerate representation p of M* on H, and a vector in H with: 11e112 =p and x =µ*(Q )
(ii) Whenever M* has an approximate unit, we have PR(IHI) = P(H).
Proof. Let us assume that x belongs to PR(H), therefore, there exists a nondegenerate representation p and Si in (A,)* such that x = p* (,fl) and for all w in M*, we have then: i(x,w)12 = I (p*(Si),w)12
=
by 1.3.4
I
iif1ii(p(,)*p(w), S)
= I* w)
for ,fl is positive
w)
Conversely, let x be an element of P(H) satisfying such an inequality. Taking
1.3.3 (ii) into account, x induces a positive form on the Banach algebra obtained by adding a unit to M*. Therefore by ([124], 4.5.11 and 4.5.5), there is a Hilbert space H, a vector in H with 11C112 = p and a representation p of M* on H such that for every w in M*, we have: (x,w) = (p(w)C 10) = (p(w), 52e) = (p*(Si),w)
Then, x is equal to y* (Se) and so belongs to PR(H), and we have: liSidi = 116112 =P
which completes the proof of (i).
22
1. Co-Involutive Hopf-Von Neumann Algebras
If M. has an approximate unit, it results from ([25], 2.1.5 (i)) that every element in P(H) satisfies the inequality of (i); the proof of (ii) follows. 1.3.9 Proposition. Let µ1 and /22 be two representations of M*. Let w be in Hom(µ1, µ2). For all vectors 1 in Hp, and 772 in ?-l1121 we have: µ2*(Qw6,772) = Y1*Al,w*q2 )
Proof. Let w in M*. We have: (122*
,712
), w) = (/22(w), Ow l ,712 ) = (/22(Li.)Wi 1772)
= (w/21(w)1 I '72 )
by hypothesis
= (izi(w)ei I w*772) = (/11(w), f
l ,w* rj2 )
= (12i*(Q1,2 ), w) which completes the proof. 1.3.10 Example. Let G be a locally compact group and µ be a non-degenerate representation of L1(G); let µG the corresponding continuous unitary repre-
sentation of G. We have, for f in L1(G) and f2 in the predual (A): ((f), S7) = IG f (s)(PG(s), Sl)ds
And so the element p* (Q) in LOO (G) is almost everywhere equal to the continuous function s
(/1G(s), (l) .
1.3.11 Theorem. Let G be a locally compact group, and f a function in L°O(G). Then, the following three assertions are equivalent: (i) The function f is positive definite, i.e.
IG
f(t)(h° * h)(t)dt > 0
for all h in Ll(G)
(ii) There exists a Hilbert space H, a vector in H, a continuous unitary representation /1G of G on H such that, for almost all t in G:
At) = GIGM 10 (iii) The function f is positive definite and almost everywhere equal to a continuous function.
1.4 Kronecker Product of Representations
23
Proof. Let us suppose (i); then f is in P(Ha(G)); as L' (G) has an approximate unit, f is in PR(Ha(G)), which gives (ii), thanks to 1.3.8 (ii) and 1.1.4. The
implications (ii) = (iii) # (i) are trivial, which ends the proof.
1.4 Kronecker Product of Representations 1.4.1 Lemma. Let H = (M, I', x) be a co-involutive Hopf-von Neumann alge-
bra. Let I be a set and, for all i, j in I, xi,j be elements in M such that: ic(xz j) = x j,i
r(xi,j) = E xi,k 0 xk,j kEI
the sum being convergent for the ultra-weak topology.
Then, for all finite subset Io C I, and all family (ai)iElo of complex numbers, the element >1,J diajxi,j belongs to PR(H). More precisely, there exists a representation p of M* and Q in (A,,)* such that:
IIII
and
E aiajxi,j = IL*(J?)
which implies:
E aiajxi,j Proof. Let w in M. We have:
E
w
E aia j (r(xi, j ), w° 0 w) E asaj (Xj,k 0 xk,j,w° ®w) i,j,k
by (ii)
aiaj (xj,k, w°) (xk, j )w) i,j,k E aiaj (K(xk), w) - (Xk,j, w) i,j,k E diaj (xk,j,w)(xk,j,w) i,j,k 2
1: kE7
ai (xk)i, w)
>0
by (i)
24
1. Co-Involutive Hopf-Von Neumann Algebras
Therefore EiJ aia j xi, j belongs to P(IHI). Moreover, we have: 2
aiaj xi2j I
=
iJ
I
l
i
I
2
<
aj(x. ,w)
(i)ai12
Z
by using the Cauchy-Schwartz inequality *w
j Z 1,
according to the above calculation. Thanks to 1.3.8 (i), it completes the proof. 1.4.2 Proposition ([79]). Let IHI = (M, F, i) be a co-involutive Hopf-von Neumann algebra, I be a set, for all i, j in I, xi,j be elements in M and {ej; i E I}
the canonical basis of the Hilbert space £2(I). The following assertions are equivalent:
(i) There exists a representation y of M* on £2(I) such that: xi,j = µ*(QEj,Ei)
di, j E I
(ii) The elements xi,j satisfy, for all i, j in I: ic(xZ ) = x j,i
r(xi,j) = E xi,k ® xk,j kEI
(the sum being ultra-weakly convergent).
(The authors are indebted to Profs. E. G. Effros and Zong-jin Ruan who pointed out a mistake in the preprint version of the following proof.) Proof. Let us assume (i). Let w be in M*; we have: (rt. (x i,j
) w) =
,Ei ), w°/
=
((w°)e j I e )--
_ (µ(w)ci I e j ) (Y* Wei "ej ), w)
and then: lc(xz ) = x j,j
1.4 Kronecker Product of Representations
25
Now, let w1, w2 be in M*. We have:
(f(x1,),1 (9 w2) _ (/t*(Qj,ej),(4.'1 * w2) _ ((W1)(2)6j I Ci )
_
>(/1(w2)ej I Ek)(Ek I kEI
thanks to {e; i E I} being an orthogonal basis w2/
Ei
) wl/
kEI
>2(xi,k 0 Xk,j,W1 0 w2) kEI
Let us assume now that M is standard in a Hilbert space H, and let r be implemented by an antilinear isometric involution 3 of H (cf. 1.2.8). We have then, for all , 77 in H:
(r(,)(Je ® ) I Jij 0 77) = (1'(x,),w33,7 0 J>ijk 0 Xk,i7
®we,y,)
kEI
by 1.2.8 kEI
_ E (ic(xj),
(Xk,j,
by 1.2.5
kEI
_ E I (x,
I2
by the result above
kEI
The same calculation proves that, for every finite subset J if I, we have:
_ 1: I(xk,i7 kEJ
kEJ
1: 1 (X k, i)
Lv "q) 12
kE7
(F(Xili)(J 0 0 1 J710 77) and, by linearity and continuity, we have, for every 1, ," 2 in the closed convex
cone c(J) introduced in 1.3.2:
xz,k ®xk,i 1 I =2 kEJ
< (r(,)s1 u2
26
1. Co-Involutive Hopf-Von Neumann Algebras
As every element S in H ® H may be written as S = L i - u 2 + i. 3 - i=4, with E in c(j) (n = 1.... , 4), and En=1 IISn 112 = Iiu II27 we then easily get that II >.kEJ xi,k 0 xk,i II < IIxi,iII.
The algebraic tensor product M* O M* being dense in (M 0 M)*, we can conclude that the sum EkEI xi,k 0 xk,i is ultra-weakly convergent, and is equal to P(xi,i ), which, by polarization, implies (ii). Let us assume (ii). Let S be an element of CM, which can be considered as a dense subset of £2(I). Therefore there exists a finite subset Io of I, and complex numbers ai (i E Io) such that:
iEIo
Thanks to the calculation of lemma 1.4.1, we know that: 2
ai(xk,i)w) E kEI iEIo i
is finite and equal to: 2
aiaj xi,j, w° * w
la,
aici
iii° * w II
Ilwll2 =11=11211w11
Since: 2
kEI iEIo
ai (xk,i, w)
2
kEI iEIo
ai (xk,i, w)ek
we have, by this way, defined a linear mapping y (w) from CM to t2 (I) such
that:
(w)u =
ai(xk,i7 w)Ck kEI iEIo
This mapping is continuous and such that II,i(w)hI < 11w II , and we have:
E(Xk,i,W)Ck kEI
which implies: (z(w)ei I
= (Xj,j,W)
Thank to its continuity, it is possible to extend µ(w) to 12(I). For all W1, W2 in M*, we have:
1.4 Kronecker Product of Representations (Y(W1 * W2)6i I Cj)
27
= (Xj,j,Wl *W2) = (1'(x,),wi (3 w2) by hypothesis
= (1: x j,k ® xk,i, w1 0 w2
Therefore: µw1 * w2) = µ(Wl)M(w2)
We have: (/1(w°)ei I kj) = (Xj,j,W°) = (K(xj,i)* I w)
= (x,,w)by hypothesis _ (ft(w)Ej I i ) _ (µ(w)*Ei I ej )
therefore y(w°) = µ(w)* and it is an involutive representation of M. on t2(I). The remaining of (i) follows immediately, which completes the proof. 1.4.3 Theorem ([79]). Let H = (M, I', i) be a co-involutive Hopf-von Neumann algebra. Let µ1 and µ2 be two representations of M*. There exists a unique
representation v of M*, such that A C Aµ1 0 Aµ2 and that, for every 01 in (Aµ1)* and S22 in (Aµ2)*, we have: v*((Q1 0
Y1*(fl1)µ2*(f22)
We shall write v = p1 X µ2 and we shall call it the Kronecker product of Y1 by 92. This operation is obviously associative. If M is abelian, this operation is commutative.
Proof. Let I1 and 12 such that fl, and fµ2 be respectively isomorphic to and £2(12). Let {e; i E I1 } and {e; m E I2} be respectively the
R2 (I1)
canonical basis of £2 (I1) and £2 (12). Let us put, for i, j in I1, and m, n in 12:
xi j = 111*(f1 i El) and xm,n =
262
28
1. Co-Involutive Hopf-Von Neumann Algebras
The Hilbert space £2 (I1 X I2) can be identified with £2 (Il) ® £2(12), its canonical basis being then {e' ® 62=}. Thus if we put x(i,m),(j,n) = x=,jxm,n) we have: IC(x*i,m ,
,n) = Iqxi,j)IC(xm*,n) 1 2 = xj,ixn,m
by 1.4.2
= x(j,n),(i,m) and:
r(x(i,m),(j,n)) = r(xz j)r(xm,n )
x ®j 1
1
xm)I ®xl)n
kEIi 1
IEI2 2
1
2
xi,kxm,l ®xk,jxl,n k,l
x(i,m),(k,l) ® x(k,l),(j,n) (k,l)EI1 X12
We are then faced with the conditions 1.4.2 (ii) applied to £2 (I1 X J2); therefore
it exists a representation v of M* on £2 (I1 x 12) such that: X(i,m),(j,n)
v*(fle,1®En,Ei®E,21)
Using the natural identification of Q
1®E2 )Ei ®E2
get : (Jnn
I2 1 *
nn
nn
with DCI)ej (9 f262
)E2
we
n
`Ej1,Ei )p2* (QEri ,Em) = v* (' `E ,Ei ® DEr2i,E n
By linearity and continuity, this implies that for all 1, 771 in 'Hy, and 2, 772 in 'we have: 12 1171)Y2*(` 2,772) = v*(Qel)
0 Q 2,772)
Let a1 be in Al and a2 in A'2. For all e1,'71 in Hy1 and 2, 712 in ?-lµ2, and w in M*, we have: (u(.,)(a1 0 a2)(6 (9
2) 1111 0 772)
(u'(w),i2011,1 0
a26,172
(z'*(Qaiei,,,i 0 Qa 2 e2 ,,72) , w ) = (121*(SIaiei,,71)/22*((1 22,172 )lw)
= (/2l*(Qi,a17i)/22*(Q2,a;172),w)
by 1.3.9
(V*(i2l,a171 0 2,a;172 )' w) = (zi(w)(e' 0'11) I (a1 0 a2)*(e2 0 112))
from what follows that v(w) belongs to Aµ1®Aµ2 'which completes the proof.
1.4 Kronecker Product of Representations
29
1.4.4 Corollary. Let H be a co-involutive Hopf-von Neumann algebra; the set PR(H) is stable by multiplication.
Proof. With the notations of 1.3.6, let x = p 1 * (Qi) and x2 = µ2* (Sl2) be in PR(H); Q1 and S?2 are therefore positive. We have: XlX2
/I1*(fl1)P2*(J?2)
by 1.4.3
_ (lei x 92)*(J?i 0 ,fta) As f21 ® S?2 is positive, it completes the proof.
1.4.5 Proposition. Let H = (M, F, rt.) be a co-involutive Hopf-von Neumann algebra. Let µ, /t1,µ2 be three representations of M*. (i) Let w be in Hom(p1, 112). Then 1®w belongs to Hom(p X 91, /i X µ2). (ii) Let 4$ be a normal morphism from Aµ1 to Aµ2 such that 4$(1) = 1 and OP o /11 = 92. Then, we have: (i (9 Ifi) 0 (Y X YO = Y X P2
-
Proof. Let w in M*, , 17 in ?,Cµ, 1 in 'Hµl , 712 in mµ2 . We have: w)(Y X Y1)(wM 0 61) I ii 0 772)
= ((µ X 91)(w)(6 0 61) 17l 0 w*772) = ((Y X POMI fk,77 (9 f26)W*q2)
= ((µ X µ2)(w), 0 '?wfl IriO = ((µ X µ2)(w)(l 0 w)(6 0 60177 (9 712)
by 1.4.3 by 1.3.9 by 1.4.3
which by linearity, density and continuity completes the proof of (i). We have also, for all 5? in (Aµ)* and Q2 in (Aµ2 )*: 49 0 (Y X YOM, fl 0 fl2) _ (Ut X Y1)Gw), ,fl & ,f12 0 4i)
(9*(fl)Y1*(fl2 0 by hypothesis
_ ((µ X µ2)(w)
0 S22)
which by linearity, density and continuity completes the proof of (ii).
30
1. Co-Involutive Hopf-Von Neumann Algebras
1.4.6 Remark. Let H = (M, I', n) be a co-involutive Hopf-von Neumann algebra. It is straightforward from what is above that the operation which associates p1 x 112 to the couple (µl, P2) is a functor from Rep M* x Rep M* to Rep M*. It is easy to check that the representation 1 (1.3.7) is a unit for that product. Since the Kronecker product is also associative, it is clear that (Rep M*, x, 1) is a strict monoidal category in the sense of [95].
1.4.7 Example. Let G be a locally compact group and p, v non-degenerate representations of L1(G). Let PG, vG be the corresponding continuous unitary representations of G. We have seen, in 1.3.10, that, for 12 in (Aµ)*, p* (Q) is a.e. equal to the function s -+ (PG(s), d2).
So, using 1.4.3, we get that, for 12 in (A,,)*, 121 in (A), the element (p x v)* (Q ®12') is a.e. equal to the function: s -' (PG(s), Sl) (VG (S), 12r) = (PG(s) 0 ZG(s),12 (9 '2r)
It is easy to conclude that p x v is the non-degenerate representation of L1(G) associated to the tensor product µG 0 VG.
1.5 Representations with Generator 1.5.1 Proposition. Let (M, I') be a Hopf-von Neumann algebra, A be a von Neumann algebra on a Hilbert space 7, V be in A 0 M. We define a bounded linear mapping p : M. -+ A by, for all w in M*: Y(W) = (i 0 WXV)
Then we have: (i) The two following assertions are equivalent: (a) p is multiplicative
(b) V satisfies the formula:
(i ®I')(V) = (V 01)(10 )(V ®1)(10 a) where o is the flip operator from H ®l to 7-1®H, or from H ® H
to HOX
(ii) If re is a co-involution on M such that (M, F,,%) is a co-involutive Hopfvon Neumann algebra, then the three following assertions are equivalent: (c) p is involutive (d) for all w in M*, we have:
(t
W 0 K)(v) = (t 0 w)(V*)
1.5 Representations with Generator
31
(e) for all (1 in A*, we have:
i)(V)) = W 0 i)(V*)
-
Proof. Let wl, w2 in M* . We have, on the one hand: Y(wl * w2) = (i 0 (.'i * w2))(V )
= (i 0L01 0 w2)((i ®r)(V)) and, on the other hand: (i 0 wl)(V)(i 0 W2)(V) (i & wl)(V((i 0 W2)(V) 0 1)) (a 0 Wil(v((i 0 w2 & ,)(v 0 i)) _ (i (9 wl)(V(( i (9 W2)(1 (9 o)(V (9 1)(1 (9 a)) (i 0 wl)((i & i & W2)((V 0 1)(10 U)(V 0 1)(10 U)) (i 0 W1 0 W2)((V 0 1)(10 a)(V 0 1)(10 a))
For any (1 in A*, the equality: WW1 * W2), D) = WWOMP2), 12)
is therefore equivalent to: ((1 0 w, 0 W2)((= 0 fl(V)) = (D 0 w, 0 Ws)((v 0 1)(1 0 o)(v 0 1)(1 0 o))
which implies (i) by linearity and density. For every w in M*, (1 in A*, we have, on the one hand:
(QOwo,c,V*)_ (t2,(i ®w o K)(V * )}
On the other hand: fin, µ(w)) 021 (i 0 W) (V))
Therefore we get that (d) is equivalent to (c). Moreover, (d) can also be written: (S2 0 W 0 x)(V) = (12 (9 w)(V*)
(S2 E A*)
32
1. Co-Involutive Hopf-Von Neumann Algebras
which is trivially equivalent to:
w o,c((,fl 0 i)(V)) = w((Q ® i)(V*)) and therefore to (e), which completes the proof. 1.5.2 Definition. Let (M, F, x) be a co-involutive Hopf-von Neumann algebra;
let p be a representation of the Banach algebra M. on a Hilbert space fl . Let U be a partial isometry in ,C(1-ip) 0 M, whose initial and final supports are equal to P 0 1, where P is a projector in .C(fl, ). We shall say that U is a generator for p if it satisfies: Y(W) = (i 0 W)(U)
for w in M*
By linearity and density, U, if it exists, is unique; for any ,fl in ,C(f,2)*, we have p* (0) = (Q 0 i)(U); by the bicommutant theorem, it is easy to see that U belongs to Ap 0 M, and, thanks to 1.5.1 (i) and (ii), U satisfies: 0 -r)(U) = (U 0 1)(10 a)(U 0 1)(10 Cr) (i 0 W 0 K)(U) = (i 0 w)(U*)(W E M*)
1.5.3 Lemma. The projection P in 1.5.2 is the projection on the essential space of ,u. So U is unitary if and only if p is non-degenerate. Proof. Let us represent M on some Hilbert space H. Let a in ?-Cµ such that p(w)a = 0 for all w in M*. This is equivalent to (U(a ®/3) 7 0 6) = 0 for all
13,6in H,7in 'hp,or to U(a®/3)=0forall0inH,i.e. to Pa®,3=0for all P in H, and then to Pa = 0; this leads to the result. 1.5.4 Lemma. With the hypothesis of 1.5.2, let µ, (/lj)IEJ be representations of the Banach algebra M*, with generators Uµ, (Up,)jEI. Then: (i) If ji' is quasi-equivalent to p, then µ' has a generator. (ii) If p" is a sub-representation of /2, then p" has a generator. (iii) The representation ®iEl/2i has a generator. Proof. Let ofi be the morphism from Aµ to Aµ/ such that 4i o p = p'. It is easy to check that (4$ 0 i)(UU) is a generator for µ', which gives (i). There exists a projection P in (A/A)' such that p" = pp. It is then easy to check that (U)(p®1) is a generator for ,u", which gives (ii). Finally, ®zEIU, is a generator for ®iElpi.
1.5.5 Proposition. With the hypothesis of 1.5.2, let iz1 and 92 be two representations of M* having generators, respectively V1 and V2. Then the element
1.5 Representations with Generator
33
(cr 01)(1 0 V1)(cr 0 1)(1 0 V2) is the generator of µ1 X µ2, and the essential space of Ill x 92 is the tensor product of the essential spaces of Y1 and µ2. So, if µ1 and /2 are non-degenerate, 1U1 X /22 is non-degenerate too. Proof. For ,f11 in (Aµ1)* , (l2 in (Aµ2)* , we get: (/21 X µ2)*(J?1 (9 D2)
= µ1*(fl1)Y2*(,f12)
by 1.4.3
_ (Qi 0 z)(V1)92*(,(22)
by 1.5.2
_ (f21 0 0(V1(1 (9 µ2*(S?2)))
by 1.5.2 _ A 0 z)(V1(z 0 ,f12 0 i)(1 0 V2)) V2))) _ (21 0 z)((i 0 ,(l2 0 z)((U 0 1)(1 0 vl)(U because (i 0 ,f12 0 i)((o 0 1)(1 0 Vl)(v 1)) = Vl (.al 0 S?2 0 Z)((a 0 1)(1 0 Vi)(a 0 1)(1 0 V2))
Let V1,2 =(o 01)(10V1)((r 0 1)(10V2). We easily get: V1, 2V1, 2 = V1, 2V1*2 = Pµ1 ® PA2 01
and then deduce that Pµ1 ® Pµ2 is the projection on the essential space of µ1 X112.
1.5.6 Lemma. With the hypothesis of 1.5.2, let yl and 112 be two representations of M*, having respectively V1 and V2 as generators. Let t be in Hom(pl, 92), and if: Aµ1 Aµ2 a morphism such that 4 o y1 = µ2; then, we have:
(tol)Vi=V2(tol)
(i)
(ii)
(4i 0 i) (Vi) = V2
-
Proof. For all w in M*, we have: 0 W)((t 0 1)V1) = t(i 0 W)(V1) = tYl(W) = 92(W)t = (i 0 w)(V2)t = (i 0 w)(V2(t 0 1))
from what (i) is easily reached. We have also: (i 0 W)((,P 0 i)(Vi)) = iP(i 0 W)(V1) = q3 0 Al (W) = Y2(w) = (i 0 UJ)(V2)
which allows to complete the proof.
34
1. Co-Involutive Hopf-Von Neumann Algebras
1.5.7 Proposition. With the hypothesis of 1.5.2, let p be a non-degenerate representation of M* on the finite-dimensional Hilbert space C", {Ei}i
xk ixk = bi j 1 k=1 n
k=1
xikxj) k=bijl
where bi, j is the usual Kronecker symbol.
The generator of It is then the matrix [xi,}i<,<, which may be considered as an element of C(en) 0 M. Proof. Let
eij be the matrix units associated to the basis {e}. Let then:
U=Eei,j®xiJ E£(Cn)®M i,3
We have for any w in M* : (s2Ej lei 0 w)(U) = (x,,w) = (p*(Qe,,ej),w) = (µ(w), S?ej,Ei )
and so, by linearity, for all w in M*: Y(W) = (i 0 W)(U)
U*=Eej,i(9 xzj id
U*U=>ei,j ®
xk)ixklj k
UU* =
ei,j ® E xi)kx j,k k
and so the conditions (ii) are equivalent to U being unitary, and the result is proved.
1.5 Representations with Generator
35
1.5.8 Corollary. With the hypothesis of 1.5.2, let {e} be the canonical basis
of Cn, and, for every i, j, 1 < i, j < n, xij be elements of M. The following assertions are equivalent:
(i) There exists a non-degenerate representation it of M* on Cn, with a generator, such that, for every 1 < i, j < n: x=,, = P*(j`Ej'Ei )
(ii) The elements xij satisfy: 'n(xi1j)
x.7,i
r(xi,j) _ ra xi,k ®xk,j k
>2X,jXk,j
= 5=,j 1
k
j*,k
E xiAx
= bid'
Proof. It results from 1.4.2 and 1.5.7. 1.5.9 Corollary. With the hypothesis of 1.5.2, let u be a non-degenerate representation of M* on the Hilbert space Cn with a generator, {e} the canonical basis of Cn; let us write down, for all i, j = 1, . . . , n: xzj = Y*(flej,ei )
Then, the representation µ defined by ji(w) = µ(w o ,)t (where t means the transposition of matrices), has a generator, which is the matrix [xz j11
xt j =
lc(xj,i) _ A*(jGEj,Ei )
Moreover, we have:
E k
xk)ixkI
j
1: 1(41k)K(xj,k)
by 1.4.2
k
_K
I: xj,kxtk
_ ic(6i,j 1)
and Ek x= kx j,k = Si, j by the same calculation.
by 1.5.7
36
1. Co-Involutive Hopf-Von Neumann Algebras
So, using again 1.5.7, we see that the matrix [x] is the generator of JL.
1.5.10 Example. Let G be a locally compact group, H. (G) the abelian coinvolutive Hopf-von Neumann algebra associated by 1.2.9. Let µG be a unitary representation of G, p the non-degenerate representation of L1(G) associated to PG by the formula (cf. 1.1.4):
µ(f) = IG (s)µG(s)ds
f
We then see that the generator of p is nothing but the bounded continuous function s -p PG(s) from G to C(fpG ), which may be considered as a unitary element of L(lµ) ® L°O(G). So, any non-degenerate representation of L1(G) has a unitary generator, and any representation of L1(G) has a generator.
1.6 Fourier-Stieltjes Algebra 1.6.1 Introduction. (i) Let us put that, being given 1HI = (M, I', ic) a coinvolutive Hopf-von Neumann algebra, C* (IHI) denotes the enveloping C*algebra of M. ([12], I §6, def. 5). We shall denote by 7r the universal representation of M. in C*(II). For every representation p (resp. antirepresentation) of M*, there exists a surjective morphism (resp. antimorphism) µ from C*(IHI) onto the C*-algebra C* (µ) generated by p(M*), such that p = p7r. It is thus possible to identify C*(p) with the quotient of C*(]HI) by Ker). Consequently, if we denote the dual space of C*(1HI) by B(]HI), the dual of C*(p) is isometrically identified with a closed subspace B(p) of B(1HI). More precisely, to each element 9 of (C*(p))*, one associates the element 9 o A of B(H) and an element of B (1H[) belongs to B (p) if and only if it vanishes on Ker µ . (ii) Let W*(IHI) denotes the enveloping von Neumann algebra of C*(1HI) ([133], th. 1.17.2) (i.e. the bidual of C*(H), or, with the notations introduced in (i), the dual of B(H)). Let us still denote by 7r the universal representation of M* in W*(lH[). For any representation (resp. antirepresentation) of M*, there exists a normal morphism (resp. antimorphism) sµ from W*(H) to A,,,
such that µ is the restriction of sµ to C* (]EI), and, so, such that p = sµir ([12], I §6, prop. 10 and [25], 12.1.5). Going over to the preduals, we get a continuous morphism:
(s,,). : (A,,)* --+ W*(H)* = (C*(H))* such that it (sµ)* = p*. Moreover 7r* implements a one-to-one linear mapping from B(IHI) to M (cf. 1.3.4).
1.6 Fourier-Stieltjes Algebra
37
(iii) When y is non-degenerate, sµ is surjective; then, we get sp (1) = 1. Furthermore, for any Q in (A/A)t, we have:
I= (1,(s)(Q)) = (s/A(1),Q) = (1,11) = IIlI Let us remark that C* (1FI) and W*(H) contain more than 0, for it exists at least one non-null representation of M* (viz. 1). 1.6.2 Proposition. Let H = (M, F, x) be a co-involutive Hopf-von Neumann algebra. We have: (i) an element x of M belongs to 7r*(B(p)) if and only if we have: supf I (X) W) 1, W E M*, Ily(w) 11 < 1} < +00
Then, we have:
sup{ 1 (x, w) 1, w E M*, I< 1 } _ I (ii) the mapping (sµ)* is an isometry from (A,,)* to B(1H[), the image of which is included in B(t).
Proof. Let 9 be in C*(p)*, that is, by 1.6.1 (i), an element such that 9 0 µ belongs to B(µ); we have, for w in M*:
(ir(9o/),.) = (ir(w),9o)
(P(W), 0)
and, therefore: sup{1(7r*(B 0 µ),w)1, w E M*, III.L(w)II < 1} _ (1811
by Kaplansky's theorem
Conversely, let x in M and c > 0 such that i(x, w) j < c jjp(w) j for all w in M*. It exists some 9 in C*(µ)* such that, for all w in M*, we have: (x) w) _ ((w), 9) _ (7 (w), 9 0 )) _ (ir(9 0 jC), w) which implies:
x=7r*(eop) and completes the proof of (i). Now, for Q in (Aµ) * and w in M*, we have:
(ir(s)(Q),w) = (*(f1),w) therefore, by Kaplansky's theorem, we have: w E M*, Il/2(W)ll : 1} = 111211
38
1. Co-Involutive Hopf-Von Neumann Algebras
From (i), we get that (s,4)(Q) belongs to B(µ) and that which completes the proof. 1.6.3 Proposition. Let G be a locally compact group; the enveloping C* -algebra
and W*-algebra of L1(G) are noted respectively C*(G) and W*(G), and will be called the enveloping C*-algebra and W*-algebra of G; let us write 7rG the continuous unitary representation of G associated to the universal representation 7r of L'(G); 7rG will be called the universal representation of G.
(i) For a non-degenerate representation p of L1(G), if µG is the corresponding continuous unitary representation of G, we have, for all t in G: sµ7rG(t) = µW
(ii) Every element f in 7r*(B(p)) C LOO(G) is almost everywhere equal to a continuous function.
(iii) ([46]) For every bounded continuous function f on G, and every representation y (including iz = 7r) of G, let us put: Ill il,. = sup{
IG'(s)w(a)ds 7 W E
L'(G)7 Ily(w)ll < 1 1
then 7r*(B(p)) (resp. 7r*(B(Ha(G))) appears as being the subset of all functions f in Cb(G) such that Ilf IIµ (resp. Ilf II,) is finite. It is a complete normed vector space, denoted Bp(G) (resp. B(G)). It can thus be identified to the dual of the C* -algebra generated by ,t(L1(G)) (resp. the dual of C* (G) i. e. the predual of W * (G)).
Proof. By a straightforward application of 1.3.10, we get (i). Let 9 be in (C*(/.t))*; it can be written as a linear combination of positive elements. It implies that 7r*(9 o /t) = p*(9) is a linear combination of positive definite elements of L°O(G), which, according to 1.3.11, completes the proof of (ii) and (iii) is just an application of 1.6.2. 1.6.4 Lemma. Let H be a co-involutive Hopf-von Neumann algebra, µ' and it" two representations of M*. We put s' = s pi and s" = s pil. We have:
x till) = µl x /1"
(i)
(S (9 i)S,XA = 3µIXµii (i ® .s") (y' x 70 = ttl X /2'I (Z 0 s II )8p'x,r = sµ1X p11
(s' 0 s")(ir x (s1
®3II)37CX7r
7r)=µ' x/1"
= sµiXµii
1.6 Fourier-Stieltjes Algebra
39
Proof. Let w be in M*, 1l' in (Ai)* and 1l" in (A,AFI)*. We have: ((s1 (9 i)(ir x
fl ®,1l") = (ir 0 p""(w),s*(RI) *(s*(f/)On ), = ((r x ®12")
w)
_ (ir*s(12')ji(Q"),w) 0 f2"), w)
x ((µi
by 1.4.3 by 1.6.1 (ii) by 1.4.3
x p1i)(w),12- ® 12")
from what, by linearity, continuity and density, (i) results; the proof of (ii) is identical and leads to (iii), thanks to the decomposition:
S/ ®S// = (s'Oi)(iOs")
.
1.6.5 Proposition. Keeping the notations used above, S7rx7r is a one-to-one normal morphism from W*(]H[) to W*(H) 0 W*(1EI) such that: (i ®s7rx ir)S7rx-7r = (s7rxjr
®i)s7rx,7c
If every representation has a generator, then Slcx7r is a co-product in the sense of 1.2.1.
Proof. Let w be in M*. We have: (i ® s,rx,r)s7rx,r(7r(w)) = (i 0 s,rx,r)(7r x 7r)(w) = (7r x (7r x 7r))(w) = ((ir x 7r) x 7r)(w) = (Sirxir 0 i)(ir x ir)(w) = (sirx, 0 i)s,rx7r(-7r(w))
by 1.6.1(ii) by 1.6.4(ii) by 1.4.3 by 1. 6.4 (i) by 1.6.1 (ii)
which by continuity and density implies the equality looked for. Considering the unit representation 1, we have also: (Si 0
(Si 0 i)(a x 7)(w) = (1 x n)(u) = u(w)
which implies, by continuity and density: (Si ®i)SWxW = i
by 1.6.1 (ii) by 1.6.4 (i) by 1.4.6
40
1. Co-Involutive Hopf-Von Neumann Algebras
from what it results that S7rxlr is one-to-one, which completes the first part of the proof. If every representation has a generator, then 7< x ,7r is non-degenerate (1.5.5) and s,r x 7r (1) = 1, which ends the proof.
1.6.6 Proposition. We define an antirepresentation *r of M*, by putting, for every w in M*: *(w) = 7r(w o K)
Then s;r is an involutive normal antiautomorphism of W*(H) which satisfies: Z>
7r*(sfir)* = K7r* S7rx7rSfr = S(Sfr 0 s7r)S7rX7r
Proof. Let w be in M. We have: s;rs*('7r(w)) = s*(7r(w)) = s*(7r(w o ic))
=-(woic) = 7r(w)
by 1.6.1 (ii) by definition by 1.6.1 (ii) by definition
by continuity and density, we get that s;r is involutive. Let d? be in (W*(H))*; we have: W) = (7r(LO), (Sfr)*('?))
(Sfr(7r(LV))' S2)
by 1.6.1 (ii)
(ir(woi),Q) _ (ir(Q),w o w) (Kr* (fl), W)
which proves (i). Let Sit, ,f12 be in (W*(H))*. We have: (sirxirs*(ir(w)), 01 (9 122)
= (S,,i.*(W), Sl 0 Si2)
by 1.6.1 (ii)
= (s,rx,rir(w o ic), Si (9 52 ) = (7r x 7r(w o rK ), i1 (9 52 )
by 1.6.1 (ii)
=((7rx7r)*(Si052),wolc) = (rc(7r*(Q1)7r*(Si2)),w) _ (Ic7r*(52)Ic7r*(Si1),w)
_ (lr*(S*)*(52)lr*(S*)*(Si1),w)
by 1.4.3
by (i)
1.6 Fourier-Stieltjes Algebra
_ ((ir x r)*((s;r)*(f22) 0 (s, _ ((7r X 7r)(w), (s*)(Q2) 0 (s*)*(Qi)) _ (s,,.x,rir(w), (s*)*(Q2) 0 (s*)*(Qi))
41
by 1.4.3
by 1.6.1 (ii)
_ ((sir 0 s*)s,rx, 7r(w), Q2 (& Q1)
_ (c(s* ®s*)s,rx,r7r(w), fl1 (3 Q2)
which, by continuity and density, completes the proof of (ii).
1.6.7 Theorem. Let H be a co-involutive Hopf-von Neumann algebra; let us suppose that every representation of the Banach involutive algebra M* has a generator; then the triple (W*(H), ssrx,r, s;r) with the definitions of 1.6.5 and 1.6.6, is a co-involutive Hopf-von Neumann algebra. If H is abelian, then W*(H) is symmetric; if H is symmetric, then W * (]HI) is abelian. Proof. The first part of the theorem is given by 1.6.5 and 1.6.6. Moreover, if M is abelian, we have S7r x 7r = 7r x 7r (1.4.3 ), and then s,r x,r = ss,r x,r; if IH[ is symmetric, M* is abelian (1.2.1) and then W * (H) too.
1.6.8 Corollary ([44]). Let G be a locally compact group, 7rG its universal r,;presentation, W * (G) its enveloping W * -algebra. There exists a unique morphism I' from W * (G) to W * (G) 0 W * (G), and a unique antiautomorphism ic in W * (G) such that, for all t in G:
r(nc(t)) =7c(t) 0 ac(f) K('7rG(t)) = 7rG(t)*
Moreover, (W*(G), r, n) is a symmetric co-involutive Hopf-von Neumann algebra, which we shall call the Ernest algebra of G. Proof. Using 1.4.7 and 1.6.3 (i), we have, for all t in G:
s,rx,ri'G(t) = iG(t) 0 G(t) s*7rG(t) = 70-1) and so, in particular s,r x,c (1) = 10 1, which ends the proof. 1.6.9 Theorem. Let H be a co-involutive Hopf-von Neumann algebra; then, the space B(H) = (W*(H))* is canonically equipped with a structure of semisimple involutive Banach algebra with unit of norm one. It will be called the Fourier-Stieltjes algebra associated to H. The mapping x7r* is a faithful
representation of this algebra in M; it will be called its Fourier-Stieltjes representation.
42
1. Co-Involutive Hopf-Von Neumann Algebras
Proof. By transposing the mappings ss,rx, and s;j., we get a product * and an involution ° on B(H). For (ll and (12 in B(H) and w in M*, we have: (Klr*(Q1 * Q2), w) = (ir(w o ti), (Ii * (12)
= (csx,.7r(o re), (ll 0 (12) by definition by 1.6.1 (ii) by 1.4.3
= ((-7r x 7r)(w o rc), f22 0 Q1)
= (lr*(Q2)lr*(Q1),w o rc) = (!7r*(Q1)!7r*(Q2),w)
Moreover, we have:
((ir(woi),Q°) = (s*(ir(w o x)*), ,fl) -
by definition
= (s*(ir(w° o 0, f2) = (7r(w° o K), i2) -
by 1.6.1 (ii)
= (ir(w°),Q) = ('7r*
(0),w°)-
= (,7r*(Q)*,w) Since 7r is non-degenerate, 7r* is faithful and B(H) is semi-simple. It results from the decomposition y* = 7r*(sµ)* that we have 7r*(B(IEII)+) = PR(H); we have seen in 1.3.7 that the unit of M belongs to PR(]HI), it follows that 7r* 1(1)
is a unit for B(H); it is easy to check that 117r* 1(1) = 1, which completes the proof. 1.6.10 Remarks. Since it results from 1.6.2 (i) that -7r* (B(]HI)) is stable by
ril, the fact used above that 7r* (B(H)+) = PR(H) implies that the image of B(H) by the Fourier representation is the set of linear combinations of elements of PR(IHI), that is, also, the set of elements which can be written µ* (,fl ), where µ is a representation of M* and (l is in (A). Since 7r* is faithful, we can deduce that B(H) is composed of those elements of the form (sµ)*((2). Thanks to 1.6.2 (i), we observe, at last, that PR(IH[) is stable by rc. 1.6.11 Theorem ([46] ). Let G be a locally compact group, and f a function on G. The following assertions are equivalent:
(i) The function f belongs to B(G), i.e. (1.6.3 (iii)), f is a bounded continuous function on G, and: sup
IG f (t)h(t)dt
I
h E L'(G), 117r(h)JI < 1 1 < +oo
where -7r is the universal representation of L1(G).
1.6 Fourier- Stieltjes Algebra
43
(ii) The function f is a linear combination of continuous positive definite functions on G. (iii) There exists a Hilbert space H, vectors 6,77 in H, and a continuous unitary representation 11G on H such that f (t) = (IG(t)e 177)
Moreover, B(G) is a subalgebra of the algebra Cb(G) of continuous bounded functions on G. Equipped with the norm:
IG f(t)h(t)at h E Ll(G), 11n(h)II
1}
B(G) is a Banach algebra, which can be identified with the dual of C*(G) (i.e. the predual of W * (G)), the duality being given, if f (t) = (,G(t)r I r7), and h is in C* (G), by: (f, h) = (p(h) 17l) (where µ denotes here the representation of C*(G) associated with jG).
Proof. Using 1.6.3 (iii) and 1.6.10, we see that B(G) is the image of the involutive algebra B(1H[a(G)) by its Fourier-Stieltjes representation. Then we have the implication (i) = (ii) by 1.6.10 and 1.3.11; (ii) = (iii) is a corollary
of 1.3.11, and (iii) = (i) is a straightforward calculation. The end of the theorem is just a re-writing of 1.6.3 (iii).
Chapter 2 Kac Algebras
This chapter deals with technical results about Haar weights, as they have
been studied by the authors in [36] and [136], and, independently, by Vainerman and Kac ([180] ). On a co-involutive Hopf-von Neumann algebra
(M, F, it), a Haar weight is a faithful, semi-finite, normal weight on M+, which is left-invariant with respect to F, i.e. such that: (i ® Sp)T(x) = W(x)1 for all x in M+
(in 2.5, we show, after Kirchberg, that this axiom may be weakened), and, roughly speaking, satisfies two other axioms involving i . The quadruple (M, T, x, p) is then called a Kac algebra. On L°O(G), the integral defined by a left Haar measure is obviously a left-invariant weight, and it is straightforward to see that it is a Haar weight (2.2.2 ). Thus, L°O(G), equipped with the convenient operations, is an abelian Kac algebra; we shall denote it K(G). Left-invariantness allows us to define the fundamental operator W, which is an isometry belonging to M ®,C(Hw), satisfying the so-called pentagonal relation (2..4.4 ):
(1OW)(cy®1)(1OW)(cr®1)(WO1) = (W ®1)(1 ®W) With the other axioms, we get that W is unitary (2.6.2 ), "implements" the co-product T, i.e. that we have, for all x in M (2.6.3 ): .P(X) = W(l 0 X)W* Moreover, the mapping: A : w --+ (w 0 i)(W*)
from M. to C(H,,) is a non-degenerate representation of the involutive Banach algebra M. (2.6.1 and 2.6.3); it will be called the Fourier representation of the (predual of the) Kac algebra.
2.1 An Overview of Weight Theory
45
Then, W and A appear both as generalizations to an abstract Kac algebra of the left regular representation AG of a locally compact group G; more precisely, for the Kac algebra Ka (G), the fundamental operator is the bounded continuous function s -+ AG(s)*, when considered as an element of L°O(G) ® £(L2(G)), and the Fourier representation is nothing but the left regular representation of L'(G) (2.4.7 and 2.5-4).
Another result strengthens the analogy with the case of locally compact groups: on a co-involutive Hopf-von Neumann algebra, the Haar weight (whenever it exists) is unique up to a positive scalar (2.8.6 ). The situation of locally compact groups is generalized, too, with the definition of square-integrable elements of M*, i.e. elements w of M* such that there exists a (unique) vector a(w) in HW such that, for all x in (2.1.6 ): (X*
)
w) = (a(w) I AV (x))
If the weight Sp is left-invariant, the set I,, of such elements is a left ideal of M* (2.4.5 ), and, if Sp is a Haar weight, we have, for all wl in M* and w2 in IV (2.4.6): a(w1 * w2) = A(w1)a(w2)
In the case of K. (G), we recover L1 (G) fl L2 (G), which is a left ideal of L1(G),
and the formula: .fi * .f2 = AG(.fi).f2
(fi E Ll(G), f2 E L'(G) n L2(G))
which links the convolution and the left regular representation of G. If K = (M, F, x, gyp) is a Kac algebra, so is K = (M, cf', ic, cp o ic), and
it is possible, too, to define a Kac algebra K' on the commutant M' of M (we suppose M is a von Neumann algebra on the Hilbert space H
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 Sp : M+ -> [0, +oo] such that W (Ax) = Aco (x) for all A
in R+ and x in M+. We shall use the notations and constructions of the Takesaki-Tomita theory associated to weights (cf. [158], [14], [114], [16], [56] or [150]) hereafter briefly summarized: Let us define:
9 ={xEM+, '(x)<+oo} _ {x E M, Sp(x*x) < +oo}
46
2. Kac Algebras
Then 91,, is a left ideal in M, and the involutive algebra 97 *9 is the linear span of 9X+; it is denoted bye, and it is possible to extend Sp to a positive linear form on 931,,, which will be still denoted by Sp.
The weight cp is said to be faithful if V(x) = 0 for x in M+ implies x = 0, semi-finite if 9XV is ultra-weakly dense in M (finite if 9XV = M), and normal if cp(supa xa) = sups Sp(xa) for all increasing directed bounded nets {Xa} in
M+. If p is normal, the set of all projections p in M such that cp(p) = 0 has a greatest element pp, and q = 1 - po is called the support of Sp. We may consider then y as a faithful weight on the reduced algebra Mq. (ii) In what follows, Sp will denote a faithful, semi-finite, normal weight on a W*-algebra M. Then, the left ideal gyp, equipped with the scalar product (x, y) -+ cp(y*x) (x, y E 91p), is a pre-Hilbert space; let us denote H. the associated Hilbert space (or H if there is no confusion), and A. the canonical injection -+ Hip.
The set 2t,, = 11w (91, fl 9l* ), equipped with the product T and the involution # defined by:
AW(x)TAv(y) = Av(xy)
d,(x)O = tl,(x*)
(X, Y E gip (1'7Ip
(x E gt, fl grp)
is an involutive algebra, dense in Hp, such that its involution is an antilinear preclosed mapping, and such that the left-multiplication representation of 2t, is non-degenerate, bounded and involutive. The W *-algebra M is then isomorphic to the von Neumann on Hip generated by this representation ([14], th. 2.13). (iii) Generally speaking, we shall call left Hilbert algebra an involutive
algebra 2t, 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 7r of Qt is non-degenerate, bounded and involutive ([158], def. 5.1). Let us then call S the closure of 0, and F the adjoint S* of S, whose domains will be respectively denoted by DO and Db. We shall say that in H is right bounded if the linear application from 21 to H defined by r, -+ 7r(77) (r7 E 2I.) is bounded; we shall then denote by the element of C (H) defined, for all ri in 21, by r'(077 = 7r (r)) . Let us then define:
21' = { E Db,
is right bounded}
Then, 21', equipped with the product e1T62 = lr'(e2)e1 (61, 62 E 21'), and the involution 6b = F6 (6 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 21", containing 21 as an involutive subalgebra, the closed operator S being again the closure
2.1 An Overview of Weight Theory
47
of the involution of 2t". A left Hilbert algebra is called achieved if 2t = fit.". Moreover, for any left Hilbert algebra 2t, the left (resp. right) Hilbert algebra 2t." (resp. 2t') is an achieved left (resp. right) Hilbert algebra, and the left Hilbert algebra 2tv constructed in (ii) is achieved. Conversely, for any left Hilbert algebra 2t, the left-multiplication representation 7r generates a von Neumann algebra M on the Hilbert space H, and the formula, for x in M+: SP(x) =
g112
if there exists
-f - oo
elsewhere
in 2t" such that x = 7r(e)
defines a faithful, semi-finite normal weight on M, with the left Hilbert algebra 2t, isomorphic to W' ([14], th. 2.11). For any in D# (resp. Db ), it is possible ([14], def. 2.1) to define a left (resp. right) multiplication by , which will be a closed operator on H, affiliated to
M (resp. M'). (iv) Starting from 2t
, the polar decomposition S = J 41/2 V gives rise to to H., the antilinear isomorphism J. (or J if there is no confusion) from and the modular operator 4, (or 4 if there is no confusion). If we consider
the elements of M as operators on HH, 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 PV generated by {xJ,A(x); x E 91,p}), and JxJ = x* for any x in the centre of M. Moreover, for all t in R, we have 4:t M4 - at = M, this last formula leading to the definition of the modular automorphism group
at by: o°(x) =
4ztx4-:t
(x E M,t E R)
Moreover, we have: SP = 5P o Olt
and the modular automorphism group satisfies the Kubo-Martin-Schwinger condition ([14] ), that is, for every x, y in TLp n *, there exists a bounded
function f on the strip Iz E C, 0 < Im z < 1}, holomorphic in its interior, such that, for all t in R, we have: f (t) = SP(at (y)x)
f (t + i) = (xoT(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 --+ at (x) has an extension to an analytic function z --+ 4(x) from C to M. We then define M' x E M, at (x) = x for all t in R}. Moreover, for any x in 91w and a in M'P, xa* belongs to 91., and we have A,,(xa*) = JaJA,(x). For technical reasons, it is useful to know there exists a maximal subalgebra 2(o of 2i" n 2t', which is both a left and right Hilbert algebra, with 2to = 2t'
48
2. Kac Algebras
and 2t0 11 = 2t!', and which is globaly invariant under the linear closed operators
az for all z in C. The algebra 2t0 is called the maximal modular subalgebra of W.
If cp is a trace (i.e. V(x*x) = yc(xx*) for all x in M), then 91V = 9*, = 1, and the modular automorphism group reduces to the identity i; the algebra 9 is both a left and a right Hilbert algebra; we recover the Hilbert algebras of [24]. If trH is the canonical trace on ,C(H), then YttrH is the algebra of Hilbert-Schmidt operators on H; it is therefore SW = J(p, z1
complete, and we may consider HtrH as H ® H, the identification 11trH being given, for all in H, by AtrH (PP) = 6 ®6, where Pe is the one dimensional orthogonal projection on C6. 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 V'(x) = cc(Jcox* Jco) defines a faithful semifinite normal weight on M', with 91.1 = J.91. J.; we identify Hi with HSp by writing If Sci (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 cpl ®SP2
on M1 0 M2 such that: (cci 0 502)(x1 ® x2) = ccl(xl)cp2(x2) (x1 E M1, X2 E M2) at att ®a5P2 t
If the weights are not faithful, we may define the tensor product of the reduced
weights on the tensor product Mlp1 0 M22, which may be identified to (Ml 0 M2) p1®p2 (where Pi is the support of Wi ), and, by composition with the reduction from M1® M2 to (Ml ® M2) p10p2 , we have a definition of the tensor product of cpl 0 502(v) If cp, b 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 R, and x in M, we have: ut+s = uto °(us )
4(x) = utat (x)ut If Sp = cp o a , for all t in R, this cocycle is a one parameter group of unitaries
in MW, which leads, thanks to Stone's theorem, to a positive self-adjoint operator h affiliated to MW, such that ut = hzt . We have then, for all x in M+. b(x) = lim W(h1/2xh1/2) C--+0
where he = h (l + eh) - l belongs to MW. Moreover: V(h1/2)
EH;
+oo} 6
2.1 An Overview of Weight Theory
49
and, for any in D(h1/2): 112
I
= sup(h. I} = lim(h, I ) C
C
For any x in
v fl 'J7.,, we have:
b(x*x) = lim cp(h1/2x*xh1/2 ) = lim I
and so JA,(x) belongs to D(h1/2), and:
?(x*x)
= IWe
shall write then 0 = in the sense of ([114], th. 5.12). The operator h will be called the Radon-Nikodym derivative of b with respect to V. If we have 0 < V, then 0 < h < 1. If cw t = Cro t for all t in R, then the operator h is affiliated to the centre of M; as then Jh1/2 J = h1/2, we get that, for any x inn, fl tYL , Aw(x) belongs to D(hl /2), and b(x*x) = I 2.1.2 Lemma. Let e j be a family of elements in M, belonging to 'ytw, analytic
with respect to Sp, such that, for all j, kill < 1, and, for all z in C, az(ej) is weakly convergent to 1 (such a family exists, thanks to ([150], 2.16). Then, for all a in'JJCL, we have: W(a)
Proof. Let x, y be in'JT
P(y*xej) = Aw(xe.7) I Aw(y)) = (SAW(ejx*) nw(y))
= = =
Aw(y)) I A,p(y))
I Aw(y))
= (jaw_i/2
3
I Aw(y))
(Aw(x) I Aw(y)) = W(y*x)
which, by linearity, completes the proof.
2.1.3 Lemma. Let / be an antiautomorphism of M (i.e. a linear mapping from M to M such that for all x, y in M, we have /3(xy) = /3(y)/3(x) and /3(x*) = /3(x)*). Let us put 0 = Sp o,Q; then 0 is a semi-finite, faithful, normal
weight on M and for all t in R, we have: at
=/-loo_to'3
.
50
2. K ac Algebras
Proof. Let x and y in 91a fl 918, then /3(x) and /3(y) belong to 91P fl 9r; therefore there exists ([14], prop. 4.4) a bounded continuous function f on the strip f z E C; 0 < Im z < 1}, holomorphic in its interior, such that for all t in R, we have: 1(t) = w(ai (R(v))P(X)) f (t + 0 = V(fl(X)at1P(fl(Y)))
It can also be written as follows: -1a, f (t) = P(#(Y)a't(0(X))) = 0(fl- -t P(X)Y) f (t + i) OWAY)) = O(Yfl-1a'tfl(X))
As, on the other hand, we have, for all x in M+: W 0(fl-law OW) = W(U-Ax)) = WOW) = O(X)
we see that 9 satisfies the K.M.S. conditions with respect to the group of automorphisms t -+ /3-1 oe t fl, then ([14] cor. 4.9) allows to conclude.
another faithful, semi finite, normal weight on M, such that b is invariant under UT for all t in R. Then A(91, fl 92,x) is dense in 2.1.4 Lemma. Let Hip .
Proof. Let h be the Radon-Nikodym derivative of 0 with respect to Sp. Let us put h = fo' sdES; en = fi/n dEB; hn = hen. Let x be in 9.; we have: b(enx*xen) = lim cp(h1/2enx*xenh1/2) E--+o
= lim
I
because en hE/2 belongs to M
_ I1Jh1n/2JAW(x)112 because hE en is weakly convergent to hn
when a goes to 0 < Jjhn1kP(X*X)
and then xen belongs to 910, and as en belongs to 9X V, by ([114], th. 3.6), we get that xen belongs to 9zp f192,,. Moreover AW(xen) = JenJAc(x) converges
2.1 An Overview of Weight Theory
51
1 when n goes to infinity. Thus, Acp (gyp fl dense in A ,( p) and therefore in H,.
910) is
to AV (x ), because efz
2.1.5 Lemma. Let 0 be a semi finite, normal weight on M, invariant by the such that AV(E) modular automorphism group of. Let E be a subspace of is dense in H. and that, for all x in E, we have: 9tV,
Sp(x*x) = b(x*x)
Then, the weights V and ,b are equal.
Proof. Let us take the same notations as in 2.1.4. If x is in E, then x is in 91, fl qi,p, therefore JAW(x) belongs to D(h1/2), and we have:
I= /(x*x) = Sp(x*x) = Ithen, using the same arguments as ([23], lemma 23), we get that Jh1 /2 J = 1, therefore h = 1 and Sc = b. 2.1.6 Definitions. (i) For every w in M*, we define:
I= sup{I(x*,w)I, x E M, Sp(x*x) < 1} Since the weight Sp is semi-finite, the condition jjwjjW = 0 implies w = 0. (ii) We put: 4,={wEM, IThus,
for every w in IV, there exists a unique vector a(p(w) in H, such that, for all x in MV: (a,(w) I AV (T)) = (x*,w) Moreover, we have:
I= IAn element of I. will be called square-integrable. If there is no confusion, we shall write a(w) instead of a, (w) .
2.1.7 Proposition. With the hypothesis and notations of 2.1.6, we have: (i) For any i in Db and in D(ir!(ii)*), belongs to Iv,, and we have: 7r'(71)%
(ii) The set IV is a dense subspace in M*, and a(I,,) a dense subspace in HV.
52
2. Kac Algebras
(iii) Let w be in I., x in M. Then, with the notations of 1.1.1 (ii), x - w belongs to I., and we have:
a(x w) = xa(w) (iv) Let w be in IV, x in 91,p. We have, with the notations of 1.1.1 (ii): `fla(w),Ap (x)
Proof. Let x
= w x*
.
We have:
(Ix)
(x*eI)
=( _
for 17 is in Vb
I ir'(r)Ap(x)) (.I(7.l)*e I A(x))
for 6 is in D(i'(,q)*)
From that (i) follows immediately. The result (i) implies that IV contains a least all states of the form w c,7 where the 6 and 77 are elements in 21', which implies that a(ICP) contains '2; so (ii) is proved. Let y be in Yt ; we have: (y*, x - w) =
(*)
= (a(w) I Aw(x*y)) = (a(w) I x*A,(y)) = (xa(w) I A (y))
so (iii) is proved. For all y in M, we have: (Y,
(a(w) I y*A (x)) _ (a(w) I
(y*x)) = (x*y, w) = (y,w . x*)
which is (iv).
2.1.8 Definitions and Notations. We shall use the constructions and notations of Haagerup's theory of operator-valued weights ([59] or [150] ): (i) Let M be a von Neumann algebra; the extended positive part M+ of M is the set of all lower semi-continuous additive functions m : M,+ --> [0, +oo], which satisfies m(Aw) = Am(w) for all A in R+ and w in W_- Given m, n in M+, A in R+ and x in M, it is straightforward to define m + n, Am and x*mx in M+, and it is clear that M+ is naturally imbedded in M+ . Moreover, ([59], cor. 1.6), for every m in M+, there exists an increasing sequence of elements of M+ converging up to m. Let H be an Hilbert space on which M is standard (1.1.1 (iii)); then, ([59], lemma 1.4), for every m in M+, there exists a closed subspace H' of H, and
a positive self-adjoint operator T on H' such that D(T) - = H' and: (6 E D(T 112))
IIT'I2e112
+00
(6
V
D(T1/2
))
2.1 An Overview of Weight Theory
53
Now, let N be a von Neumann subalgebra of M; an operator valued weight on M with values in N is a mapping E : M+ -' N+ such that:
E(m + n) = E(m) + E(n)
(m, n E M+)
(A E ][Y+, m E M+) E(Am) _ AE(m) E(a*ma) = a*E(m)a (a E N, m E
M may be considered as an operator valued weight on M with values in the von Neumann subalgebra C1 M . Faithfulness, semi-finiteness and normality of operator valued weights are defined exactly the same way as for weights. Those operator valued weights such that E(1M) = 1N are called conditional expectations; as we have then
E(M+) = N+, E can be extended to a positive linear map from M to N, which satisfies E2 = E. (ii) Let M be a von Neumann algebra, N a von Neumann subalgebra of N, V (resp. 0) a faithful semi-finite normal weight on M (resp. N). Then,
if at (x) = °t (x) for all x in N and t in R, there exists a unique faithful semi-finite normal operator valued weight E on M with values in N such
that So= OR If the restriction of Sp to N+ is semi-finite, we may take 0 = V N; the operator valued weight E is then a faithful normal conditionnal expectation which is associated to the projection P of HV on the closure of Ap( fl N),
that is, for any x in M, Ex is the unique element of N such that (Ex)P is equal to PxP. Moreover, for all x 91., we have ([160]): PelW(x) = Av(Ex)
Let A be a von Neumann algebra and w be a positive element in A*, such that w(1) = 1. We may then define a conditional expectation (i ® w) from M ® A to M ® C M, such that, for any faithful semi-finite normal weight on M, we have:
wo(i®w)=5P®W For any w in A*, we may define (i ® w) by linearity.
(iii) The tensor product of faithful semi-finite normal operator valued weights may be defined, using the tensor product of weights. Using this, it is possible to define a faithful semi-finite normal operator-valued weight i 0 from A 0 M to A ® C r-o A, where A, M are von Neumann algebras, and Sc a faithful semi-finite normal weight on A. Moreover, for any positive element w in M*, we shall have:
wo(i0c) =w®(
54
2. Kac Algebras
2.1.9 Lemma. For all w in M*, we have: (w
® i)( i®w) C 9'tw.
Proof. Let us suppose w is positive; for any X in 9fl:Ow , then (w ® i)(X) is positive and we have:
i)(X)) = W(i 0 0)(X) < +00
from which it results that (w ® i)(X) belongs to 9R+, and (i) follows, by linearity. Let us suppose now that w is positive and w(1) = 1. Then w ® i is a normal conditionnal expectation from M 0 M to M, and we have, for all X in M ®M: ((w 0 i)(X ))*(w 0 i)(X) < (w ® i)(X *X )
whence, if X belongs to SP(((w 0 i)(X))*(w 0 i)(X )) < W(w 0 i)(X *X) < +oo
by (i)
from which (ii) follows, by linearity.
2.1.10 Lemma. Let x be in 9
and y be in 9XLA,(.)®(P; then (x* 0 1)y(x ®1)
belongs to 9, w®Sp, and we have:
(w4,() 0 Sp)(y) = (SP 0 p)((x* 01)y(x ® 1))
Proof. For all a in M+, we have: WA,(x)(a) = Sp(x*ax)
and this equality remains true if a is in the extended positive part of M; in particular, for all y positive in M 0 M, if we replace a by (i 0 )(y), we get: WA,(2:)((i 0 WXY)) = V(X*(i 0 V)(Y)X) = P((i 0 W)(X* 0 ')Y(x (D 1))
which may be written, using ([59], 5.5): (w4j,(Z) 0 SP)(y) = (5P (9 W)((x* (9 1)y(x (9 1))
Then, both sides are finite if y belongs to 9R by linearity.
W
(X)®W, and the result comes
2.2 Definitions
55
,2.2 Definitions 2.2.1 Definitions. Let (M, r) be a Hopf-von Neumann algebra and Sp be a faithful semi-finite normal weight on M. We shall say that W is a left-invariant weight with respect to r if it satisfies the following property:
(i 0 )r(x) = P(x)1
(LIW)
(x E M+)
Then (p satisfies the weaker property: (HWi)
r(9zW) c 91iow
Let H = (M, r, x) be a co-involutive Hopf-von Neumann algebra, and (p be a faithful semi-finite normal weight on M. We shall say that Sp is a Saar weight on H if it complies with (HWi) and the two following axioms: (HWii) (i 0 0)((' 0 Y*)F(x)) = 14(i 0 W)(-r(Y*)(1 0 XM
(x,yE4YL,)
(left and right-hand sides of the equation in (HWii) make sense, thanks to (HWi)) (HWiii)
Kct =7!t
(teR)
2.2.2 Proposition. Let G be a locally compact group and Ha(G) = (L°°(G), ra,'ca) the associated abelian co-involutive Hopf-von Neumann algebra (1.2.9). Let us consider the faithful semi-finite normal weight Spa on L°O(G), which is the trace arising from the Saar measure on G by:
IG
f(3)ds
(f
E L°O(G)+)
It is a left-invariant and a Saar weight on Ha(G). Moreover, the weight SO is finite if and only if the group G is compact. Proof. We identify HWa with L2 (G), 9 a with L2 (G) n L°O (G), 9AWa with L'(G) n L°O(G); let f in LOO(G)+; since JG If 112(st)dt = fG If 112(s)ds, it is clear that (LIW) holds. The translation of (HWii) is: IG
(t)f(st)dt = J (s1t)f(t)dt
(f, g E L2(G) n L°O(G))
which results straightforwardly from the left-invariance property of ds. And
(HWiii) is trivial, Sp being a trace.
56
2. Kac Algebras
2.2.3 Proposition. Let H = (M, r, ic) be a co-involutive Hopf-von Neumann algebra, and Sp be a faithful semi-finite normal weight on M. If Sp satisfies (LIW) (resp. (HWi), (HWii), (HWiii)), then cp' satisfies the same property. Proof. Let x be in M'+, and So a left-invariant weight; then:
(i 0 p')((J 0 J)f(J5J)(J 0 J)) = J(i 0 )r(J5J)J = cp(J5J)1 = cp'(x)l
(i
So cp' is a left-invariant weight. Proving axiom (HWi), (HWii) and (HWiii) is just a straightforward cal-
culation of the same kind, using 91,i = J9p J. 2.2.4 Proposition. Let H = (M, I', rc) be a co-involutive Hopf-von Neumann algebra, and Sp be a faithful semi-finite normal weight on M. If Sp is a leftinvariant (resp. a Haar) weight on H, then cp o x is a left-invariant (resp. a Haar) weight on
Proof. In 1.2.10 (ii), we have seen that the application u(x) = Jrc(x)*J from M to M' is an H-isomorphism from W to H'. Furthermore, we have gyp' o u (x) = Sp o n (x) for all x in M+; so this Hisomorphism exchanges the weight W o ic with the weight W'. The result is then a corollary of 2.2.3. 2.2.5 Definitions and Notations. Let H = (M, r, rc) be a co-involutive Hopfvon Neumann algebra, and cp a Haar weight on H. We shall then say that the quadruple K = (M, r, rc, Sp) is a Kac algebra. By 2.2.2, for any locally compact group G, (L°°(C), ra, rya, cpa) is a Kac algebra, denoted K, ,,(G). For any Kac algebra K = (M, I', rc, W), we denote K' = (M', I'', rc', cp') and
K1 = (M, cr, rc, Sp o i) the Kac algebras associated respectively in 2.2.3 and 2.2.4. They will be called respectively the commutant Kac algebra of K, and the opposite Kac algebra of K. Evidently, we have Ka (G)' = Ka (G) and Ka (G)' = Ka (G°PP ).
Let K = (M, I', rc, cp) be a Kac algebra, and a > 0. It is clear that (M, I', rc, aSo) is a Kac algebra, which will be denoted alts. Let K1 = (M1, , Vi) and K2 = (M2, F2, K2, Sp2) be two Kac algebras. We shall say that K1 and K2 are isomorphic if there exists an H-isomorphism: I'1, id
u:(Ml,ri7ic1)-+(M27F2,K2) and a > 0 such that W2 o u = acpl. Clearly, then, u is implemented by a unitary U from H(1 onto HW2 defined for all x in 'Awl, by: UJLp1(x) = a_1/2A p2 (U W)
Therefore K and aK are isomorphic.
2.2 Definitions
57
2.2.6 Proposition. Let K = (M, r, x, W) be a Kac algebra and R a projection in the centre of M, such that:
F(R) > R 0 x x(R) = R We shall denote by KR the quadruple (MR, rR, KR,'PR) where, for all x in M: (i) MR is the usual reduced algebra; the canonical surjection M - MR will be denoted by r;
(ii) (iii)
.PR(r(x)) = (r 0 r)(-P(x)) KR(r(x)) = r(K(x))
(iv) the weight coR on MR is obtained by reduction from co on M as in ([16], def. 3.2.4).
Then KR is a Kac algebra, called a reduced Kac algebra of K, and r is a surjective H-morphism.
Proof. It is trivial to check that (MR, -PR, KR) is a co-involutive Hopf-von Neumann algebra and that r is an H-morphism. We have 9XV, = r (9J1 , ) which implies:
ra(9XnA) = ra(r(9Rw)) = (r 0 r)r(m,,) C (r 0
9Ri,&nA
Let x, y be in'7Lp. We have:
(i 0 'VR)((1 0 r(Y)*)rR(''(x))) ° (e 0 'PR)(T 0 r)((1 0 Y*)r(x)) (i 0 V)(r 0 r)((l 0 y*)-r(x)) _ (i 0 W)((R 0 Ry')r(Rx)) R(i o V)((1 0 Ry*)-P(Rx))
Rk(; 0 »)(r(RV')(1 0 rte)) tqi 0 V)(F(Ry*)(R 0 Rx))
= rqa 0 w)((R 0 a)r(v`)(1 0 x)) KR(i 0 WR)(r 0 r)((-P(y*)(1 0 x))
= kR(l 0 'OR)(r(T(Y)x)(1 (9 r(x)))
As rat = atR r, the axiom (HWiii) is trivially proved.
2.2.7 Proposition. Let K = (M, I', ic, Sp) a Kac algebra and M a sub von Neumann algebra of M such that:
58
2. Kac Algebras
iM ® M N
()r(1cf)c N
N
N
(ii) ic(M)= M
(iii)a'°(M)=M (tER) N
(iv) the restriction Sp I M+, which NwillN be denoted S ,N is a semi finite weight. M W e shall denote K the quadruple (M, r, R, ,P) where r and R. are respectively N N
the restrictions of F and ic to M; the canonical one-to-one morphism M -+ M will be denoted by j. Then, K is a Kac algebra, called a sub-Kac algebra of K, and j is a oneto-one IRE-morphism.
Proof. Everything is proved by restriction. In particular, using ([16], 3.2.6),
we have at = at J
In what follows, (M, r) is a Hopf-von Neumann algebra and V a faithful, semi-finite, normal weight on M, satisfying (HWi).
2.3.1 Lemma. For all w in M*, we have:
(ii)(w®i)r(9l,)c9t4,. Proof. It is clear that (HWi) implies r(971,) C 9721®,; so the proof results immediately from 2.1.9 (i) and (ii). 2.3.2 Definition. Thanks to 2.3.1, we can define an unbounded linear operator
£(w) on H. by: V(f(w)) = AV(91p) t(w)AV(x) = Aw((w 0 i)-P(x))
2.3.3 Proposition. For all w, w' in M*, we have:
i(w * w') = i(w Aw)
.
Proof. Let x be in 91,; we have: t(w * wJ)AV (x)
_ AV((w * w' ® i)r(x)) = A ((w ®w' ® i)(r ® i)r(x))
by 2.3.2
2.3 Towards the Fourier Representation
59
=An((w0 oi 0 i)(i 0 flf(x)) = AW((WI 0 i)(W 0 i 0 0(i 0 F).r(x)) AW((W' 0 i)F((w 0 i).r(x))) = e(w by 2.3.1 and 2.3.2 0 OrW) =Q(w )f(w)A,p(x) by 2.3.2
2.3.4 Lemma. For all w in M*, x, y in 91., we have: (t(w)A,(x) I A ,(y)) = (w 0 p)((1 0 y*)r(x)) Proof. We have, if w is positive and w (l) = 1:
(e(w)e(=) I Fe(y)) = (nv((w (9 .)(r(=)) I ne(v)) = W(Y*(w 0 OFW)
by 2.3.2
= (W 0 W)((1 0 Y*).r(x))
and so the result is proved, by linearity.
2.3.5 Definition. Let H = (M,1', ic) be a co-involutive Hopf-von Neumann algebra, and cp a faithful, semi-finite, normal weight on M, satisfying (HWi). Then, we define, for all w in M*, an unbounded operator A (w) by: A(w) = t(w o K)
We have then: V(A(w)) = nn(`nw)
a(w)A,o(x) = 4((w o K 0 i)P(x)) A(w * w') = A(w)A(w')
(x E 7to)
(W, W' E M*)
(A(w)A4,(x) I ma(y)) = (w o x 0 w)((1 0 v')r(=))
(w E Ma X, Y E NW)
2.3.6 Example. Let G be a locally compact group; let us consider the Kac algebra &(G) defined in 2.2.5; for all f in L1(G) the operator A(f) defined in 2.3.5 is equal to the restriction of AG (f) to L2 (G) fl L°O (G), where AG is the left regular representation of L1(G) (cf. 1.1.5).
Proof. By 2.3.5, we have, for f in L1(G), g in L2 (G) fl Lo' (G), tin G:
(A(f)g)(t) =
IG
f
(s)g(s-1 t)dt
= (AG(f)g)(t)
by 1.1.5
60
2. Kac Algebras
2.3.7 Example. Let us compute the mapping A' associated to the co-involutive Hopf von Neumann algebra El' and the weight Sp'. Let w be in. M*; we note
w' the element of M' defined, for all x in M', by:
(x,S) = (Jx*J,w) We easily get (w o rc)' = w' o ice, and w'(JxJ) = w(x) for all x in M. Then, we have, for x in gyp:
JA'(w')JA,(x) = JA'(wi )Aw'(JxJ) = JAW, (P 0 K11 0 i).r'(jXj)) 1
= JAW, MW o rc)' (& 0(1 0 T)P'(x)(J 0 J)) = AW((ZU -Orv 0 i)-F(X))
by 2.3.5
- A(6j-)AW(x)
and then: A'(WI) = JA(W)J
2.4 The Fundamental Operator W In what follows, (M, T) is a Hopf-von Neumann algebra and V a left-invariant weight with respect to T (so Sp satisfies (HWi)).
2.4.1 Lemma. For all x, y in gyp, we have: (9 0)((X* 0 ').P(Y*Y)(x 0 1)) =
Proof. By definition (2.2.1), we have:
(i 0 w)r(y*y) = V(y*y)1 whence, by ([59] 2.1 (3); cf. 2.1.8 (i)): V(Y*Y)X*X = X*(i 0 V)F(Y*Y)X *
= (i 0 VWX 0 l)-P(Y*Y)(x 0 1)) Then: W(X*X)W(Y*Y) = (V 0 w)((x* 0 l)-r(y*y)(x 0 1))
2.4 The Fundamental Operator W
61
2.4.2 Proposition. (i) There exists a unique isometry W such that, for every x, y
91., we have:
W(A,(x) ® jtW(y)) = Al,®,(r(y)(x ®1))
(ii) For all x in M, we have:
I'(x)W = W(1®x) (iii) W belongs to M ®,C(HV) Then, W is called the fundamental operator associated to (M, F, Sp).
Proof. By polarization and linearity it follows from 2.4.1 that, for all Si, x2, Y1, y2 in 91, we have:
0 1)r(y2y1)(x1 0 1)) = P(x2x1)'P(y2y1) which can be also written as:
(A®,(r(y1)(x1 ®1)) n ,®c,(r(y2)(x2 0 1))) = (A,(xi) 0 AS0(yl) I AV(X2) 0 A (y2))
p) is dense in H,, 0 H,,, there exists an isometry W in ,C(HW 0 H,) such that, for all x, y in, we have: As AW(91W) ® A,p(
W(Aw(x) 0 AW(y)) = AVOW(F(y)(x (9 1))
which is (i). Let x be in M, y, x in 91.. We have:
.P(x)W(A,(y) 0 A,(z))
-P(x) Avow (IV(z) (y 0 1))
by (i)
Aw(&w(-P(xz)(Y 0 1))
= W(AW(y) 0 Aw(xx)) W(l 0 x)(AW(y) 0 AV(z))
by (i)
By linearity, continuity and density, we obtain (ii). Let a, b be in gyp. We have:
(JaJ 0 JbJ)W(Aw(x) 0 AW(y)) = r(y)(x (9 1)(JA,(a) ® JA9p(b)) = r(y)(JaJA0(x) 0 JAW(b))
= r(y)(JaJ 0 1)(A,(x) 0 JAW(b)) This equality still holds, by continuity, when AW(x) converges to any vector e in H,, and when a strongly converges to 1. Therefore, we have:
(10 JbJ)W( 0 AV(y)) = -P(y)( 0 JAV(b))
2. Kac Algebras
62
which is valid for all y in 91, and
in Hg,. Let x' be in M' and let us replace
by x' in the above formula: (1
JbJ)W(xi 0 1)(C 0 iv(y)) = r(y)(x 0 1)(E 0 JnW(b)) = (X 0 1).r(y)(C 0 JAW(b)
= (x' 0 1)(1 0 JbJ)W(C 0 An(y)) which still holds when b strongly converges to 1, then we get:
W (x' 01)( 0 A(y)) =
(x'
01)W(6 0 AW(y))
By linearity, density and continuity we can conclude that W and
x' ® 1
commute, therefore W belongs to M 0 ,C(HI). 2.4.3 Lemma. We have, for all x, y in 91V and w in M* :
w((i 0 )(r(y*)(1 0 x))) = (A,(x) I (w 0 i)(W)A,(y))
Proof. Let a be in; we have: WA,(a)((i 0 P)(F(Y*Xl 0 x)))
_ (w 0 w)((a` 0 i)r(v')(a 0 x))
_ (A1,(a) 0 An(x) I AWOn(r(y)(a 0 1)))
by 2.1.10 by (LIW)
0 aW(=) I N'(nw(a) 0 ew(v)))
by 2.4.2 (i)
_
(AV W I (wA, (a) 0 0 (W) Aw W)
and so the result is proved by continuity and linearity arguments.
2.4.4 Corollary. Let (M,1') be a Hopf-von Neumann algebra, and ( a leftinvariant weight with respect to 1. Then: (i) for all w in M*, the linear mapping £(w) defined in 2.3.2 by:
An(X) -'
0 i)I'(T))
(x E NW)
is bounded; we shall still note £(w) its unique continuous extension to Hg,; it satisfies: £(w) = (w 0 i)(W) and I_< IIwII
(ii) the fundamental operator W satisfies:
(r 0 i)(W) = (10 W)(a 0 1)(1 0 W)(a 0 1)
2.4 The Fundamental Operator W
63
which can also be expressed as the so-called "pentagonal relation":
(10 W)(o ® 1)(1 0 W)(cr 0 1)(W 01) = (W 0 1)(1 0 W) .
Proof. Let w be in M; , x, y in'JIp; we have: (AW(x) I i(w)AW(y))
by 2.3.2
(AW(x) 14((w 0 i)-P(y)))
V(((w 0 i)f(y*))X) W((i 0 O)TWX1 0 X)) (A,(x) I (w 0 i)(W)AW(y))
by 2.4.3
By linearity, we see that, for all w in M* , £(w) is the restriction of (w 0 i) (W ) to A ,(')1,); so (i) is proved. If we apply (1.5.1 (i)), to the linear mapping w -+ £(w) = (i 0 w)(oWo) and to the Hopf-von Neumann algebra (M, cI'), we get:
(i 0 sr)(awa) = (awa (9 i)(1
)(iWa
a)
from which we deduce:
(1' 0 i)(w) = (10 wXU 0 1)(1 0 w)(a 0 1) The pentagonal relation comes then from 2.4.2 (ii).
2.4.5 Theorem. Let (M, T) be a Hopf-von Neumann algebra, and V a leftinvariant weight with respect to T. Then IV (defined in 2.1.6) is a left ideal in M*; moreover, we have, for all w in M,, and wo in I.: a(w * wo) = t(w)*a(wo)
and then: lw * wollSP :5 lIwlI llwolko
Proof. Let x
91,; we have:
(x*,w*w0) = (X'Co
Cc1o)-
by 1.2.1
= ((CA) 0 i)T(x),LOO)
= (((w ® i)I'(x))*,wo) = (a(wo) I A ,((w 0 i)I'(x))) = (a(w0) l e(w)AvW) = (C(am)"a(wo) I A,o(x))
by 2.3.2 by 2.4.4 (1)
64
2. Kac Algebras
which gives, by density, the first result; the norm inequality is just then a staightforward application of 2.1.6 (ii) and 2.4.4 (i).
2.4.6 Proposition. Let H = (M, I', ic) be a co-involutive Hopf-von Neumann algebra, and cp a left-invariant weight with respect to F. Then: (i) For all w in M*, the linear mapping A(w) defined in 2.3.5 by: A(w)Aw(x) =
(x E
A(p((w o re 0 i)r(x))
)
is bounded; we shall still note )(w) its unique continuous extension to H.; it satisfies: A(w) = (w o ,c ® i)(W)
and
IIA(w)II C 11wII
(ii) For all w in M* and all wo in I,, we have: a(w * wo) = A(w°)*a(wo)
(iii) I , fl i., is a dense involutive subalgebra of M*. (iv) The weight W satisfies (HWii) if and only if A is involutive; we have then, for all w in M* and all w' in I,,: a(w * w') = A(w)a(w')
(v) The weight cp satisfies (HWii) if and only if, for all w in M*, we have:
(wOi)(W*) = (woicOi)(W) . Proof. The assertion (i) is just a rewriting of 2.4.4(i). Thanks to 1.2.5, (ii) comes from 2.4.5. By 2.1.7 (ii), we see that I,, is a dense left ideal of M*; so is a dense right ideal of M*, and the algebraic tensor product I Q IV is a dense subspace of (M ®M)*. Let x be in M such that (x, w1 * w2) = 0 for all w1 i w2 in IV. We get (F(s), wi 0 w2) = 0 which therefore implies x = 0. Thanks to the Hahn-Banach theorem, it follows that the subspace generated by Iw' * I, is dense in M*. But, by 2.4.5, this subspace is included in Ip fl and so (iii) is proved. Using 2.3.5, we get, for any w in M*, x, y in 9lw:
(A,(x) I A(w°)A ,(y)) _ (A(w°)A,(y) I A,(x))
_ (w° o rc 0 )((1 0x)I'(y)) _ (w
x*)f(y))-
_ (w ®SP)(r(y*)(1 0 x)) Using again 2.3.5, we have: (AW(x) I A(w)*Aw(y)) = (w o x 0 p)((l 0 y*)F(x))
by 1.2.5
2.4 The
Fundamental Operator W
65
And, by density and continuity, we get (iv). If we apply 1.5.1 (ii) to the linear mapping w -4 A(w o r%) = (i (9 w)(cWa) and to the co-involutive Hopf-von Neumann algebra W, we get that A is involutive if and only if, for all w in M*, we have: (i 0 w o rc)(o*Wc) = (i 0 w)(oW*cr)
and so (v) comes from (iv). 2.4.7 Proposition. Let G be a locally compact group. (i) The function s -+ AG(S), considered as an element of Cb(G,,C(L2(G))), which can be identified to a subspace of ,C(L2(G))®L°O(G), is equal to cWGc, where WG is the fundamental operator of the Kac algebra Ka(G). (ii) The ideal IVa of the Kac algebra Ka(G) is L1(G)nL2(G); the mapping a is then the canonical injection from L1(G) n L2(G) into L2(G). Proof. Let 91, 92 be in Loo (G) n L2(G); by definition, we get, for s, t in G:
(W(g1 ® 92))(s, t) = 92(st)gl(s) where we identified L2(G) ® L2(G) with L2(G x G). By continuity, for all f in L2(G x G), we get:
(WGf)(S,t) = f(s, St) and:
(crWGaf)(s,t) = f(ts,t)
We identify L2 (GI L2(G)) with L2(G) ® L2(G) (or L2 (G x G)): OP in L2(G, L2(G)) will be identified to the function f : (s, t) -+ (t)(s). We have then:
(AG(F1)(t))(S) ='P(t)(ts) = f (ts, t) = (crWGof)(S,t) = (aWGo(t))(s) so we get (i); and (ii) is trivial. 2.4.8 Proposition. The fundamental operator W' associated to (M', I'', Sp') as defined in 2.2.3 is:
W, _ (10 J)W(J 0 J). Proof. It is an easy calculation, using the identification AV(JxJ) = JAW(x), for all x in 91W.
2.4.9 Lemma. Let (M, I') be a Hopf-von Neumann algebra, co a left-invariant weight with respect to r, W the fundamental operator associated by 2.4.2; let A be a von Neumann algebra, 0 a faithful semi finite normal weight on A,
66
X in
2. Kac Algebras
,and y in; then, (i 0 r)(X)(1 0 y 0 1) belongs to
and we have:
0 r)(X)(1 0 y & 1)) = (10 W)(1 0
0 Ap(y)) -
Proof. We have:
( 0V ® p)((1®y* ®1)(i ® r)(X *X)(1®y ®1)) = SP(y* (O ®z ®W)(i 0 r)(X *X )y)
_
_
(*(O
® co)(X *X )y)
(y*)(O
0 co)(X*X) < +oo
which gives the first part of the result. After polarization, we get, from the calculation above, the existence of an (9 F)(X)(1 0 y ®1)). isometry U sending A,,®,(X) 0 A,,(y) on Now, let xl be in 970, x2 be in 91W. We have: AjO®WOSp((i 0 r)(x1 0 x2)(1 0 y 0 1))
= A,,(xl) ® A W(r(x2)(y ®1)) by 2.4.2 = A,(xl) 0 W (Aw(y) 0 A,(x2)) = (10 W)(1® o)(A,(xi) 0 Aw(x2) 0 Ap(y)) Therefore U coincides with (1® W)(10 a) on all the vectors which are of the form A, , (x i) 0 AW(x 2) 0 A , (y ), which, by linearity, continuity and density, completes the proof.
2.5 Haar Weights Are Left-Invariant In the sequel, H = (M, r, ic) is a co-involutive Hopf-von Neumann algebra, and co a faithful, semi-finite, normal weight on M, satisfying (HWi) and (HWii); so, we can use the results 2.3.5 (but not 2.4 which is about leftinvariant weights, which may not satisfy (HWii)). 2.5.1 Lemma. For all w in M*, and x in 91,,, we have: AW(91w) C D(A(w)*)
and
A(w)*Aw(x) = A(w°)A,(x)
Proof. Let us assume that w is positive and w(1) = 1. We have, for all y in
(a(w)l(=) I Aw(v)) = (w 0 K 0 w)((1 0 v')r(=))
by 2.3.5
2.5 Haar Weights Are Left-Invariant
67
_ (w
0 X))) W((w (9 i).P(y*)x)
(AV(x) AV(w 0 i).P(y)) (A,,(x) A(w")AW(y)) by 2.3.5 and the hypothesis So, we get A(w)* D A(w°) which is the lemma.
2.5.2 Lemma. Let C be in 11,(91,p). The linear form vi on M*, defined by vo(w) = (a(w)e I C) is continuous, and IIii
11 <
IICII2.
Proof. By adding a unit e, let us consider the involutive Bannach algebra M* ® Ce. We can extend vv to i , by putting, for all w in M*, a in C:
E(w+ae)= F(w)+allEll2 Then, we have: vt((w° + de) * (w + ae))
=- (WO*W+dW+aWO+da) - v{(w° * w) + avo(w) + avt(w,) + aagI12
(,X(w' w) I ) + (A(w) I Ila(wg112 + (a(w)l I JJA(w) + a6112 > 0
(A(Lo*)a616) + (aE I
Ilgli2 by 2.3.5 and 2.5.1
Therefore v is positive on M* ® Ce, so is continuous, and we have: IldI = V-<(e) =
which implies the lemma, by restriction to M*. 2.5.3 Theorem. With the hypothesis of 2.5, for all w in M* the operator A (w) defined in 2.3.5 by:
A(w)Aw(x) _ A,((w o Ic 0 i)r(x))
(x E
p)
is bounded; we shall still note A(w) its unique continuous extension to Hw. The mapping A from M* to ,C(H.) then defined is a representation of the involutive Banach algebra M*, which will be called the Fourier representation. We shall write M the von Neumann algebra on H,p generated by A.
68
2. Kac Algebras
Proof. Thanks to 2.5.2, for all in A4,(1,), we have: (A(w° * w)g I e) <
11w° * wit
which implies, using 2.3.5 and 2.5.1: II a(w)e 112 <_ lIwil2g112
So A(w) is bounded, and I< 11w II . The whole proof is given by 2.3.5 and 2.5.1.
2.5.4 Example. Let G be a locally compact group; let us consider the Kac algebra K ,,,(G) defined in 2.2.5; using 2.3.6, we see that the Fourier represen-
tation of ][(G) is the left regular representation of L1(G), and the algebra L°O(G)^ is then the von Neumann algebra ,C(G) defined in 1.1.7.
2.5.5 Lemma. For all w in M*, and t in R, we have: (i) A(w)z =L1 A(w) (ii) JA(w)*J = A(w o x). Proof. Let x be in Y, fl 9j,. Let us assume that w is positive. Then, we have:
S,a(w)Aw(x) = S ,A ,((w o rc 0 i)r(x)) _ !1,((w o IC 0 i)r(x*))
by definition
_ A(w)Aw(x*) _ A(w)SWA,(x)
Since A (qi, fl qr) is a core for S,, we get: A(w)S, C S ,A(w)
By substituting w° to w (w° is still positive), we get: A(w°)S, C SpA(w°) or, using 2.5.3: A(w)*Sw C SWA(w)*
and, by transposing: a(w)F,p C
Then, we get :
a(w)ay = A(w)F0S, C FFA(w)SS C FpSpA(w) = A,A(w) from what (i) follows for w positive and by linearity in the general case.
2.5 Haar Weights Are Left-Invariant
69
We have also, for all w in M* and x in gi, f 1 qr : by 2.5.3
SwA(w)*SSpAp(x) = S A(w°)Acp(x*)
S Av((wo
= o ,c ® i)r(x* )) = A,,((w 0 i)r(x))
by 1.2.5
_ A(w o n)A,(x) Since 11,(gi, fl qi*) is a core for Sip, we get:
Ja1 /2A(w)*a-1/2J =Sa(w)*S, C A(w o n) which, by (i), implies: JA(w)* J = A(w o K)
which completes the proof.
2.5.6 Corollary. For all t in R, we have:
rat =(i®at)r Proof. Let x be in 91w, w in M*;we have:
A,((w o,c 0 E)(rat (x))) =
by 2.3.5
(x)) a,;A(w)nw(w)
by 2.5.5 (i)
= Aw((w o K 0 i)(I'(x))) = Aa(atMw o K 0 z)(I'(x)))
by 2.3.5
_ Aa((W o K 0 i)((i 0 °t )(r(te))) which implies:
(w o x ® i)(rat (x)) _ (w o,c 0 i)((i ®cr )r(x))) therefore, for all x in Map, we get :
rat (x) _ (i ® ot)r(x) which completes the proof, by density and continuity.
2.5.7 Corollary. Let x in M, analytic with respect to Sp. Then, for all w in M*, the element (w ® i)r(x) is analytic with respect to p, and, for all z in C, we have: 01,P((w 0 OFW) = (W 0 OF(aftx)) -
70
2. Kac Algebras
Proof. The function z --+ (w ® defined on C is analytic, and, by 2.5.5, it extends to C the function defined on R by t -- a '((Lo ® i)I'(x)).
2.5.8 Theorem. Let H = (M, I', rc) be a co-involutive Hopf-von Neumann algebra, Sp a faithful, semi-finite, normal weight on M, satisfying (HWi) and (HWii). Then co is left-invariant.
Proof. Let w be in M* , y in and x j a family of elements satisfying the hypothesis of 2.1.2. We can write down: S0(((w ® i)1'(y))(xj )) _ W((w 0 i)(r(y)(10 Xj)) _ (w (D p)(I'(y)(1 0 x j ))
_ (w o tc 0 V)((1 0 y)r(x;)) = cp(woK ®i)((1 ®y)I'(xj)) _ cp(y(w o n 0 i)(I'(xj )))
by (HWii)
It follows from 2.3.1 (i) that (w 0 i)I'(y) belongs to 9, and from both 2.3.1 (ii) and 2.5.7 that (w o x ® i)F(x j) satisfies the hypothesis of 2.1.2. By passing to the limit, we get, for all w in M* and y in : w((w 0 a)r(v)) = W(Y)W(l)
with w being fixed, these two expressions will define two faithful semi-finite,
normal weights on M which coincide on second one is a and we have: (w
(w
9X+,,.
w)(
The modular group of the
o, )r(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 all y in M+ and w in M* : w((i 0 )r(y)) = w(Sp(y)1) By linearity it is true for all w in M*, which completes the proof. 2.5.9 Example. The Fourier representation A' of ]HI' satisfies:
A'(w')=A(w), VwEM* where w'(x) = w(Jx*J), for all x in M'. Proof. In 2.3.7, we have got A' (w') = JA(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 WIs Unitary We keep on with the same hypothesis as in 2.5. The weight Sp 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, A, and the ideal I,, (2.4.6). Let us recall that M is the von Neumann algebra generated by the Fourier representation A (2.5.3). 2.6.1 Proposition. The fundamental operator W, the Fourier representation A and the ideal I p are linked by: (i) For all w in M*, we have:
A(w)=(wo1 0Off) _(w®i)(W*) (ii) For all Q in M*, we have: Q)(W*) = i(i (9 Q)(W) A*(Q) = (i 0 (iii) For a, #,,y, 8 in Hip, we have the connection formula:
(W(a 0 P) I ^10 8) = (/3 I A(w-Y,a)6)
(iv) The fundamental operator W belongs to M 0 M (v) For all w. in M* and w' in lip, we have: A(w)a(w') = 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 3 of H,p implementing x (i.e. such that
ic(x) = Jx*3 for all x in M), we have:
W* = (10 J)W(J ®J)
.
Proof. Let us recall that, for all a, /3 in Hip, wJa,JQ = w"Q (1.2.8). So:
(w 0 i)((3 0 J)W (3 0 J)) = J(w° 0 i)(W )J = JA(w° o ic)J = JA(w o K)*J = A(w)
= (w 0 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)
2. Kac Algebras
72
2.6.3 Corollary. (i) The Fourier representation A is non-degenerate, and crW*c is its generator in the sense of 1.5.2. (ii) For all x in M, we have:
F(X) = w(l 0 X) w*
.
Proof. The assertion (i) is clear by 2.6.1(1), 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 0 M, we have:
(i 0 -P)(X) = (10 W)(1 0 a)(X 0 1)(10 a)(1 0 W*) . Proof. Let a be in A, x in M. We have:
(iOfl(aOx).aOr(=) = a0 W(l0 x)W' =(10 W)(a0 10 x)(10 W')
by 2.6.3 (ii)
=(1®W)(1 0 a)(a 0 x 0 1)(10 a)(1 0 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 0 M such that:
(i 0 )(V) = (V 0 1)(1 0 a)(V 0 1)(10 o) Then, for every w in M, we have: (i ®w o ic)(V) = (i 0 w)(V * )
The mapping T : M* -- A defined by r(w) = (i 0 w)(V) is a non-degenerate representation of M, and V is the generator of r. Proof. By 2.6.4, we have:
(10 W)(1 0 9)(v 0 1)(10 a)(1 0 W*) = (V 0 1)(1 0 U)(v 0 1)(1 0 a) which can also be written:
(V® 1)(1 0 °)(1 0 W*)(1 0 a)(v* 0 1) = (10 Q)(1 0 W*)(V 0 1)(1 0 a) (*)
2.6 The Fundamental Operator W Is Unitary
73
Let us consider the representation of M* in A ® £(H) defined by: µ(w) = V(1 0 A(w))V* Therefore, for all w in M* 1, we have, using 2.6.1 (i):
µ(w) _ (i 0 i ®w)((V ® 1)(10 o.W*cr)(V* ® 1))
(i 0 i 0 w)((1 ® o)(1 0 W*)(V 0 1)(10 a))
by (*)
For all f11 inA*1, (12 in£(H)*1, and w in M, we have:
ta(m), n, 0 nz) = (Qi 0 nz 0 W)((1 0 o)(v 0 10 0 WO 0 o)) _ (n1 0 w 0 n2)((v* 0 1)(1 0 w)) = Wl 0 w)((i 0 i 0 fl2)((V* 0 1)(10 W))) A (9 w)(V*(1 (2) (i (9 fl2)(W))) W((121 (9 i)(V*(l 0 (i 0 S?2)(W)))
W(A 0 i)(V*)(i 0 fl2)(W)) Therefore, by 1.5.1 (ii), we have:
K((S?1 0 i)(V*)(i 0 Q2)(W)) _ ((Qi 0 i)(V*)(i 0 f12)(W))* which can be written as follows:
tc((J?1 0 i)(V))ic((i (9 Q2)(W*)) = (Qi 0 i)(V*)(i (9 Q2)(W) since f11 and (12 are positive. If we apply 1.5.1 (ii) again, we get: K(f2l ® i)(V)(i ® Q2)(W) = (121 0 i)(V *)(i ® Q2)(W ) or:
(i 0 122)(0(121 ® i)(V) ®1)W) = (i 0 Q2)(((fli 0 i)(V*) ®1)W ) which, by linearity, will still holds for all f12 in .C(H)*, and therefore implies:
(K(Q1 0 i)(V) ®1)W = ((Qi ® i)(V*) ®1)W As W is unitary, we have:
IC(i21 (9 i)(V) = A 0 i)(V*)
2. K ac Algebras
74
and by 1.5.1 (i) and (ii), r is a representation and we have, for all w in M*: (i ®w o rc)(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, T, KK). Then, we have:
n(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, T, ic) the Kron-
ecker 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 y is a representation of M*, we have, for all w in M*, with the notations of 1.1.1 (ii): (/1 x u)(W) = N(u W) (u x µ)(w) = µ(w - u) .
Proof. By 1.2.3, u is a unitary and satisfies T(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 y and u.
2.6.7 Proposition. Let R be in the centre of M, such that:
F(R) > R OR x(R) = R Then, we have:
(i) W(R R)=(R0 R)W (ii) I'(R)(R 0 1) = I'(R)(1 0 R) = R 0 R. Proof. By hypothesis, we have, using 2.6.3 (ii):
R®R= (R0R)T(R) = (R 0 R)W(1 0 R)W* Using 2.6.2, we get:
R®R=(R®R)(J®J)W*(J®J)(1®R)(J®J)W(J®J)
2.6 The Fundamental Operator W Is Unitary
75
and, using the fact that JRJ = R (because R belongs to the centre of M) and that JRJ = R (because c(R) = R), we have: R ®R = (R ®R)W*(1®R)W Taking adjoints, we get:
R®R=W*(10R)W(R®R) and, as W is unitary:
W(R0 R)=(10 R)W(R0 R) = (R 0 R)W(1 0 R)
_ (Re R)W
by 2.6.1 (iv) by (*)
So, (i) is proved. We have then:
I'(R)(R ®1) = W(1® R)W*(R ®1) = W (R 0 R)W*
=ROR
by 2.6.3 (ii) by 2.6.1 (iv) by (i)
Applying this result to (M, cr, N1 Sp o r.), we get:
I'(R)(1 0 R) = R 0 R which ends the proof.
2.6.8 Corollary. Let P, Q two projections in the centre of M, such that:
F(P)>P®P
K(P)=P
F(Q) > Q ®Q
K(Q) = Q
P-+Q>1 Then, either P or Q is equal to 1. Proof. From 2.6.7 (ii), we have:
r(P)((1- P) ®P) = o r(Q)(Q®(1 - Q)) = 0 which implies, as 1 - Q < P and 1 - P < Q, by hypothesis:
FWX(i - P) 0 (1 - Q)) = 0
FMO - P) 0 (1 - Q)) = 0
76
2. Kac Algebras
Taking the sum of these equalities, as
(P) + 1T(Q) > 1, we get:
which ends the proof.
2.7 Unicity of the Haar Weight Let (M, r, ic) be a co-involutive Hopf-von Neumann algebra and gyp, 0 two faithful semi-finite normal weights on M+, satisfying both (HWi) and (HWii).
2.7.1 Lemma. Let z be in M such that f(z) = z 0 1; then z is scalar. Proof. We have, for all t in R:
r(at (x)) _ (i 0 at')-r(z)
by 2.5.6
(i oat'P)(Z =x0 1 = r(x) As r is one-to-one, or(z) = z and z belongs to My'. Let x, y be in 91,; we have:
W(1 0 Jz*J)(Av(x) 0 W(Aw(x) 0 Av(yz)) = A,p0W(1'(yz)(x 0 1)) = Av(&v(F(Y)-r(Z)(x 0 1))
by 2.4.2 (i)
= nw(&sa(I'(y)(zx 0 1)) = W(Aw(zx) 0 AW(y)) = W(z 0 1)(Aw(x) &
by 2.4.2 (i)
by linearity and density, we get:
W(1 0 Jx*J) = W(z 0 1) and then:
10Jz*J=z®1
which implies z being a scalar.
2.7 Unicity of the Haar Weight
77
2.7.2 Proposition. The relative position of M and M is such that:
(i) Mflicf' =C (ii) M' n M' = C. Proof. Let x be in M n M'; we have, then:
1'(x) = W(l 0 x)W*
=10x
by 2.6.3 (ii)
by2.6.1(iv)
and then: r(rt,(x)) = c(rc 0 K)r(x) = c(rc 0 rc)(1 0 x) = c(x) 01
Using lemma 2.7.1, we see that .(x) is a scalar, and so is x too; therefore (i) is proved. If we apply result (i) to H', and to the weight 'p' which satisfy the same hypothesis by 2.2.3, we get (ii), using 2.5.9. 2.7.3 Corollary. (i) Let A be a von Neumann algebra. Let P be a projection
of A 0 M such that (i 0 sr)(P) < P 0 1 (resp. (i (9 sr)(P) > P 0 1). Then there exists a projection Q in A such that P = Q 0 1.
(ii) Let P be a projection of M such that r(P) < P 0 1; then, we have
eitherP=O, orP=1. Proof. From 2.6.4, we have (i ®sr)(P) = (10 cWa)(P 0 1)(1 0 crW*o). So, the hypothesis (i 0 sr)(P) < P 0 1 may be written:
(10 oWQ)(P 0 1)(1 0 oW*Q)(P 0 1) = (10 aWo)(P 0 1)(1 (9 oW*o) or:
(P 01)(10 aw*a)(p 01) _ (P 01)(10 oW*c) For allwinM*l,weget: (i 0 i 0 w)((P 0 1)(10 aw*a)(P 0 1)) = (i 0 i 0 w)((P 0 1)(10 aw*a)) Or, by 2.6.1 (i):
P(10 A(w))P = P(1 0 )t (w)) which, by linearity, is true for all w in M* . Taking the adjoints, we get, thanks to A being a representation: P(1 0 A(w)) = (10 A(w))P
which, by continuity, implies that P belongs to A 0 if'; the result (i) comes then from 2.7.2 (i). With the hypothesis (i 0 sr)(P) > P ® 1, the proof if analogous; taking A = C, one gets (ii).
78
2. Kac Algebras
2.7.4 Proposition. (i) Let x in £(H). Then, x belongs to Mr if and only if.-
(10 x)W = W(1 ®x) (ii) Let x be in M', such that:
W*(10 x)W = 10 x then x is scalar. Proof. We have the following sequence of equivalences: x E 1Vl r t x* A(wy,a) = A(wy,a )x* Va, y E H (/31X*A(W7,a)8) = (/31A(W7,a)x*8) Va, l3, 7 E H V a, fl, 'T E H 4* (x,@ I = (3 I A(w7,a )x* 8)
by2.6.1(iii)
W(1®x)=(1®x)W So, (i) is proved; (ii) is then clear from (i) and 2.7.2 (ii).
2.7.5 Proposition. For all t in R, we have: aW om
t atoms
o
= loo'__t rt.
_
(iv) The space A<,(qiw fl o,,) is dense in HW. (v) The space A ,(9-L. fl giwon fl 9r fl 9i,* is dense in H.. Pox)
Proof. The assertion (i) is just an application of 2.1.3. We have:
fa`t OM = r,ca_t,c = c(ic 0 ic)(i 0 a!t)rrc
by (i) by 1.2.5 by 2.5.6
= c( 0 i)(i 0 o!t)(lc 0 K)sr
by 1.2.5
= c(ic 0 r.)ra!trc
= c(i
t
(aT ok ®i)r
So (ii) is proved.
by (i)
2.7 Unicity of the Haar Weight
79
Let x be a positive element in M. We have: o'c(X) Oor.
(i P)(at 0 i)F(x) atlor'vi 0 V).r(x))
by 2.5.8
by (ii) applied to 0 by 2.5.8
W(x)l
So (iii) is proved; then (iv) is then an application of 2.1.4. Now, let x be in 9 , n po,c . As n givo,. = r. and as, because of (iii) and ([114], prop. 5.6. and 5.10), the weight cp + W o rc is semi-finite,
there is a net of projectors ei ingiw n givor. monotonely converging up to 1. It is clear that ei x belongs to 'p n gr n qiwor,, n por and that A(ex) = e i A , (x) converges to A , (x) . Then Aw ( n gr n 9vo rc n for.) is dense in !1w(
no,,); using (iv), we get (v).
2.7.6 Theorem. Let K = (M, r, x, () be a Kac algebra, let W be its fundamental operator. Then, for all t in R, we have: (z)crj or = at
(ii) rot = (i ®oT)r = (ot ®i)r
(iii) r(M) is a crt ®4-invariant subalgebra of M 0 M.
(iv)(1'(M)UMOC)" = MOM and (r(M)UCoM)" = MOM (v) Let x be in ,C(H). Then x belongs to M' if and only if:
(xOl)W=W(xOl) (vi) W*(ait®1)W =z1 O'Ait (vii) W*(aw®1)W =QV®ap. Proof. The assertion (i) is a straightforward consequence of (HWiii) and 2.7.5 (i). Then (ii) comes from 2.5.6, 2.7.5 (ii), and (i).
As we have at ®'P = at (ii) implies f®®'r = ra2 and so we get (iii). Let us put N = ((M) U (M 0 C))". It results from (iii) that N is a at ®'P-invariant subalgebra of M 0 M. For x and y in 9, by 2.4.1 we get that r(y)(x (9 1) belongs to )t'®9, n N. From what it is easy to deduce that the restriction of W 0 Sp to N is semi-finite. Therefore, by 2.1.8 (ii), there is a faithful normal conditional expectation E from MOM to N, and, if P stands
for the projection from HH®w onto Avow (,®flN), E is such that for any x in M 0 M, Ex is the unique element of N such that (Ex)P = PxP. But the closure of 4o,(9iow n N) contains the closed subspace (using 2.4.2 (i)):
{n®(r(y)(x o i)).=,v E
m.,}- = {W(A(x) 0 nw(v)),=,v E 7u,}
80
2. Kac Algebras
As W is unitary (2.6.2), this subspace is H, 0 Hop. Therefore we have suc-
cessivelyP= 1, E=i and N =MOM. Let us apply this result to Ks (2.2.5), and we get (iv). Let now x in M'; we have W(x 0 1) = (x 0 1)W by 2.4.2 (ii). Conversely, let us suppose W (x 01) = (x ®1)W . For any y in M, we have: (x 0 1)W(1 0 y)W* = W(1 0 y)(x 0 1)W* = W(1 0 y)W*(x 0 1) = 1'(y)(x 0 1)
(x
by 2.6.3 (ii) by hypothesis by hypothesis by 2.6.3 (ii)
So (x 0 1) commutes with F(M); as it commutes with C 0 M, by (iv), it commutes with M 0 M and we get (v). Let z, y in 9lW. We have:
(gy 0 1)W(A,a(i) 0 iLa(Y))
_ (A'W 0 1)hvOn(r(Y)(z 0 1))
by 2.4.2 (i)
AW04at, 0 i)((F(Y)(z 0 1))) = nvoa(r(ai (Y))(ai (z) 0 1)) W(Ajo(atp(z)) 0 A.(at,(Y))) = w(oit Oa=t)(n,(z) 0 AW(y))
by (ii) by 2.4.2 (i)
from what follows (vi), by density and because W is isometric. It leads directly
to (vii) because then the infinitesimal generators of those two continuous groups of unitary operators are equal. 2.7.7 Theorem. Let K = (M, I', ,c, Sp) be a Kac algebra, and let b be a faithful,
semi-finite, normal weight on M+, satisfying (HWi) and (HWii). Then, and Sp are proportional.
Proof. We have, for all t in R:
0 =00 aSpoIC = oQw
by 2.7.5 (iii) applied to the weights b and y
by2.7.6(i)
then, using ([114], prop. 5.6 and 5.10), we see that the weight 9 = W + 0 is semi-finite. As 91e = 91, f101 , it is clear that 9 satisfies (HWi) and (HWii); we have, then, for all t in R:
9 = 9 o a °" = 0 0 ov
by 2.7.5 (iii) applied to the weights 9 and W by 2.7.6 (i)
2.7 Unicity of the Haar Weight
81
As Sp < 9, there is an injective positive operator h in M8, 0 < h < 1, such that cp = 9(h.). It implies, for x in 910: sO(X*X) = I
w)((x* (9 l)-F(Y*Y)(x & 1))
= I(9 = I(9
Jeh112Je)Ae0e(r'(y)(x 0 1))112 Jeh112Je)WeAe(x) (9 ne(y)112
by 2.4.2 (i) applied to (M, I', 9), where We stands for the fundamental operator associated to (M, I', 9). By applying 2.4.2 (i) to (M, r, (p), it is also worth: II=
I(9 Jeh112Je)(Ae(x) (9 Ae(y))112 Therefore, we have:
IIW(Joh"2Jo 0 Jeh112Je)We(Ae(x) 0 ne(y))112
= II(Joh'12Jo 0 Jgh112Je)(A8(x) 0 Ae(y))112 and, by the unicity of the polar decomposition, we get: WB (JBhTe 0 JBhJe)WB = JBhJe 0 JBhBJe
As We belongs to ire (M) ® £(H9) by 2.4.2 (ii) applied to (M, I', 9), and Je h1/2 J8 belongs to -7r9(M)', we have:
(J0hJ9 01)W(1 ® JB hJ9 )W0 = J0 h J0 0 J0 h J9
and, since h is one-to-one: W0 *(1 ® Je h Je) We = 1® Je h Je
By applying 2.7.4 (ii) to the quadruple (M, r', rc, 9) and the operator JO hJ0, we see that this operator is a scalar, so is h too; it means that cp and 9 are
proportional, and so are ( and t too. 2.7.8 Corollary. Let K1 = (M1, F1, K1, SP1) and K2 = (M2, I2, X2, Sp2) be two
Kac algebras, u a surjective H-morphism from M1 to M2, Pu the greatest
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2. Kac Algebras
projection of the ideal Ker u, Ru = 1 - Pu. Then, there exists a > 0 such that: 'p2 o u(x) = a5p1(Rux)
Vx E M1 .
Proof. From 1.2.7 we get Ti (Ru) > Ru 0 Ru and x 1(Ru) = Ru. So, from 2.2.6, the quadruple K1Ru = (M1Ru , P Ru ) K1Ru , SP1Ru) is a Kac algebra. Let r be the canonical surjection M1 --> M1Ru 'We can define a bijective H-morphism
v from M1Ru to M2 by v(r(x)) = u(x). Clearly, 'P2 0 v is a Haar weight on K1Ru and then, using 2.7.7, there is a > 0 such that, for all x in M1Ru : '2 o v(x) = acolRu (x)
So, for all x in M: '2 0 u(x) = c02 o v(r(x)) = a'P1Ru (r(x)) = aSP1(Rux)
2.7.9 Corollary.Let K1 = (M1, I'i, i1, 'l) and K2 = (M2, I'2, K2, 'P2) be two Kac algebras, u an H-isomorphism from (Mi,rl,Ki) to (M2,r2,ic2). Then there exists a > 0 such that S02 0 u = acp , and so the Kac algebras K1 and K2 are isomorphic in the sense of 2.2.5.
Chapter 3 Representations of a Kac Algebra; Dual Kac Algebra
In this chapter, we shall use the notations hereafter: K = (M, F, ,c, gyp) will be a Kac algebra, A its Fourier representation, W its fundamental operator and M the von Neumann algebra generated by A. This chapter deals with the representations of the Banach algebra M*, following Kirchberg ([79]) and de Canniere and the authors ([21] ), and the construction of the dual Kac algebra, as found independently by the authors ([34]) and Vainermann and Kac ([180]).
This chapter begins with a Kirchberg's important result on Kac algebras: every non-degenerate representation of the involutive Banach algebra M* has a unitary generator (3.1.4 ). For the Kac algebra K. (G) constructed with L°O (G), one recovers the well-known result that every non-degenerate representation of L1(G) is given by a unitary representation of G. As a corollary, we get that, for any non-degenerate representation p, the Kronecker product A x p is quasi-equivalent to A (3.2.2 ); in the group case, that means that, for every unitary representation µG of G, the tensor product AG 0 PG is quasi-equivalent to AG, which is Fell's theorem ([48] ). When we choose p = A, we then get a coproduct T on M (3.2.2 ). A co-involution k on M is then defined, for all w in M*, by the formula: ik(A(w)) = A(w o rc)
The triple (M, T, k) obtained is a co-involutive Hopf-von Neumann algZbra. For locally compacts groups, that means (8.3.6) that the von Neumann algebra £(G) generated by the left regular representation AG has a co-involutive Hopf-von Neumann structure given by a coproduct P. and a co-involution is such that, for all s in G: Fs(AG(S)) = AG(S) ® AG(S) 1c8(AG(S)) = AG(S-1)
By predualizing the canonical surjection from the von Neumann algebra generated by M* onto M, one obtains an isometric, multiplicative and involutive morphism from the Banach involutive algebra M* in the Fourier-Stieltjes
84
3. Representations of a Kac Algebra; Dual Kac Algebra
algebra defined in Chap. 1, the image of which will be called the Fourier algebra of the Kac algebra, and is a self-adjoint ideal of the Fourier-Stieltjes algebra. For locally compact groups, we recover the situation of the Fourier algebra A(G) and the Fourier-Stieltjes algebra B(G) defined and studied by Eymard in [46]. Using the canonical Tomita-Takesaki construction, we define, starting from
the left ideal I. of M*, a left Hilbert algebra dense into the Hilbert space Hip, which generates the von Neumann algebra M and a faithful semi-finite normal weight WP on M (3.5.2 ), satisfying a Plancherel-type relation: (A(w)*A(w))
= If a(w)IP
for all w in I.
Moreover, the modular operator A, is affiliated to the centre of M, and is, in the sense of [114], the Radon-Nikodym derivative of the weight 50 with respect to the weight soon (1.6.7). We prove that this weight is a Haar weight (3.7.4 ), and we have so defined a dual Kac algebra K = (M, I', ik, cp), the fundamental
operator W of which is aW * a, and the Fourier representation A of which is given by icA* (where A. : M* -p M is obtained by predualizing A). As A is non-degenerate, A is faithful. ,o.
On ,C(G), the weight so constructed is equal to the Plancherel weight studied by Haagerup in [58]. So, Ks (G) = (r(G), I's, K, , W9) is another
example of a Kac algebra (3.7.5 ), which is symmetric and will be studied in Chap. 4. Another essential result about Kac algebras is the following: characters
on M* (that is, elements x of M such that x 54 0 and P(x) = x ® x), are unitaries, verify x(x) = x*, and, with the weak topology of M, form a locally compact group, called the intrinsic group of the Kac algebra (3.6.10). See also (1.2.2), (1.2.3) and (2.6.6).
3.1 The Generator of a Representation 3.1.1 Lemma. With the definition of 1.3.6, we have: PR(K) ni 91* C A* (M*)
Proof. Let x be in PR(K) f1 s31 . It follows from 1.3.6 that there exists a Hilbert space 71, a non-degenerate representation y of M* on H and a vector 6 in ?-1, such that, for all w in M* : (X7 W) = WWX 10
3.1 The Generator of a Representation
85
As the space II fl I is norm dense in M* by 2.4.6 (iii), the algebra µ(II fl I ) is dense in p (M* ). Thanks to Kaplansky's theorem, it exists a sequence {Wfl}nEN of elements of IV such that I< 1 and that (W) strongly converges to 1. Let us consider the linear forms on M* defined by: (w E M*)
!Pn(w) = (x,w n * w * wn)
We have, by hypothesis on x: !Pn(w° * w) = (x, (w * wn)° * (w * wn)) >_ 0
Therefore 4 n is positive definite. Moreover, since x belongs to by 2.4.5, wn * w * wn belongs to Iv,, we get:
91* and since,
by 2.1.6 (ii) by 2.6.1 (v) by 2.5.3
4'n(w) = (a(wn * w * wn) = (A(wo * w)a(wn) I
= (A()a(wn) IA(wn)11V(x*)) (A(w), rIa(Wn),A(Wn)AV(x*))
Let us put On = fla(Wn),a(Wn)Ap(x*). We have then Ifin(w) _ (A(f1),w), and therefore: IPn = A*(fn) E A*(M*) From 1.6.10, we get the existence of x (I in B(K)+ such that x = 'r* (x f2). Let us also consider the decomposition A* = -x*(sa)* (1.6.1 (ii)). We have:
(x fl - (sA)*(Qfl),lr(w))l = I(ir*(xQ) I(x - 4n, W) I I (x, w) - (4n, W) I I (X, W) - (X, Wo * W * Wn) I
J(Y(W) I
(Y(Wn)*Y(W)Y(Wn)6 16)1
_ I(,1(w),Q f' flN(W., )E) 1 :5 JIM(W)II jj
ti(Wn)611 11 + IL(Wn)611
by ([571 Proof 2.11) < 211y(w)II 2117r(W)II 11611116
F(Wn)611
Therefore, thanks to Kaplansky's theorem, we get:
I- (s)(Q)II < 2IICI1 IC - i(wn)CII n --+oo 0 )
As, by 1.6.2 (ii), (SA)* is an isometry, its image is closed in B(K), therefore x Sl belongs to it. Which is to say that there is an element ,fl in M* such that x,fl = (SA)*(Q) which implies that X = ir*(xf2) = A*(sl).
86
3. Representations of a Kac Algebra; Dual Kac Algebra
3.1.2 Lemma. (i) Let x be in,, fl 9lcpo,, and w in I,,; let us write 77 = a(w). Then
belongs to Yt
,
_ 4(x),
and we have: A(w)Ap(x(x*))
(ii) The set {i7 E M*, A*(Sl) E 91*} is dense in M. Proof. For all w' in M*, we have:
(A*w') = (A(S)I) = (A(x) I A(w')*a(w)) = (A,(x) I A(w'O)a(w) = (iL,0(x) a(w'O * w))
= (x*,wO * w)= (ic(x),w° * w') = ((w° ® i)I'i(x),w')
by 2.5.3 by 2.6.1 (v) by 2.1.6 (ii)
Therefore: ((w° ® i)I'r,,(x))* _ (w o x ®i)I'(r,.(x*)) As KK(x*) belongs to
by 1.2.5
belongs to 91, by 2.3.1 (ii), and 2.3.5 gives
,
the completion of (i). By using 2.1.7 (ii) and 2.7.5 (iv), (ii) is an immediate corollary of (i). 3.1.3 Proposition. Let y be a representation of M*; let us suppose that the set {9 E (Aµ)*; µ*(9) E 91* } is dense in (Aµ)*; then, y is quasi-equivalent to a subrepresentation of A.
Proof. Let 0 be in (A)t such that p* (9)* belongs to 91.. Therefore, by lemma 3.1.1, we get the existence of an element (1 in M* such that: A*(Q) = µ*(e) Then, we have, for all w in M* :
(/A(w),9) = (j(O),w) which can be written as well: (Sp(W(W)), 0) = (S,\(7r(W)), J?)
(A(w),Q)
3.1 The Generator of a Representation
87
By density, for all x in W * (K), we shall have: (S/A(X), 0) = (SA(x), Q)
Let us assume s,\ (x) = 0; then we shall have (s,(x), 0) = 0 for all 0 satisfying
the above hypothesis; by linearity and density, it implies that sµ(x) = 0; therefore Ker s,\ C Ker sµ; so, there exists a morphism 4$ from M to Ap such that 4$(A(w)) = µ(w) and the lemma is proved.
3.1.4 Theorem. Any representation p of M* has a generator. Proof. Let w be in M*, Q in (Aµ)*; we have: (A x p)*(w ® ,'l) = A*(w)p*(A2)
As sl, is a right ideal, we see, using 3.1.2 (ii), that the representation A x p satisfies the hypothesis of 3.1.3. So, A x p is quasi-equivalent to a subrepresentation of A, and, by 1.5.4 (i) and (ii), we get the existence of a partial isometry U in AA x µ ® M C M 0 A p ® M such that, for all w in M*: (A x y)(w) = (i 0 w)(U)
UU"=U*U=Paxµ0 1 where PAxµ is the projection on the essential space of A x p. Now, let w be in M* , w(1) = 1, and 0 in (Aµ)*; we have: W)(U)
(0 X p)(W),
D)
by 1.4.3 Therefore:
i)(U)
A*(W*"*)p*(S2)
P 0 i)(aW*a)Y*(S2) (W"* 0 i)(oW"o(1 0
and, by linearity and density:
(i 0 n 0 i)(U) = aw*a(i 0 p*(fl)) So:
10 µ,(.R) = oWo(i 0 R (& i)(U) (i 0 D 0 i)((a 0 1)(10 CrWcr)(a 0 1)U)
by 2.6.1 (i)
88
3. Representations of a Kac Algebra; Dual Kac Algebra
Therefore, we have:
(10 /1(w), w ®17) = (1)(/1(Si),) _ (2' 0 Si 0w)((a ®1)(1 ® aWa)(a ®1)U) and, eventually:
10 Y(W) = (i 0 i 0 WWU 0 1)(10 awa)(U 0 1)U) Let x be in C (HW), by multiplying this equality (valid for all w in M*) by x ®1
on the left and on the right, we get that (a ®1)(1® aWa)(a ®1)U commutes with x 0 10 1, therefore it belongs to M ® A, ®M f1(,C(H) 0 C 0 C)', i.e. to C 0 Aµ 0 M, and there is a partial isometry V in A,, 0 M such that:
(a & 1)(10 awa)(U 0 1)U = 10 V and then, for all w in M* : Y (LO) = (i 0 W) (V)
Then, we easily get:
10 V*V=U*U=PPxp01 and so we deduce that the projector PA x µ may be written 10 Q, where Q is in AW Moreover, we get :
10 vv' = (a 0 1)(1 & owo)(o 0 0 0 0 1)(10
i) 0
0 1)(10 0 1)(a 0 1)(10 aw*a)(a
1)
And so we get that V * V = VV* = Q ®1 and the theorem is proved. 3.1.5 Corollary. (i) Let jl and 112 two non-degenerate representations of M*; then /21 x /22 is non-degenerate. (ii) The triple (W*(K), ss, x,r, s;r), with the definitions of 1.6.5 and 1.6.6, is a co-involutive Hopf-von Neumann algebra; it is symmetric if K is abelian, and abelian if K is symmetric.
Proof. The assertion (i) is a direct corollary of 1.5.5 and 3.1.4, as 7r is nondegenerate and (ii) is a trivial application of 1.6.7 and 3.1.4.
3.2 The Essential Property of the Representation A
89
3.,2 The Essential Property of the Representation A 3.2.1 Lemma. Let µ be a non-degenerate representation of M. with a generator V. For all w in M*, we have:
spa x µ)(w) = v(i 0 a(w))v*
.
Proof. It results from 1.5.5 that the generator U of A x µ is equal to: (c 0 i)(i 0 oW*o)(1 0 V) _ (Cr 0 1)(1 0 a)(1 0 W*)(1 0 cT)(o 0 1)(1 0 V) _ (Cr 0 1)(1 0 a)(1 ® W*)(V ®1)(1 0 ar)(a 0 1) Therefore, we have, for all w in M* :
t(a x µ)(w)V O(i0
Lo)((l 0 U)(1 0 W*)(V 0 1)(10 U))V u)(1 0 W*)(V 0 1)(10 U)(V 0 1)) 0
w')(=Or)(v)(l0 a)) (+0 i 0 w)((1 0 o)(i 0 W*)(1 0 W)(1 0 a)(v 0 i) (1 0 0)(1 0 W')(1 0 a)) i 0 w)((V 0 1)(1 0 oW'o)) = V(1 0 (i 0 LO)(oW'o)) = V(1 0 A(te))
by 1.5.1(; by 2.6.4
by 2.6.1 (i)
which completes the proof.
3.2.2 Theorem and Definitions. Let µ be a non-degenerate representation of M* with a generator V. Then: (i) The mapping which sends any element x of M to V(1 0 x)V* is a oneto-one normal morphism from M to Ap ® M which shall be denoted by %y,; (ii) The representations A and A x µ are equivalent; moreover, we have, for all w in M*: ryp(A(w)) = t(a x µ)(w)
(iii) We have: 'Yµ sa = csA x p
(iv) The mapping ya is a coproduct on M; it shall be denoted by 1' and by transposition it induces a product *^ on M*. For all x in M. we have: F(x) = aW*cr(1 0 X)oWa
90
3. Representations of a Kac Algebra; Dual Kac Algebra
(v) For all u in the intrinsic group of K, and w in M*, we have: yu(A(w)) = A(u w) Proof. As V is unitary, it is enough to check in which space belongs V(1 Ox)V *
for any x in M. It results from 3.2.1 that for all w in M*, we have: Yµ(A(w)) = t(A x µ(w)) E Ap 0 M
which by density and continuity gives (i) and (ii) on our way; the assertion (iii) is straightforward, (iv) results immediately from the associativity of the Kronecker product and (v) is the application of (ii) and 2.6.6 (iii). 3.2.3 Lemma. Let /11,112 be two non-degenerate representations of M*, with, respectively, generators V1, V2; then: (i) For all x in M, and t in Hom(/11, /22), we have: (t 01)'5
(x) = 7"' µ2 (x)(t (9 1)
(ii) For any morphism if: Aµ1 -+ Aµ2, such that 4P o pl = 112, we have:
(Oi)5',1=
7µ2
Proof. Let w be in M*. We have:
(t
(t 0 1)4;(A x µ1)w) = s(a X 92)(w)(t 0 1) = 7µ2(a(w))(t 0 1)
by 3.2.2 (ii) by 1.4.5(1) by 3.2.2 (ii)
by continuity and density, we get (i). Let x be in M; we have: z)'Yµi(x) = (4i 0 z)(Vi(l 0 x) V1) = V2(1 0 x)V2
=
7,02 (X)
by 3.2.2 (1) by 1.5.6 (ii) by 3.2.2 (i)
which completes the proof. 3.2.4 Lemma. Let y a non-degenerate representation of M*, W^ in M* and in (Aµ)*. We have:
A* (W 0 w)o7µ) _ A*Cw)µ*C)
3.2 The Essential Property of the Representation A
91
Proof. Let w be in M*. We have:
(A*((Q®')o#5',),c4.)
=(Aw®,1l)
by 3.2.2 (ii) by 1.4.3
which completes the proof.
3.2.5 Lemma. Let y1 and 112 be two non-degenerate representations of M*. We have: '7141 xµ2 = (c ®i)(i ® yµ1 )yµ2
Proof. Let w be in M*, fll in (Au1)*, (22 in (Aµ2 )* and w in M*. We have:
(siX
2(/\(W)) i J?1 (9 122
- (A X Ill x 112(w), w 0 f2l 0 02)
_ _ (A((Q1 0 w)
)(112)*(Q2),w)
by 3.2.2 (ii) by 1.4.3 by 3.2.4 by 1.4.3
_ ((A x 92)(w) (171 0 w) o ryµl .f12) _ (c(A X µ2)(w) (S22 0 ,(ll (9 cv) o (i (9 'YPi )) = (5',2 (A(w)), ((12 (9 ,fll (9 w) o (z D10 S22 0 ^tIA 1 )71A 2
by 3.2.2 (ii)
which completes the proof by linearity, density and continuity. 3.2.6 Proposition. Let p be a non-degenerate representation of M*. We have, with the notations of 3.2.2:
(oi)i'=(iot). Proof. We have: by 3.2.2 (iv) by 3.2.3 (ii) by 3.2.2 (ii) by 3.2.3 (ii) by 3.2.5 by 3.2.2 (iv)
92
3. Representations of a Kac Algebra; Dual Kac Algebra
3.2.7 Fell's Theorem ([48]). Let G be a locally compact group, AG the left regular representation of G, µG a unitary representation of G. The representation IIG ®AG is then quasi-equivalent to AG. More precisely we get, for any
s in G: V(1 (9 AG(s))V* = PG(s) 0 AG(s)
where V is the unitary in C(fl) 0 L°O(G) defined by the continuous bounded function s -+ PG(s).
Proof. Let us apply theorem 3.2.2 to the Kac algebra Ka (G) defined in 2.2.5. We have seen that V is the generator of the non-degenerate representation p associated to PG (1.5.10), that M is then the von Neumann algebra £(G) generated by the left regular representation A of L1(G) (2.5.4). So the morphism
y satisfies: yp(X) = V(1 0 X)V*
-ym(A(f)) = (A x y)(f)
(X EC(G))
(f E L' (G))
From this last relation, we deduce, using 1.4.7, for the unitary representations of G, associated to y o A and cA x p respectively: 'Yµ(AG(s)) = PG(s) 0 AG(S)
3.2.8 Corollary. Let G be a locally compact group, ,C(G) be the von Neumann algebra generated by the left regular representation AG. There exists a unique
normal injective morphism I's from £(G) to £(G) 0 £(G) such that, for all s in G, we have: I's(AG(s)) = AG(S) ® AG(S)
and I's is a coproduct (in the sense of 1.2.1) on C(G).
3.3 The Dual Co-Involutive Hopf- Von Neumann Algebra 3.3.1 Proposition and Definitions. The mapping from £(H 0) onto itself defined by x --+ Jx*J for all x in £(H,) is an involutive anti-automorphism
of £(H,). The restriction of this mapping to 1 ! is an involutive antiautomorphism of M in the sense of 1.2.5. It shall be denoted ic. Moreover, the involutions rc and hl are linked by the following relations: A(w o rc) A*(w o Ic)
(w E M* ) (W^ E M*) .
3.3 The Dual Co-Involutive Hopf-Von Neumann Algebra
93
Proof. For all x in At, let us put k(x) = Jx*J. Let w be in M*. We have: k(A(w)) = JA(w)*J = A(w o ic)
by 2.5.5 (ii)
which altogether provides the first equality and ensures, by continuity, that for all x in M, k(x) belongs to M; the involutive character of k is trivial. For all w in M*, we have:
(A(2),wo x) _ (A(w o x), W^)
_ (k(A(w)),Z) _ (A(w),2 o k)
_ (A(2' o
by the first equality
w)
which completes the proof.
3.3.2 Theorem. The triple (M, I', h) is a co-involutive Hopf-von Neumann algebra. It will be called the dual co-involutive Hopf-von Neumann algebra of K.
Proof. Let w be in M*, W1, W2 in M*. We have:
(I'kA(w),2'i ®w2) = (I'A(w o K), w1 (9w2)
=
((A x A)(w o x), W2 (9 w1)
= (A(c22)A(2i),w o /C)
by 3.3.1 by 3.2.2 (iv) by 1.4.3
= (*(w1 o by 3.3.1 0 k), w) = ((A x A)(w),(w1 ®w2) o (k0 k)) by 1.4.3 by 3.2.2 (iv) = (c(k 0 k)I'a(w), ('2'i ®w2)) Therefore we have:
I'ka(w) = c(k 0 k)I'A(w) and we can complete the proof by continuity. 3.3.3 Proposition. The mapping sa is an ]H[-morphism from (W*(K), Ss7rx7r, s*)
to (MI F, ic). If K is abelian, (M, I', I) is symmetric, and if K is symmetric, (M)I', ic) is abelian. Proof. Let w be in M*. We have: I'sA7r(w) = yas'\Ir(w)
= csaxalr(w) = c(A x A)(w) = c(sa 0 sa)(7r x ir)(w) _ (s,\ ® sa)csirxirlr(w)
by 3.2.2 (iv) by 3.2.2 (iii) by 1.6.1 (ii) by 1.6.4 (iii) by 1.6.1 (ii)
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3. Representations of a Kac Algebra; Dual Kac Algebra
and, we get :
FS,\ = (S,\ 0 sa)cs,x, And: kA(w)
=A(woic) = sa7r(w o K)
= svr(w)
by 1.6.1 (ii) by 3.3.1 by 1.6.1 (ii) by definition of * (1.6.6)
= sAs;r7r(w)
and we get: s,\ = s,\ s;r
At last, since A is non-degenerate, we have s,\(1) = 1, by 1.6.1 (iii); as s,\ is one-to-one, it completes the proof, together with 3.1.5 (ii). 3.3.4 Definition and Notations. Let us denote C,\(K) the C*-algebra generated by the Fourier representation A of K. By 1.6.1 (i), we may identify its dual (C(K))* with a closed subspace B,, (1K) of B (IK) . More precisely, to each
element 9 of (C (K))*, one associates the element 9 o A of B(K) (where A denotes the restriction of s,\ to C*(K); ,cf. 1.6.1 (i) and (ii)). By 1.6.2 (ii), the mapping (s,\)* is an isometry from M* into B(K), the image of which is contained in BX(K). By transposing 3.3.3, (s,\)* is then an isometric Banach algebra morphism, the image of which will be denoted by A(K) and called the Fourier algebra associated to K. Every element of A(K) vanishes over Ker s X; conversely, let 9 be in B(K), such that (x, 9) = 0 for all x in Ker s,\. We can define a linear mapping w on M by writing: (SA(z),w) = (z,9) (z E W*(K))
In fact, w appears as the composition of the restriction of 9 to the reduced algebra W * (IK)supp s,\) with the canonical isomorphism between W * (K)supp 8a and M. Therefore w is ultraweakly continuous and belongs to M*, and we have 9 = (s, )* (w). So, an element of B(K) belongs to A(K) if and only if it vanishes over Ker s,\.
3.3.5 Proposition. The mapping xA* is a non-degenerate faithful representation of M* in M. Its generator is W and we have:
(i) (ii)
(i ®1')(W) _ (W ® 1)(10 C)(W ® 1)(10 a) (i ®w o k)(W) _ (i ® w)(W*)
(w E M*) .
Proof. By 1.6.1 (ii) we have xA* = x7r*(s,,)*, it then results from 1.6.9 and 3.3.4 that it is a faithful representation. By 2.6.1 (i), for all w in M*, and all
3.3 The Dual Co-Involutive Hopf-Von Neumann Algebra
95
w in M*, we have: (A* (W^ ), w o x) = (A(w o x), w) = (w ®w)(W )
therefore: ICa*(w) = (i ®w)(W)
and we get (i) and (ii) through a straightforward application of 1.5.1 (i) and (ii).
3.3.6 Theorem. Let G be a locally compact group, £(G) be the von Neumann algebra generated by the left regular representation AG. There exists a unique
normal morphism I's from £(G) to £(G) 0 £(G), and a unique normal antiautomorphism r., in £(G) such that, for all s in G.: te(at(s)) = AG(S) 0 AG(S) KS(AG(s)) = AG(S_ 1)
Then, (,C(G), I's, tc,) is a symmetric co-involutive Hopf-von Neumann algebra; we shall denote it 1,(G). It is the dual co-involutive Hopf-von Neumann algebra associated to the Kac algebra Ka(G). Moreover, the morphism sA from the envelopping W * -algebra W * (G) to £(G) such that we have, for all
s in G: SA(G(S)) = AG(S)
is an H-morphism from the Ernest algebra of G (cf. 1.6.8) to Hs(G).
Proof. The existence of x, is the only non-trivial result; applying 3.3.1, we see there exists a co-involution k on (C(G), I', ), defined in 3.2.8, such that,
for all f in L'(G): (A(f )) _ A(f 0 Ica)
From 1.1.3, we have, for any s in G, f in L 1(G) : 1
(f 0 rva)(S) = f (S - ),AG(S-
1)
and so, we have:
(JG f(5 G(5)dS)
-
IGf(s-1),AG(3-1)AG(s)ds = IG f(s)(s1 )ds
We then get, for any s in G:
K(at(e)) = ac(a-1)
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3. Representations of a Kac Algebra; Dual Kac Algebra
3.3.7 Theorem. Let K = (M, I', ,c, gyp) be a Kac algebra, K' the commutant Kac algebra. Then the dual co-involutive Hopf von Neumann algebra (M', 1^7 IC'^) is equal to (M, cI', IC).
Proof. Using 2.5.9, we see that the Fourier representation A' generates M;
so M'^ = M. By 2.4.8, the fundamental operator W' associated to K' is W' = (J 0 J) W (J 0 J). The coproduct P'" is, then, using 3.2.2 (iv), such that:
P'"'(x) = (J ® J)cW*o(J 0 J)(1® x)(J 0 J)crWar(J 0 J) = (k ® = SP(x)
by 3.3.1 and 3.2.2 (iv) by 1.2.5 applied to (M, P, k)
As) by the identification of H,p with H, pi, the associated antilinear isomor-
phism J, and J ,, are equal, we see, by 3.3.1, that rc' = k and the theorem is proved. N
N
N
3.3.8 Proposition. Let K = (M, P, I, Sp) be a Kac algebra, and K = (M, P, k, gyp)
be a Kac subalgebra in the sense of 2.2.7. Let us denote j the canoniN
cal imbedding from M into M. which is an IHt-morphism. There is then a canonical surjective H-morphism r from (M, I', k) to (M", P"', R) such that r(AAw)) = A(w o j), for all w in M*, where A is the Fourier representation of K.
Proof. Let usNcall I the isometry from H0 to H. defined, for all element x in
91, nMby:
(i)
As j is an H-morphism, the application w -+ w o j is multiplicative and involutive from M* to (M)*. This application is surjective because, for any a, -y in Hip, we have wy,a o j = wI* y,I* a. Let x in w in M*. We have:
j o k 0 )r())
I,X(w o
=
by 2.3.5
N
o IC o j 0 i)P(x)) = A ,((w o re o i))PU(x))) = A(w)AWU(x)) = A(w)IA ,(x)
by (i)
therefore:
la(w o j) = A(w)I
(ii)
Let us put, for x in M, r(x) = I*xI. We have r(1) = 1 and r(a(w)) = A(wo j), because I is an isometry.
3.4 Eymard Algebra
97
N
Let now w be in (M^)*. We have: N
(j(L(2)),w) = (*(c2)),w o j) (5(w oj),w) _ (r(A(w)), w) _ (A*r*(w), w)
then:
N
A*r*(W) = j A*(w)
As A* is injective, it can be then easily deduced that r* is involutive and multiplicative, and therefore that r is an H-morphism.
3.4 Eymard Algebra 3.4.1 Lemma. The set BA(K) is a self-adjoint part of B(K). Proof. For all x in M and w in M*, we have: (x*, w) = (x,w° o lc)
and, by 3.3.1:
by 1.2.5
I° r) II = I= IBy
using 1.6.2 (i) we then see that 7r*(BA(K)) is stable by involution, and, thanks to 1.3.4, we get the result. 3.4.2 Proposition. (i) Any norm-one positive element of BA(K) is the limit, for the ar(B(K), C*(K)) topology, of norm-one positive elements of A(K). (ii) The space BA(K) is the o(B(K), C*(K)) -closure of A(K). (iii) The space BA(K) (resp. 7r*(BA(K))) is composed of the elements of the form (s,)(Q) (resp. M* (Q)), where p is a representation of M* weakly contained in A. in the sense of [47], and Q an element of (All)*.
Proof. Any element of BA(K) vanishes on Ker (cf. 1.6.1(ii)). By ([25], 3.4.2 (i)) any norm-one positive element of BA(IK) is thus the limit, for the cr(B(K), C*(K)) topology, of elements of the form (SA)*(i7), where 11 is norm one positive in M*; at last 3.3.4 allows to deduce (i). It results from (i), by linearity that BA (K) is contained in the closure, for the ar(B(K), C* (K)N) topology, of A(K); on the other hand, since B,\(K) is the annihilator of Ker A, it is cr(B(K), C*(K)) closed, which completes the proof of (ii).
3. Representations of a Kac Algebra; Dual Kac Algebra
98
Let p be a representation of M* weakly contained in A and 5? in (AM)*.
By ([25], 3.4.4), (s )* (fl) is the a(B(K), C* (K)) limit of elements of the form (sA)*(w) where w belongs to M* , i.e. of positive elements of A(K). By belongs to BA (K); by linearity this using (ii), we can conclude that (sµ )* conclusion still holds for any Sl. To prove the converse, let us note that those elements of the form (sµ) *
with p weakly contained in A and 51 is in (A,)* compose a vector space. Indeed if p (resp. p') is weakly contained in A and 5 (resp. fl') belongs to (A,,)* (resp. (Ai)), it is easy to check that:
(s)*(Q) + (s')(Q') = (s,,A')*(Q ® ST) and that p ® p' is weakly contained in A. By linearity, it is therefore enough to consider x in BA(K)+. Let y be the positive linear f o r m on C (K) such that x = y o A (cf. 3.3.4). The Gelfand-Naimark-Segal construction allows to associate to y a triple (?-(, 6,,u). Then p o A is a representation of M* and we have: ((Y 0 A)* ('20, W)
WA(WN 10 (Y' A M)
_ (yo5,ir(w)) (X, -7r(W))
(7r* (X), W)
Therefore, we have: 7r* (x)
o A)*
and: X=
since it is clear that p o A is weakly contained in A, it completes the proof. 3.4.3 Lemma. The product of B(K) is a(B(K), C*(K)) separately continuous on the bounded parts.
Proof. Let ei be a bounded family of elements of B(K) converging to an element 9 in B(K). For all 9' in B(K) and w in M*, we have: (ir(w),9*9') _ (r* (Oi * 8'), w) by 1.6.9 _ (ir(w .
(9')) , O j)
which converges to: (ir(w i*(9')), 9) = (ir(w), 9 * 9')
thanks to the same computation.
3.4 Eymard Algebra
99
Since the 8i * 8' are bounded by I O' l I sup (8i I I which is finite, by the
density of 7r(M*) in C*(K), we get that 8i * 8' converges to 8 * 8' for c(B(K), C*(K)). The left multiplication is dealt with in the same way.
3.4.4 Theorem. The sets A(K) and BA(K) and are norm-closed self-adjoint ideals of B(K). Specifically, BA(K) is an involutive Banach algebra which we shall call the Eymard algebra associated to K. Proof. Let cw be in M*, 8 in B(K) and w in M*. We have:
(ir*(8 * (sA)*(w)),w _ (A*(.')ir(8),w)
by 1.6.9 and 1.6.1 (ii) _ ((A x ir)*(w ® 8), w) by 1.4.3 _ ((A x 7r)(w), w ®8) _ (%A(w), 8 ® w) by 3.2.2 (ii)
_ (A(w),(8®)o%) _ (A((8®) o'Y7r),w) From what we obtain that: 0 * (S,\)*(^) = (S,\)*((O 0 ^) 0 ^ )
By 3.3.4, it follows that A(K) is a left ideal of B(K). As A(K) is self-adjoint and norm-closed, the first part of the theorem is secured. Now, let 8 be in BA(K)+1. By 3.4.2 (i), 8 can be cr(B(K), C*(1K)) approximated by norm-one positive elements of A(K). Applying the first part of this proof, as well as 3.4.3 and 3.4.2 (ii), we find that, for all 8' in B(K), 8' * 8 belongs to BA(K). By linearity, we can conclude that BA(IK) is a B(K)-left ideal; since, by 3.4.1 and 1.6.1 (i), BA (]K) is norm closed and self-adjoint, the proof is completed. 3.4.5 Proposition. The restriction to BA(1K) o f the Fourier-Stieltjes representation o f B(K) is the transposed o f the mapping kA from M * to C (K) (once BA(K) is identified with the dual of C *(K)).
Proof. Let 8 be in (C (K))* and w in M. Then by 3.3.4, 8 o A belongs to BA (1K) and we have:
(icir(8o),w) _ (7r(woK),8o) _ (A(woi),8) _ (kA(w))),8) which completes the proof.
by 3.3.1
3. Representations of a Kac Algebra; Dual Kac Algebra
100
3.4.6 Eymard's Theorem ([46] ). Let G be a locally compact group, and f an element of B(G). The following assertions are equivalent: (i) There exists a Hilbert space H, vectors , 77 in H, and a unitary representation ,G of G on H, weakly contained in the left regular representation, such that, for all t in G:
f(t) = GIOX 17l) (ii) We have:
If (t)h(t)dtI, h E L1(G), IIA(h)II < 1
sup
GI
< +oo
The space of such functions is noted Ba(G); it is a closed ideal of the FourierStieltjes algebra of G, and will be called the Eymard algebra of G. Moreover,
if f is in BA(G), its B(G)-norm is equal to: sup
I
G
h E L1(G), I< 1
The space B,\(G) can be identified with the dual of C *(G) (the C*-algebra generated by the left regular representation of L1(G)), the duality being given, i f f (t) = (,G(t)e I 'i), and h in C (G), by:
(f, h) = (1 77) where µ denotes again the associated representation of C *(G) (recall that µ is weakly contained in A). Moreover, for every w in £(G)*, the set:
A(G) = Is -+ (A(s1),w)
(s E G)}
is a norm-closed ideal of BA(G). It will be called the Fourier algebra of G, and its dual is ,C(G). Proof. Using 3.4.2 (iii), we see that property (i) characterizes 7r*(BA(Ka(G))); by 1.6.3 (iii), it is the same for property (ii).
So, we get BA(G) = 7r*(BA(Ka(G))) = Ka7r*(B.\(Ka(G))) and all other properties of BA(G) come then from 1.6.3 (iii) and 3.4.4. Let us now consider ica 7r* (A(Ka (G))) . Using definition 3.3.4, it is the set of all elements which may be written, for all fl in £(G)*: Ka7r*(sa)*(f2) = KaA*(S?)
In 1.3.10, we have seen that A*(fl) is the function s -+ (AG(s), fl); by then iaA*(fl) is the function s (AG(S'), fl).
3.5 Construction of the Dual Weight
101
So, we get A(G) = ica 7r* (A(Ka (G)) ), and all properties of A(G) come from 3.4.4.
3.5 Construction of the Decal Weight In that paragraph, we consider the set Z = a(Ip fl i3). 3.5.1 Proposition. Let w, w1 be in I( fl I,. The formulas:
(i)
a(w)T a(wf) = a(w * w') a(w)#
(ii)
= a(w°)
allow us to equip B with a structure of left Hilbert algebra, dense in H.. Let us denote by *r the left multiplication of Z. We have, for all w in I , fl 10:
(iii)
ir(a(w)) = A(w)
and the von Neumann algebra generated by *r(Z) is equal to M.
Proof. (a) We have seen in 2.4.6 (iii) that I, fl I3 is an involutive subalgebra of M. As a is a bijection from I, fl I3 to , we see that 93, equipped with T and # is an involutive algebra. (b) Let be in H,, orthogonal to 93; by 2.4.5, we have, for all L01, L02 in
I:
0 = (a(wi * w2) I ) _ (A(wi)*a(w2) I ) _ (a(w2) I A(w1)e)
by 2.4.6 (ii)
because of the density of a(I0) in Hp (2.1.7 (ii)), it implies A(w1) = 0 for all w1 in I,; because of the density of I. in M* (2.1.7 (ii)), it implies, for all w in M*, A(w)e = 0, which, in turn, because of A being non-degenerate (2.6.3 (i)) implies 0. Therefore Z is dense in H.. (c) For all w1 fixed in I, fl I3 the mapping a(w) -+ a(w1)T a(w) is continuous from 93 to 93. In fact, we have: a(w1)T a(w) = a(w1 * w) = A(w1)a(w)
by definition by 2.6.1 (v)
(d) For all wl, w2, w3 in I, fl I3, we have: (a(W1)ta(w2) I a(w3)) = (A(wi)a(w2) I a(w3)) = (a(w2) I A(wi)a(w3))
by (c) by 2.5.3
= (a(w2) I a(w1)T a(w3))
by (c)
= (a(w2) I a(wl)#Ta(W3))
by definition
3. Representations of a Kac Algebra; Dual Kac Algebra
102
in H, orthogonal to 93T93. We have, for all wl, w2 in I, fl
(e) Let
o = (a(wi)Tm(w2) If) _ (A(wfla(W2) If)
_ (a(w2) I a(wj)F)
by (c) by 2.5.3
By (b) it implies A(wl )e = 0, since a(I, fl 1) is dense in H by (b); by continuity, it implies A(w)e = 0, for all w in M*; therefore because of A being
non-degenerate, it implies e = 0; so,T (f) Let w be in I, fl I and x in giv f
is dense in H.. ¶fl ,01, . We have:
(A (x) a(w)#) = (A,(x)Ia(w°)) = (x*,wo)_ = (IC (x), w)
= (a(w) I A ,(rs(x* )))
because
cpoK =
by definition by 2.1.6 (ii) by 1.2.5 by 2.1.6 (ii)
x(91,). Therefore, the mapping # has an adjoint, the restric-
tion of which to A,(
fl Ovo,,) is the mapping Ap(x) -- Ap(K(x*)). By
2.7.5 (iv), this adjoint mapping is densely defined, therefore 0 is closable. Following ([158], def. 5.1; cf. 2.1.1 (iii)), if we remark that (iii) has been
proved in (c), and that, thanks to 2.4.6 (iii), *(Z) generates M, we have completed the proof. 3.5.2 Definitions. We shall denote B' the right Hilbert algebra associated to 93, 95" the achieved left Hilbert algebra, and cao the maximal modular subalgebra
of B" (cf. 2.1.1(iii)). We shall still note T and # (resp. ) the product and the involution on 93" A
A
(resp. 95'). A
A
We shall note S and F, the closures of # and b, with respective domains denoted to D# and Db . In particular, we have, for all x in flpo1 : FAW(x) = Av(r(x*)) If in Hp is left bounded with respect to Z, we shall still note u1 ) the "left multiplication" by ([14], def. 2.1). 3.5.3 Definitions. We shall note cp the faithful, semi-finite normal weight on M canonically associated to ([14], th. 2.11), and call the dual weight associated to K. For all w in I , fl II, by 3.5.1 (c), A(w) belongs to 7^W B) and therefore to fl OZ . Moreover, for wl and w2 in I, fl Imo, we have: SP(A(w2)*A(wl)) = (a(w1) 1 a(w2))
3.5 Construction of the Dual Weight
103
To the weight S we associate the Hilbert space H. and the canonical oneto-one mapping Ao : Ho. We shall note 21 the left Hilbert algebra associated to Vp, i.e. 11(91, fl fir), which is isomorphic to B" (2.1.1 (iii)). More precisely, the mapping which, to every C in B", associates the vector: A
can be uniquely prolonged into a unitary operator from H(2 to HO still denoted by F. It will be called the Fourier-Plancherel mapping and will allow us to identify HO and H., and, through this identification of H., we
have 93" ='. Using the definition 2.1.6 (ii), we shall note a instead of a(. 3.5.4 Proposition. For all w in I., a(w) is left-bounded with respect to 95, and we have:
(i) *(a(w)) _ A(w)
(ii) a(w) = 1iW(a(w)) (iii) for all a, ,y in 21', yTab is left-bounded with respect to 95 and we have:
(7Tat) = A(wy2a) . Proof. Let w1 be in Ip, w2 in I, fl I,,,
in 93'. We have:
w1)
by 2.6.1 (v)
_ 1(a(w2 * w1))e because w2 * w1 belongs to Ip fl Io _ \(w2 * w1) by 3.5.1 (c)
_ 1(w2)a(w1) As 1 is in the closure of A(I, fl Io) = *(Z), we have: i'(e)a(w1) = A(w1)C
which yields (i), and (ii) immediately, then (iii) follows by applying 2.1.7 (i).
3.5.5 Remark. Thanks to 2.1.1 (iv), we know that M is in a standard position in H; so, by 1.1.1 (iii), every element of M* can be written M for some vectors a,# in H.. This element shall be written wa,Q
3.5.6 Corollary. (i) The algebra ,C(G) is in a standard position in L2(G). (ii) The predual ,C(G)* is equal to the set { Sl f,g ,C(G), f, g E L2 (G) } .
104
3. Representations of a Kac Algebra; Dual Kac Algebra
(iii) For all f in L2(G), let us put f(s) = f (s-1)- for all s in G. The set of all If * g, f, g E L2(G)} is the Fourier algebra A(G) defined in 3.4.6. Proof. The assertions (i) and (ii) are just applications of 3.5.5 to Ka(G). In 3.4.6, A(G) has been defined as the set of functions s -+ (AG(S), w) , for all w in ,C(G)*. But we have: (AG(S-')7 W^f,g) = (AG(S -1)f 19) =
IG.f(st)9(t)dt
- JG f
(t)9(s-it)dt
f(t)9(t-is)dt
= G
(f * 9)(S)
And so, (ii) implies (iii).
3.6 Connection Relations and Consequences 3.6.1 Proposition (Connection relations). The operations T and T are linked by the following relations: (i) For any a, -y in 2L' and b in H., we have: (W(a ® /3) I ^10 b) = (/3 I (^'Tab)b)
(ii) For any a, 'y in 2t' and 0, b in B', we have: (W(a 0 0) 17 ®b) = (/3t8b I yTab)
(iii) The set 93'T93' is included in AW(91,,) and, for any a, 'y in HW and b in TB', we have:
(W(aO/3)f706) = (lr(/3t6)al7). Proof. Combining 2.6.1 (iii) and 3.5.4 (iii), we get (i). Therefore, with the hypothesis of (ii), we have: (W(a 0 6) 17 0 a) = (13 I * (b)(7Ta")) = (n
_ ('(6)/3 which is (ii).
I 7Ta') (/3t6b I'YTab)
3.6 Connection Relations and Consequences
105
It can also be written as follows: A
A
(W(a ®/3) 1'Y ®b) = (/3t6b jir'(ab)'Y) = (ir'(a)(/3t6) I 'Y))
It follows that:
II'(c)(/3t6)ll
suv{J(w(a 0 p)
0 a)j, ry E W, 11711 s 1} < IllilIVbll
Thus, /3T8 is left-bounded with respect to 2t (cf. 2.1.1(iii)), and we can write: A
(W(a 0 13)1 'Y 0 b) = (ir(/3t61')a 17)
which, by continuity, still holds for any a, y in Hgp. This completes the proof.
3.6.2 Lemma. (i) The set B'TB' is included in Aw('tv fl'Yt,,c). More pre93', we have: cisely, for /, 6 ir(/3Tbb) = A*
and:
)*
(wb,a A
A
rz(7r(#TbI )*) = 7r(bt /3b )
(ii) The space A,('Yt7p fl qi o,c) is a core for F.
Proof. Let /, b be in 93', a, y in Hip. We have:
(A*(Z's,)*al7) _ (A*(^b,a),w'Y,«)
_ (W(a®/3)1y®b)
by 2.6.1(iii)
_ (7r(#Tbb)a y)
by 3.6.1 (iii)
from what follows the first equality. For w in M*, we have, then:
(ic(ir(flt6)),w) _ (A*
w°)
= Ca(w)wQ,b
by 1.2.5
106
3. Representations of a Kac Algebra; Dual Kac Algebra
Therefore, we have: A
A
n('r(/3T8b)*) = A*(wa,b)* = ir(oti36)
by the first equality; as As, by ([158], p. 17),
the proof of (i) is completed. 'TB' is a core for F, (ii) is immediate. 0K
3.6.3 Lemma. Let x be in M. (i) For any a in Db, xa belongs to Db, and we have:
Fxa = n(x)*Fa A
A
(ii) For any /3 in DO, x# belongs to D#, and we have: Sx# = IC(x)*S#
Proof. Let a be in A9,(w n 9iw°,c), w in I, n I.0, x in M. We have: (xa I a(w)#) _ (xa I a(w°))
by 3.5.1 by 2.1.6 (ii)
_ (KA(xa),w) _ (ic(xA ' (a)), w}
_ 1(Fa))*, w} _ (A(,c(x*)Fa)*, w)
by 3.5.2 (i)
_ (a(w) I ,c(x*)F^a)
by 2.1.6 (ii)
Therefore xa belongs to Db and:
Fxa = ,c(x*)Fa
As 4(m. n
is a core for F by 3.6.2 (ii), we have proved (i). Let a be in Db , 0 in D# and x in M. We have: A
A
(x# I F"a) _ (/3 I x*F^a) _ (,C3 F^ic(x)a) _ (ic(x)a I ,5`"P) _ (a I ,c(x*)S^3)
which completes the proof.
by (i)
3.6 Connection Relations and Consequences
107
3.6.4 Proposition. The modular operator A =A, is affiliated to M'. Proof. Let a be in Bo and x in M. We have:
xAa = xFSa = Eic(x*)c = FSxa = Axa
by 3.6.3 (i), because Sa belongs to Db by 3.6.3 (ii)
as 93Q is a core for A, we have xA C Ax, which completes the proof.
3.6.5 Corollary. For all t in R. we have:
Proof. For all x in M, and t in R, we have: Fat (x) = UW*(crt (x) 0 1)Wo _ or
W*(Ditxa-it
_ crW*(Qit 0
by 3.2.2 (iv)
0 1)Wa
1)(x (D
1)(A-it 01)Wi
Now, by 2.6.1 (iv), W belongs to M ® M and by 3.6.4, Alt belongs to M', therefore, we have:
i'cr°(x) _ o(oit 0 1)W*(x 'Alt
(10
®1)a' )aw*(X (D 1)Wo(1 (0 A-it) 01)W(A-it
_ (10 At)I'(x)(1 0 A-It )
by 3.2.2 (iv)
®Qt r'(x) which completes the proof.
3.6.6 Corollary. For any x in M, we have: (i) (ii)
,(x*)=JxJ (.1 o J)W(3 0 J) = W*
.
Proof. Let a be in 93o and x in M. We have: " xJa = xa12Sa
= Al/2xSa = Al/2S,c(x*)a = Jn(x* )a
by density, we get (i). By (i) and 2.6.2, we get immediately (ii).
by 3.6.4 by 3.6.3 (ii)
108
3. Representations of a Kac Algebra; Dual Kac Algebra
3.6.7 Theorem. The modular operator A is affiliated to the centre of M; moreover, it is the Radon-Nikodym derivative of the weight W with respect to the weight cp o k, in the sense of [114] (cf. 2.1.1 (v)).
Proof. By 2.7.6 (i), we have atVOIC = 0 t for all t in R. The theorem 5.4 of [114] gives then the existence of a unique positive self-adjoint operator h, (cf. 2.1.1 (v)). Let x be affiliated to the centre of M, such that co o # = qtpor,,; then A ,(x) belongs to D(0/2), and we have: in fl IIh1V2AW(x)112 =
o K(x*x) = I=
INow, lI1'iL(x)lI2
= I=
by 3.5.2 (i)
by 3.6.2 (ii), we see that A ,(m. fl gqcpo.%) is a core for F, thus also for a-1/2; on the other hand, 3.6.4 implies that h1/2 and a-1/2 commute.
Using the same arguments as in ([23] lemma 23), we can conclude that h1/2 = A-1/2, and so h = A-1. The operator A is therefore affiliated to the centre of M and y o gyp, which completes the proof. 3.6.8 Corollary. For all t in R, we have: /LV 1 = c
tk
.
Proof. By 3.6.7, for all t in R, A' belongs to the centre of M. We then have for all x in M:
kCr (x)=Jot(x)J = = 'A a it JxJAit = A t k(x)Ait JAitxA-it J
by ([14], 4.10) by 3.3.1
which completes the proof.
3.6.9 Lemma. Let x in M. x 54 0, such that F(x) = x 0 x. Then, we have, for all w in M*:
t)
A(w)x = xA(x(x) - w)
xA(w) -a(s w)x A(x(x) - w) = x(x)A(w)x
3.6 Connection Relations and Consequences
109
Proof. Let y be in 91W. We have: A(w)xdW(J)
_ .x(w)4(xJ) = Av((w o K 0 i)l(xy))
by 2.3.5
= AP((w 0 rt. 0 i)((x 0 X).P(Y)))
= XAW((w 0 10 x (9 OFM)
= XAW(((t4x) w) o k 0 z)r(y)) XA(x(x) ' w)Asv(y)
by 1.2.5 by 2.3.5
which yields (i), by continuity. Taking the adjoints in (i), one gets, using 2.5.3: x* A(w°) =
w)° )x* WOW _ A(x* .
by 1.2.5
and, changing w to w°, x to x* (which satisfies the same hypothesis), we get (ii).
Let us now assume that w is in I., and let w' be another element of I.. We have:
A(rc(x) w)Ja(w') = V (Ja(w'))a(K(x) w) = JA(w')Jic(x)a(w) = JA(c.gi)x*ja(w)
= Jx*A(,c(x*) wJa(w) = r%'(x)JA(r4x*) w')Ja(w)
= rc(x)f'(Ja(rc(x*) w'))a(w) = rc(x)A(w)Ja(rc(x*) w') = K(x)A(w)JK(x*)a(w') = rc(x)A(w)xfa(w')
by 3.5.4 (ii) and 3.5.1 (c) by 3.5.4 (i) and 2.1.7 (iii) by 3.6.6 (i)
by (i) applied to w' and x by 3.6.6 (i) by 3.5.4 (i) and 3.5.1(c) by 3.5.4 (i) by 2.1.7 (iii) by 3.6.6 (i)
By continuity, we get: A(K(x) w) = rc(x)A(w)x
for all w in I,, and by continuity again, for all w in M*, which is (iii).
3.6.10 Theorem. Let K be a Kac algebra. The intrinsic group of K is equal to the set of characters on M*, that is the set of all x in M, such that x 0
and F(x)=x®x.
Proof. Let x be a character on M*, that is, x belongs to M, and is such that x 0 0 and r(x) = x 0 x. As rc(x) satisfies the same hypothesis, we have, for
110
3. Representations of a Kac Algebra; Dual Kac Algebra
allwinM*: xn(x)A(w) = xA(rc(x) - w)rc(x) = A(w)xrc(x)
by 3.6.9 (ii) applied to ic(x) by 3.6.9 (i)
So, by continuity, xK(x) belongs to M', and, then, by 2.7.2 (i), it is equal to a scalar a. But then, for all w in M*, we have: A(w)x = xA(rc(x) - w) = x,c(x)A(w)x
by 3.6.9 (i) by 3.6.9 (iii)
= aA(w)x
By continuity, we get x = ax, and, as x 54 0, we have a = 1. So xr%,(x) = 1, x is invertible, and the theorem is proved. 3.6.11 Proposition. Let G be a locally compact group, Ka(G) the abelian Kac algebra associated to G in 2.2.5; the dual co-involutive Hopf-von Neumann algebra associated to Ka(G) is (,C(G), F,, rc9) (cf. 3.3.6), and the dual weight
(on .C(G) is the Plancherel weight cp8 studied in [58], associated to the left Hilbert algebra ,C(G) of continuous functions on G with compact support.
Proof. By definition 3.5.2, the weight (cpa)" is associated to the left Hilbert algebra a(I. n i3), that is, by 2.4.7 (ii) and 1.1.2 (ii), the set:
If E L1(G) n L2 (G); the function s -+ f '(s) = J(s)ZG(s') belongs to L1(G) n L2 (G) }
equipped (by 3.5.1 and 1.1.2 (ii)) with the usual convolution product and the involution °. We have 1C(G) C a(I , n I3), and the operations on 1C(G) being the restrictions of those on a(Iw n I,p). Both generate the same von Neumann algebra,C(G). Thanks to 3.6.7 and 2.2.2, it appears that the modular operator associated to (cpa)" is the Radon-Nikodym derivative of the left Haar measure with respect to the right Haar measure, that is the modular function Z1G. It is also the modular operator associated to cos. Therefore, using ([114], prop. 5.9), we have: S03 = (cpa)"
3.6.12 Theorem. Let G be a locally compact group; the set of continuous characters of G (i. e. continuous multiplicative functions from G to C, except the function 0), is a locally compact abelian group, which is the intrinsic group of the Kac algebra Ka(G).
Proof. The intrinsic group of Ka(G) is, by 3.6.10, the set of all f in L°O(G)
such that f 0 0 and I'a(f) = f ® f (i.e. f(st) = f (s) f (t), a.e.)
3.7 The Dual Kac Algebra
111
Let now g in 1C(G) such that (g, f) 34 0; we have:
(g, f) f (s) = (JG (tf (s) = JG f (st)g(t)dt = JG f (t)g(s-1t)dt We then see that f is almost everywhere equal to a continuous function, and we get the result.
3.7 The Dual Kac Algebra 3.7.1 Lemma. Let x, y in 910; then I'(y)(x 01) belongs to A
and we have:
-(I'(y)(x 01)) = aW*o(A,(x) 0 AO(y))
Proof. Let X31, 02 and 6 in B'; let w be in I,, and x in 910. We have:
((1 0 ''(6))aW*a(A,(x) 0 a(w)) I Al 0 02) _ (cTW*o(A,(x) 0a(w)) I01 0A2T8 )
_
(W*(a(w) 0 A,(x)) I #2 T"'6" (& 01) (A(fl a(w)Atbb)A,(x) I Al)
by 2.6.1 (iii)
_ (A(w . (/32t6)*)A(x) I /31)
by 2.1.7 (iv) by 3.6.2
= (A(w ' A* (wb,A2)) A , (x) I A1)
= (A(w
A*(t`'b,02 ))I wAQ(x),A1)
(A*(2'A,(),,31), w ' A* (wb,A2 ))
_ (A(Z'6,2 )A*(^
, w)
(A* (``'AC,(x),Al * wb,A2 ) w)
by 3.3.5
(A(w), "A- (x) 1#, * wb,02 )
_ (I'(A(w))(A,(x) (9 8) I Al (9 02)
So, by linearity and density, we have:
(10 Let 61 be in
(*'(Si) 0
0 a(w)) = t(A(w))(A,(x) 0 6) V. We have: i'(6))aW*c(A,(x)
0 a(w)) _ (*'(o1) 01)(r'(A(w))(A,(x) 0 6) = ®6) = 1'(A(w))(x ®1)(61 ®6)
112
3. Representations of a Kac Algebra; Dual Kac Algebra
Then, we can deduce that crW*a(A,(x) 0 a(w)) is left-bounded with respect to the weight cP 0 , i.e. that it belongs to A0®0(910; so I'(\(w))(x (9 1) belongs to 91000 and:
0 a(w)) Let i/i the weight on M defined for y in M+ by: O(Y) = P 0 P"O(W 0 l)-P(Y)(x (9 1))
We have, for all w in M* : (A(w)*A(w))
_ (5P 0 )((x*01)r(A(w))*r(A(w))(x ®1)) _ IIAO(x) 0 a(w)112
by (i), since W* is isometric by 3.5.2 (ii)
_ W(x*x)V(A(w)*A(w))
The weight b is therefore semi-finite. Moreover Sp ® S is normal as well as I'
and the mapping Y -* (x* (9 1)Y(x (9 1) from M 0 M into itself. Therefore is normal. For all t in R and y in M+, we have: O(Ut (y)) _ (SP 0c)((x* 0 1)t(ct (y))(x 0 1)) _ P (9 SP)((x 0 1)(i 0 at )I'(y)(x 0 1))
by 3.6.5
_ (SP 0 S)((x* 01)r(y)(x 0 1)) _ ib(y)
Therefore 0 is o °-invariant, we can apply the proposition 2.1.5 to the von Neumann algebra M, the two weights c (x * x) and tb, with E = A (Ip ), because, by 2.1.7 (ii), A(,(E) = a(I,) is dense in HO. Therefore we have, for all y in M+: S
b(y) = c (x*x)SP(y)
that is, for all x in 910 and y in M (SP 0 S)((x* 0
(Y) (x ®1)) = S (x*x)V^(y)
By polarization, for x 1, x2, y1, Y2 in 910, we shall have: (SP 0 S )((x2 0 1)I'(y2yi)(xi 0 1)) = S (x2x1)5P(y2y1)
Therefore, there is some isometry U in C(Ho 0 Hg,) such that, for all x, y in 910, we have:
3.7 The Dual Kac Algebra
113
A1,®c(r(y)(x ®1)) = U(Ao(x) 0 An(y)) By (i) and by density, we see that U = crW * cr, which completes the proof. 3.7.2 Proposition. The weight 5P is left-invariant.
Proof. Let x be in 'A-, y in 9t . By 2.1.8 (i), it exists a sequence {afl}flEN of positive elements in M which are monotonely converging up to (i ® S)(I'(y)). We have:
= lim T (an, wA (x) )
= 1 m T (anAc(x) I A. (x))
= lim Tv(x*anx) n
= SP (x*((i 0 S F*"' W)
_ V((i 0 )((x* 01)I'(y)(x (9 1)))
by ([59], 2.1(3))
_ (SP 0 )((x' ® 1)P(y)(x 0 1))
by 3.7.1
= c (x*x)SP(y)
_
by ([5917 5.5)
(i)
I A,(x))
H' C H, and a Now, we know by 2.1.8 (i) that is exists a closed subspace positive selfadjoint operator T on H' such that: 'D(T)- = H' and:
(i ®SP)(r(y))(w) = IITV2eII2 (i 0 SP)(I'(y))(we) = +oo
E D(T 1/2)) (
It follows from (i) that D(T112) contains 11v( therefore H' = HH.
D(T1/2))
), which is dense in Hp;
Moreover, for all 6 in AW(9L), we have:
I= (e(y)e I )
jj(AY)1/2jgjj2
Thanks to the unicity of the polar decomposition, it follows: T1/2 = SP(y)1/21 or:
T=Ay)1
by (i)
114
3. Representations of a Kac Algebra; Dual Kac Algebra
Therefore (i 0 c) (P(y)) is bounded and is worth ^(y)l, which completes the proof.
3.7.3 Theorem. The quadruple (M, T, R-7 W^) is a Kac algebra. Its Fourier
representation is A = xA*; its fundamental operator is W = aW*cr (once Hip and H0 have been identified). Proof. We got in 3.3.2 that (M, T, k) is a co-involutive Hopf-von Neumann algebra and in 3.7.2 that Sp is a left-invariant weight on it. We can therefore apply 2.4 to (M, P, gyp); the fundamental operator associated to (M, P, Sp) appears to be o W * a, by 3.7.2. Applying 2.4.6. to (M, P, ic, c) we can then define a bounded linear mapping A(2), for all w in M* , by: A(w) = (Z' o I 0 i)(crW*a) _ (i 0 w o k)(W*) Using 2.6.1 (ii) and 3.3.1 we get: A(w) = A*( W"" o Ic) = rcA*(w)
Using 3.3.5, we see that A is a faithful non-degenerate representation; so, by 2.4.6 (iv), the weight Sp satisfies (HWii); it satisfies (HWiii) by 3.5.3, and then (M, P, Ic, Sp) is a Kac algebra.
3.7.4 Definition. The Kac algebra K = (i121, F7 Ic, c) will be called the dual Kac algebra of K. 3.7.5 Theorem. Let G be a locally compact group. Then the quadruple (C(G), P,, rC,, 5P,), where ,C(G) is the von Neumann algebra generated by the left regular representation AG of G, P,, rc, and W, have been defined respectively in 3.2.8, 3.3.6 and 3.6.11, is a symmetric Kac algebra, denoted K,(G), and we have: K8(G) = Ka(G)^
Its fundamental operator is equal to the function s -+ AG(s) (s E G), considered as an element of £(G) ® L°O(G). Its Fourier representation A(w) is defined as being, for all w in £(G)*, the function s -+ (AG(S),tZ') (s E G); the Fourier representation A of K,(G) is then surjective on the Fourier algebra defined in 3.4.6 and 3.5.6 (iii).
Proof. By 3.7.3 and 3.6.11, we see that (L(G), P,, 1c,, W.) is the dual Kac algebra of Ka(G). Therefore, we get, by 2.4.7 (i) that o.W*a is the function s --+ AG(s), when considered as an element of C(G) 0 L°O(G). By 3.7.3, it is the fundamental operator of K, (G). By 2.6.1 (i), applied to 1K, (G), we get then the Fourier representation of K, (G).
3.7 The Dual Kac Algebra
115
3.7.6 Proposition. Let K1 = (M1, I'i, xl, cot) and K2 = (M2, I'2, K2, W2) be two Kac algebras, U a unitary H.1 --> H(,2 which implements an IREisomorphism from K1 to K2 (cf. 2.7.9 and 2.2.5). Then U implements an H-isomorphism from K1 to K2, too.
Proof. Let u an ]H[-isomorphism from M1 to M2, such that SP2 o u = awl (cf. 2.7.9) and let U be defined by (cf. 2.2.5): U11p1(x) = a-1/242 (U W) Then, f o r a n y w in I, . 2 , w o u belongs to I.1, and we have:
Ual(w o u) = a1/2a2(w) So, we get, using 2.6.1 (v), for all w' in M2*:
UA1(w'o u)al(w o u) = Ua1(w'ou*wou) = Ua1((w'*w)ou) = a1/2a2 (w' * w) = al/2A2(w')a2(w) = A2(w')Ual(w o u)
from which we get that the application u defined, for x in M2, by:
u(x) = U*xU is a von Neumann isomorphism from M2 to Ml such that, for all w' in Ml*: u(A2(w')) = Al(w' o u)
From that, by predualizing, we get u*, a Banach space isomorphism from M1* to M2*, such that, for all w1 in M1*: ,\2*u*(wl) = uA2*(w1)
and, as A2* is injective (3.3.5), we see that u* is multiplicative and involutive, and, so, u is an ]8[-isomorphism. Moreover, we have then, for all w in 1602: S l(Al(w o u)*)11(w o u)) = 11al(w o u)112
by 3.5.2 (ii)
= aIIa2(w)112
= from that, we deduce, by 2.7.9, that W^ 1 o u = a^ W2.
by 3.5.2 (ii)
3. Representations of a Kac Algebra; Dual Kac Algebra
116
3.7.7 Proposition. Let K be a Kac algebra, W its fundamental operator, A the modular operator associated to the dual weight gyp. Then: (i) We have:
W(1®A)W* =,A^ OA (ii) For all t in R, ait belongs to the intrinsic group of K. (iii) For all t in R, w in M*, we have: A(w o Ra=t )
o
(iv) For all x in M+, we have: (SP ®i)(r(x)) =
SP(x)4-1
(v) Let x be in 9Zv fl 9Zcpo,. and i in 93o. Then (i 0 w,7)F(x) belongs to 9
fl 91,po,, and we have:
(x)
A W((' ®w,)(I'(x))) =
Proof. The fundamental operator of the dual Kac algebra K is uW*o (3.7.3); so (i) comes from 2.7.6 (vii) applied to K. From (i), we get W(1 (&a:t )W * = .' ®Qi; so (ii) comes from 3.2.2 (iv). By 3.6.6 (i) we have K(z 't) = JaztJ = so, 3.6.9 (iii) applied to A-it yields (iii). Let w be in M* +. We put w' = w(LX ) in the sense of ([114], prop. 4.2). By 3.6.7 and ([16], 1.1.2 (b)), we have:
Therefore, for all x in M+, by ([114], prop. 4.2), we have: (SP 0 w)(I'(x)) = lim (p o . 0 w')((a 0 i)CF(x)) C--,0 where: We have:
W(1 0 (1 + CIA))W* = 10 1 + eW(1 O'A^)W*
= 10 1 + E( 0 (i +
Ea)-')w*
=
(1®1+(AeA)y1
3.7 The Dual Kac Algebra
117
and:
(A 0 A)'C = (A 0 A)W(i 0 (1 + CA)-l )W
= W(i OA^)W*W(l 0 (1 +,eA)-')W*
= w(i Oa^(l + = r(aE)
Ea)-l)yy*
where:
therefore, we get:
lim(cp o ic 0 w')f(a&x) (v 0 W)(r(X)) = C-->o by 2.2.4
lim SP o
= SOW UJ(a-1)
by 3.6.7
which is (iv). We assume 117711 = 1. Then (i 0 w,,) is a conditional expectation and we have:
(i 0 w7)(r(x*x))
(i 0 w77)(r'(x*))(i 0 w7)(f (x)) which implies:
SP((i ® w77)(f(x*))(i 0 WO (I'(x))) : (SP 0 w7)(r'(x*x))
=I
therefore (i (9 w,,) (F(x)) belongs to
((i 0
by (iv)
. Furthermore, we have:
(i 0 w7 o ic)(rc 0 ,)f(x*) _ ("71 o
o
which, by hypothesis, and by 2.3.1 (ii), belongs to 9l,; therefore (iOw77)(f(x)) belongs to and by 3.5.2 (i), Ap((i ®w77)f(x)) belongs to Db, and we have:
EA
= = a(un)F'AV(x) Therefore, we have:
!!v((i 0 w,7)1'(x)) = FA(w,7)FA,(x) = Ja-1/2A(w,7)A1/2JAW(x)
by 2.3.5
by 3.52 (i)
118
3. Representations of a Kac Algebra; Dual Kac Algebra
On the other hand, for any
(A(WAl/4)e
in 950, we have:
(W -1/4,
_ (W*(z/4?i ®
by 2.6.1 (i) Q-1/477
(W *(a-1/277 (& C) Ii (& C)
by 3.6.7
(W*(A_h/2?i ® A/2A1/2) 17) 0 ((1 ® A1I2) W*(?7 ® A1/2e) 177 ®
by (i)
®4-1/26)
_ (W *(r/ ® 41/26) 177
by 2.6.1 (i)
Therefore we have A-1/2/\(wj7)A1/2 C A(wa-1/4n) which allows to complete the proof of (v).
3.7.8 Theorem. Let K = (M, .P, n, gyp) be a Kac algebra, K' the commutant Kac algebra. The dual Kac algebra K'" is equal to K. Proof. We have seen in 3.3.7 that the dual co-involutive Hopf-von Neumann algebra (M'^, I''^, rc"") is equal to (M, cr, ic). Let us now compute the dual weight Sp'^; let us recall (2.3.7) that, if w belongs to M*, if we define w' in M* by w'(x) = w(Jx*J), we have A'(w') = A (w). Let us now suppose that w' belongs to IV . We have, then, for any x in 91*, :
(JxJ) = w(Jx*J)= w'(x) _ (a'(w') I Aw,(x*))
(a'(w') I JAW(Jx*J))
_ (Ja'(w') I A,(Jx*J)) so w belongs to I., and a(w) = Ja' (w') . For w', w' in Iv, we have: (A/
A'(w')) = (a'(w') I a'(w'))
by 3.5.2 (ii) applied to cp'
= (Ja(2) I Ja(w1)) = (a(1) I a(w2)) = (A(2)*A(ii))
by 3.5.2 (ii) applied to v = (Mz'2 o rc)A(wl o K)*) by 1.2.5 and 2.5.3 (k(A(w2))k(A(wi)*)) by 3.3.1 = = S0 o SP
Using 2.7.7, we get gyp' = W o ic, which ends the proof.
3.7 The Dual Kac Algebra
119
3.7.9 Proposition. Let K = (M, r, K, Sp) be a Kac algebra and i = (M, I', k, SP) be a Kac 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 SP o E = Sp. Moreover, E satisfies:
roE= (E0E)I' RE=Etc and the projection P defined by PA,(x) = A4p(Ex) belongs to the centre of M; (ii) the canonical surjective H-morphism r from M to M^, defined in 3.3.8, has P as support, and identifies K^ to Kp (cf. 2.2.6).
Proof. As M is at -invariant and cplM semi-finite, by [160], there is a unique normal conditional expectation E such that Sp 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 91, n M by: AW(j (x))
Let P = II*. We know, from [160] that, for all y in 91.: Asp(Ey) = I*A,,(y) Acp(Ey) = PAW(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 IV; it is easy to see that w o j belongs to Ir., and that: a(w o j) = I*a(w) We have then, using 3.3.8: i)'#C
w""'*G'\(w 0
_
A(w 0 M
j)112
by 3.5.2 (ii) applied to
=I
=allRa(w)112
by 3.5.2 (ii)
120
3. Representations of a Kac Algebra; Dual Kac Algebra
= a II Ra(w) I I2 for all w in I.; so we get P = R (and so P belongs to the centre of M), a = 1, S = S Op, which ends the proof of (ii). Now, E 0 E is a faithful normal conditional expectation from M 0 M to N N M 0 M such that, for all X in WOW:
So, we have I
Av04E 0 E)(X)) = (P 0 P)Avosa(X ) So, for x, y in 91W, we shall have:
AW04E 0 E)(-P(y)(x 0 1))) = (P 0 P)AO,(.P(y)(x 0 1)) = (P 0 P)W(nw(x) 0 AW(y)) = (P 0 1)W(l 0 P)(Aw(x) 0 AW(y)) because P belongs to Z(M) = (P 0 1)W(elw(x) 0 dW(Ey)) = (P 0 1)A,O,(I'(Ey)(x 0 1))
As E ® i is also a faithful normal conditional expectation from M 0 M to M ® M such that, for all X in
0 i)(X)) = (P 0 We have then:
AW04E 0 E)(.P(y)(x 0 1)) =
0 i)(-P(Ey)(x 0 1)))
and, therefore:
(E 0 E)(r(y)(x ®1)) = (E 0 i)(r(Ey)(x ®1)) By continuity, we get, for all y in %,:
(E 0 E)(r(y)) = (E ® i)(r(Ey)) = r(Ey) and, by continuity again, we have:
(EoE)r=rE Let now x be in M; we have:
k(Ex)P = J(Ex)JP = J(Ex)PJ
by 3.6.6(i) because P belongs to Z(M)
= JPxPJ = PJx JP = Pic(X )P = Ex(X )P
because P belongs to Z(Jcf ) 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 K = (M, I', x, Sp) be a Kac algebra, R be a projection
of the centre of M such that I'(R) > R 0 R, n(R) = R and KR be the reduced Kac algebra in the sense of 2.2.6. There is a canonical one-to-one H-morphism j from (KR)^ to K which identifies (KRY with a Kac subalgebra of K (cf. 2.2.7). More precisely, if r denotes the reduction x -+ xR of M on MR, we shall have, for all w in (MR)*: j(aR(w)) = A(w o r)
where AR denotes the Fourier representation of KR. Proof. Let us call I the projection R, considered as an element of C(HW, HVR ). We have, then:
I*I = R II*=1HpR IA. (x) _ A R (r (x))
r(x) = IxI*
(x E 91p)
(x E M)
w-,,a o r = WI*a,I*7
(a, ^t E HWR)
Moreover, if w is in ')R' it is easy to check that w o r is in I. and that: a(w o r) = I*a(w) The reduction r is an H-morphism, thus the mapping w -+ or from (MR)* to M* is multiplicative and involutive; therefore, the set {A(w o r), w E (MR)* } is an involutive subalgebra of M; let us call N its weak closure.
Let /3 in Hp such that (1QIx6)=0forallxinNandbinH,.Wehave, for all a, in Hc,R, all bin Hp: 0 = (/3 I A(wi*7,I* a )b)
= (W(I*a 0 i3) I I*y 0 8)
= ((IO1)W(I*a®/3)I708) which implies, for all a in Hs,R :
(10 1)W(I*a 0 0) = 0 or:
(R 0 1)W(I*a 0
0
by 2.6.1 (iii)
122
3. Representations of a Kac Algebra; Dual Kac Algebra
as R is in the centre of M, W belongs to M ® M and RI* = I * , it implies:
W(I*a®/3)=0 As W is unitary, it gives I * a ®,Q = 0, for all a in HER , which implies a = 0. Then, N is a non-degenerate algebra on Hg,; it is a von Neumann subalgebra
of M. Let y be in
,win(MR)*. We have:
raw o
r o k 0 Z)r(y))
by 2.3.5
= nWR((w o ic 0 E) (r 0 r)r(y)) = AWR((W o k 0 r)r(ry)) = AR(w)AwR(r(y)) = AR(w)IAW(y)
therefore:
IA(w o r) _ Ag(w)I and: RA(w o r) = I* 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 , 77 in 2t,, we have:
(R&=Rib
RAE%I
RfTRIi=R(eTiJ)
Now, let z be in M such that zR = 0. For all , 77 in zA(wR,R,7) = zir(ReTRi ')
= zR*(eT rib)
we shall have: by 3.5.4 (iii)
by the above remarks
which implies, by continuity, for all , 77 in Hp: zA(wR ,R,7) = 0
or, also, for all -y, a in H(PR :
zA(wi*7,i*a) = 0
that is, for all w in (MR)*: zA(w o r) = 0
and zN = 0, which ensures z = 0 by the above results on N. So, the reduction N -> NR is an isomorphism. Let us call the inverse isomorphism, and for x
3.7 The Dual Kac Algebra
123
in (MR)", let us put j(x) =(I*xI). It is clearly a one-to-one homomorphism from (MR)" to M such that i(l) = 1. Moreover, we have, for all w in (MR)*:
i(aR(w)) = (I*AR(w)I) _ 3(Ra(u o r)) = A(w o r) The range of j is therefore equal to N. Now, let be in M. We have:
o r) = (A(w o r), w) = (jAR(w),27) = (AR*j*(Z'), w) (rA*(2'),w) = (A(Z'),w * and then rA* = AR* j*.
From what it is straightforward to prove that j* is involutive and multiplicative and therefore that j is an H-morphism. For all t in R, we have: OT (A(w
by 3.7.7 (iii) o r)) = A(w o r o Loi:) = A(w o Lr(Qit) o r) which belongs to N
therefore N is ut -invariant. Let w in I(R . We have: WWAR(w)*AR(w)) = P(A(w o r)*,X(w o r)) = I=
I=
I'OROR(W)*ARMJ
Therefore j (AR(I(PR) C 910, which implies that cp I N is a semi-finite weight.
Finally, we see that N is a Kac subalgebra of M, j is an H-isomorphism from (MR)^ to N and, by 2.7.9 and the above calculation S o j = VPR, which completes the proof.
Chapter 4 Duality Theorems for Kac Algebras and Locally Compact Groups
In that chapter, we obtain a duality theorem for Kac algebras, namely that the bidual Kac algebra is isomorphic to the original Kac algebra (4.1.1). From that, we can successively deduce that the Fourier representation A is faithful,
and that M* is semi-simple (4.1.3 ). We also see that the dual Kac algebra of the Kac algebra K8 (G) constructed in Chap. 3 is the Kac algebra K, ,(G) constructed in Chap. 2 (4.1.2 ). These results were found, independently, by the authors in [36], and Vainermann and Kac in [180]. Moreover, we obtain that the relative position of the von Neumann algebras M and M is such that (4.1.5 ):
MnM=MnM' =M'nM^ =M'n1V1' This result, from [136], leads, in the case of Ka(G), to Heisenberg's theorem (4.1.6 ).
The crucial link with duality of locally compact groups is given by Take-
saki's theorem ([157]), which states that every symmetric Kac algebra K is isomorphic to the symmetric Kac algebra constructed from the intrinsic group of K (4.2.5 ). By duality, we get that every abelian Kac algebra K is isomorphic to the abelian Kac algebra constructed from the intrinsic group of K. Applied to a standard Borel group with a left-invariant measure (4.2.6), we get A. Weil's theorem [197].
Applied to K3(G), Takesaki's theorem leads immediately to Eymard's duality theorem ([46] ), which states that G is the spectrum of the Fourier algebra A(G) (4.3.3 ), and, which, in turn, contains, in the commutative case, Pontr j agin's duality theorem (4.3.8). Eymard's duality theorem allows us to give a precise description of all the objects constructed from the symmetric Kac algebra K8 (G); in particular, the Fourier algebra of K8(G) is L1(G), and its Fourier-Stieltjes algebra is M1(G) (4.4.1), which leads, in the commutative case, to Bochner's theorem (4.4.3 ). We then, after [38], characterize all the morphisms which realize the quasi-
equivalence of A with A x y, for all non-degenerate representations /u, as
4.1 Duality of Kac Algebras
125
proved in Chap. 3 (1.5.6 ). When i is of dimension 1, we then get another characterization of the intrinsic group of a Kac algebra (4.5.8 ), due to De Canniere ([18] ), which leads to Wendel's duality theorem ([199]) for locally compacts groups (1.5.9 ). To each couple of non-degenerate representations of M* and M* respectively, we functorially associate a unitary operator belonging to the tensor product of the von Neumann algebras generated, as it was done in [40]; it is called the Heisenberg's pairing operator (1.6.2 ). For one-dimensional representations, i.e. for the intrinsic groups, we get a bicharacter in a situation similar to Heisenberg's commutation relation (1.6.7). This Heisenberg's pairing operator allows us to construct the extension of any non-degenerate representation of M* to a representation of the Fourier-Stieltjes algebra B(K), just the same way non-degenerate representations of L1(G) are extended to M1(G) (1.6.8 ). This will be essential to define the arrows of the category of Kac algebras in Chap. 5. Chapter 4 ends with a Tatsuuma type theorem about Kac algebras (1.7.2 ), which gives, as corollaries, Ernest's duality theorem ([44]) and Tatsuuma's duality theorem ([168]) on locally compact groups.
4.1 Duality of Kac Algebras 4.1.1 Theorem. Let K = (M, I', r., gyp) be a Kac algebra. The bidual Kac algebra
K^^ is isomorphic to K (equal if we identify Hp and Hip), and the Fourier representations A and A are linked by:
A=KoA*
A=k oA*.
Proof. The von Neumann algebra M"" is, by definition, generated by the Fourier representation A which is equal to rcA* (cf. 3.7.3), up to the isomorphism between H, and Hip; we thus have:
Mrn"CM The fundamental operator W associated to K is equal to aW*u (cf. 3.7.3); so 2.7.4 (i) applied to K gives that, for any x in ,C(H), x belongs to M^^' if and only if (x 0 1)W = W (x 0 1). So, by 2.7.6 (v), we get M^^' = M' and
then M = Similarly, for any x in M-^ = M, we have: aW*Cr(1 0 x)oWCr
= W(10 x)W*
= r(x) and:
by 3.2.2 (iv) applied to K by 3.7.3 by 2.6.3 (ii)
4. Duality Theorems for Kac Algebras and Locally Compact Groups
126
by 3.3.1 applied to K
K"'"(x) = Jx * J
= k(x)
by 3.6.6 (i)
By 2.7.7 the two Haar weights Sp and Sp^^ are proportional. Let fl, 6 be in 2t', such that /3T8b 54 0. We have:
(A(z5)* by 3.7.3
,b)) A
A
by 3.6.2 (i)
= I,/3t6fI2 by 2.1.7 (i) applied to Sp
=I
= Therefore
3.5.2 (ii) applied to K
and the theorem is proved, using 3.7.3 applied to K and
K.
4.1.2 Corollary. Let G be a locally compact group, and K3(G) the symmetric Kac algebra associated by 3.7.5. The dual Kac algebra K3(G)^ is equal to Ka(G) (when L2(G) and H(,, are identified).
Proof. It is a combination of 4.1.1 and 3.7.5. 4.1.3 Corollary. Let K = (M, r, x, Sp) be a Kac algebra, K = (1V1, I', dual Kac algebra. Then:
k,
5P), its
(i) The modular operator A is affiliated to the centre of M, and is the Radon-Nikodym derivative of SP with respect to WP o ik.
(ii) The Fourier representation A is injective and, therefore, M* is semisimple.
(iii) For all t in R, a:t belongs to the intrinsic group of K. (iv) Let x in gio fl 9l... and 77 in Mo. Then the element (i (9 2,,)1'(x) belongs to
. fl 9toog and we have:
A0((i ®w17)I'(x)) = JA(wJA ,(x) (v) Let A be a von Neumann algebra, 0 'a faithful semi-finite normal weight on A+, and let X in
,®O n 910(&(,o, and 77 in 2.o. Then the element
I')(X) belongs to®- fly,®Spo, and we have: i 0 W- w 17)(i
0 r)(X)) _ (1
Ja(wo-114,,)J)AOOW(X)
4.1 Duality of Kac Algebras
127
Proof. As, by 4.1.1, K"^ = K, the first assertion results of 3.6.7 applied to K. By 3.3.5, ica* is faithful, therefore so is A, thanks to 3.7.3. Applying this result to K and using 4.1.1, we get the second assertion. The third assertion is 3.7.7 (ii) applied to K = K^^. By applying 3.7.7 (v) to K, we get (iv). Let us assume II 77 11 = 1. As i 0 i 0 w,7 is a conditional expectation, we get: O 0 rt')
W17)
F) (X 0
-P) (X))))
W17)
rt.)(X*X)
because cp o k is left-invariant with respect to s ' (2.2.4). So (i ® i (9 w,?)(i (&I')(X) belongs to 91,p®-o-. Similarly, we have: 0 W^17)(i 0 F)(X
0 17)
P^) (X))))
W?7)(' 0 r)(X*X) 1/2 7711
2(00 W )(X*X) < +00 by 3.7.7 (iv) applied to K
and the operator which sends Then (i 0 i ® W^ 77)(i ®I')(X) belongs to ((i ®i ®w,q)(i ®I')(X )) is bounded and its norm is smaller A,p®,(X) on
than IkX'2II2. For x j in
910 and x 2 in 'Y , n
it is immediate, by (i),
that we have: Alpoo((i 0 i Ow- t7)(i 0 -P)(Xl 0 X2)) = (10
JA(wa-,14,)J)(AO(xj) 0
which, by continuity and density, completes the proof of (v).
4.1.4 Corollary. (i) Let K = (M, r, n, Sp) be a Kac algebra, and let K; be (M, cl', x, W o x) the opposite Kac algebra as associated in 2.2.5. Then the dual Kac algebra K;^ is equal to K'. (ii) The Kac algebra K is abelian if and only if the dual Kac algebra K is symmetric.
Proof. Let us apply 3.7.8 to the Kac algebra K; we get (K)" _ (K)` = Ks by 4.1.1; the second assertion is clear from 3.3.3 and 4.1.1. 4.1.5 Corollary. Let K = (M, I', ,c, cp) be a Kac algebra, K = (M, I', k, V*') its dual Kac algebra. Then:
MnM=MnM'=M'nM=M'nM'=C.
128
4. Duality Theorems for Kac Algebras and Locally Compact Groups
Proof. By 2.7.2, we have:
Mn1V1'=M'nM'=C Applying 2.7.2 to the Kac algebra
we have, using 4.1.4:
MnM=M'nMA = C and the result is proved. 4.1.6 Corollary (Heisenberg's Theorem). Let G be a locally compact group. If we consider the elements of L' (G) as operators on L2 (G), we have: LOO(G) nC(G)l = LOO(G) n L(G) = C
-
Proof. We get the result by applying 4.1.5 to the Kac algebra Ka(G).
4.1.7 Proposition. The isometries J and J commute.
n gi..,, n * o,, ; by 3.5.2, i.e. to D(al 2) n D(a-1 2). We have then: Proof. Let x in
n tY
(x) belongs to D# n Db ,
3JA,(x) = Ja1/2Ap(x*) = Q1/2JAw(x*)
by 4.1.3 (i)
A 1/2 A -1/2 (Aw(x*)), ,A1/2A
by 3.5.2 (i)
A'2Ll"2A(p(ic(x)) by 3.6.7 because tlp(r,,(x)) belongs to D(A1/2) _ 'A _ 'A
by 3.6.7
JA -112t1W(t4x*))
= JJ(A(,(tC(x*)))I
= JJtlp (x)
by 3.5.2 (i)
by density (2.7.5 (v)), it completes the proof.
4.1.8 Lemma. The set Y = {xJA,(x); x E 91, selfdual cone PW defined in 2.1.1 (iv).
qiorj
is dense in the
4.1 Duality of Kac Algebras
129
Proof. Let us put A-1 = f °O sdE,; e1z = J'° n dE,; from 3.6.7, en belongs to the centre of M. Let x in TV. We have:
enx*x) < +oo
Sp o x(enx*x) = Sp(d
and enx belongs ton, n givo,Moreover: enxJAw(enx) = enxJenAw(x) = enxenJAW(x) = enxJAcp(x)
because en belongs to the center of M. Thus, this vector, which belongs to Y, converges to xJA,(x) when n goes to infinity. Therefore Y is dense in {xJAw(x); x E 91W}, the closure of which is P0.
4.1.9 Theorem. The isometry J is the canonical implementation of ic in the sense of [57] (cf. 1.1.1 (iii)). Proof. Thanks to ([57] th. 2.18), 3.6.6 (i) and 4.1.7, it remains to prove that P, is invariant under J. Using the above lemma, it is enough to prove that JY C Psp . As App (gig n
qion) C Db =
D(.A-1 /2) and
as A is affiliated to the centre n qi,.,
of M, we have Ja-1/2 C A-1/2J, and therefore, for all x in JA,(x) belongs to Db; then, we have: JxJAW(x) = Al/2FxJAp(x)
= a1/2Ic(x*)FJAW(x) = dl/2K(x*)a/2JJAW(x) = al/21(x*)a1/2JJAy,(x) = a1/2K(x*)J'Al/2 J-AW(x) =
by 3.6.3 (i)
by 4.1.7 by 3.6.7 by 3.5.2 (i)
Let us remark that K(x*)JA,(K(x*)) belongs to Y C Pip. Let us put: oo
Q1/2 =
sdE 9 0
dn =
n
1/n
sdE s
It results from 3.6.7 that dn belongs to the centre of M, and then that do = do/2Jd1/2J. By ([57] 1.9.(3)), we get that dnP is included in P and therefore dnK(x*)JA,(x(x*)) belongs to PW. As Pip is closed, it implies, when n goes to infinity, that A1/2 n(x*)JAW(ic(x* )) belongs to PP, which completes the proof.
4. Duality Theorems for Kac Algebras and Locally Compact Groups
130
4.1.10 Corollary. The isometry J is the canonical implementation of I (in the sense of 1.1.1 (iii)). Proof. By 4.1.9 and 4.1.1.
4.2 Takesaki's Theorem on Symmetric Kac Algebras In all this paragraph K = (M, I', rc, gyp) is a symmetric Kac algebra (i.e. cI' = I'), G(K) (or G) is its intrinsic group. 4.2.1 Lemma. The algebra M* is abelian, and by 3.6.10, G is its spectrum. Let 9 be its Gelfand representation, defined by, for w in M*, and s in G: gw(s) = (s*, w)
We have, for s, t in G: (i) 9(w o re)(s) _ gw(s-1)
(ii) g(s w)(t) _ 9w(s1t)
(iii) 9(w°) = (gw) (iv) The representation CJ can be identified with the universal representation of M* in C*(M*). Therefore C*(M*) is equal to the algebra Co(G) of continuous functions on G, vanishing at infinity. (v) The representation 9 is one-to-one. Proof. We have, by definition: g(w o rc)(s) = (s*, w o IC) _ (K(s*),w)
_ (s,w)
by 2.6.6(i)
by definition of c and 1.2.3 which gives (i). Moreover:
g(s w)(t) _ (t*, s w) _ (t*s, w)
_ ((s_ht)*,w) by definition of g _ gw(s-1t) which gives (ii). We have: 9(w°)(s) = (s*, w°) (K (S), W)
(9w) - (S)
which gives (iii).
by 1.2.5 by 2.6.6 (i) by definition of 9
4.2 Takesaki's Theorem on Symmetric Kac Algebras
131
By (iii) we see that all characters on M. are hermitian; using ([12], I, §6), we get (iv). By 4.1.3 (ii) we see that the Fourier representation A is one-to-one; there-
fore the same holds for the universal representation, that is 9, by (iii), and it completes the proof. 4.2.2 Lemma. There exists on G a Haar measure ds such that, for all wl, w2
in I.: (A(w2)*A(Wl))
(9w2) (!9wi) ds
_ fCG.
Moreover, the space 9IW is dense in L2(G) and (gI()2 is dense in L1(G).
Proof. Let a be the canonical extension of A to CO(G) (cf. 1.6.1 (i)). The space A(C0 (G)) is included in M which is an abelian von Neumann algebra. Therefore, the weight W is a trace, and S o A is a lower semi-continuous trace on C0(G).
Let w in I.. We have: A
o a(14Wj2) = AA(w)*A(w))
=I
Therefore !9W belongs to 91 ..X. The ideals 91,Oa and 92,.a are dense in Co(G); by ([114], 5.6.3) they include the algebra )C(G) of continuous functions on G
with compact support. Therefore, Sp o A is a positive linear form on 1C(G) and, by [11], a Radon measure on G which will be noted As Sp o A is lower semi-continuous, we shall have, for any f in C,o (G)+: Y.
0 a(f) = IG .fdy
and, by (*) we get, for all w in IV:
IG
IswlZdv
= I< +00
therefore, 9w belongs to L2(G, p). Polarizing, we shall get, for wi, w2 in IV:
IG
(a(w1) I a(w2))
^(A(W2)*A(ujl))
132
4. Duality Theorems for Kac Algebras and Locally Compact Groups
Finally, 91. is dense in Co (G), it is then also dense in L2 (G, P) and (gI4II will be dense in L1(G, p). And, we have: JG
(c2)(st)cwi(st)d1t(t) (9(s - W2))_(t)(9(S - WO)(t)
IC =(a(s.w1)I a(s-w2)) = (sa(w1) I
= (a() I a(02)) (GL02)-(Gwl)dp
by 4.2.1 (ii)
by (**) by 2.1.7 (iii) by 1.2.3 by (**)
G
By density, it follows that y is left-invariant, which completes the proof.
4.2.3 Proposition. There exist a Haar measure ds on G and an isomorphism U from HV to L2(G, ds) defined by, for all w' in I.: Ua(w') = gw'
such that we have, for all s in G, and w in M*: (i) (ii)
UsU* = AG(S) UA(w)U* = gw
Proof. Thanks to 4.2.2 it is clear that U as defined in the proposition is unitary. Moreover, we have, for all s, t in G and w in I,: (AG(s)gw')(t) _ gw'(s-1t)
_ (g(s w'))(t)
by 4.2.1 (ii)
Therefore:
AG(s)cw' = g(s w')
= Ua(s w') = Usa(w')
by definition of U by 2.1.7 (iii)
which gives (i).
Let w in M*, w' in I, . We have: UA(w)U*Cw' = UA(w)a(w') = Ua(w * w')
_ ww'
by 2.6.1 (v)
4.2 Takesaki's Theorem on Symmetric Kac Algebras
133
Since 91. is dense in L2(G), we have: UA(w)U* = gw
which completes the proof.
4.2.4 Takesaki's Theorem. Let K = (M, r, n, cp) be a symmetric Kac algebra,
G(K) its intrinsic group. There exist a Haar measure on G(K) and an isomorphism U from Hip to L2(G(K)) such that the mapping x -+ UxU* is an isomorphism of the Kac algebra K to Ka(G(K)). Proof. Using 4.2.3 (ii) and considering the generated von Neumann algebras,
we see that x -+ UxU* is an isomorphism from M to L°°(G(K)). More accurately, we have, for all w in M* : UA(w)U* = gw
and we can deduce, thanks to 1.4.5 (ii), that we have: (U 0 U)rA(w)(U* 0 U*) _ (U ® U)c(A X A)(w)(U® ® U*)
= s9 x 9(w) Let f , fl, f2 in L1(G(K)). We have:
(9*f,w) = (cw,f) =
G ((s)4w(s)ds
c* f =
= j)H
f (s)s*ds JG
and:
(cc X g(wi), fl ®f2) _ (cf2gfi,w) = J
I
xG xG
f2(t)fi(s)(t*s*,w)dsdt 9w(St)fi (3)f2(t)asdt
= (l'acw,fi ®f2) Thus we have, for all w in M* :
(U ® U)r(A(w))(U* ® U*) = ra(cw) and, by density, for all x in M:
(U 0 U)r(x)(U* 0 U*) = ra(UXU*)
4. Duality Theorems for Kac Algebras and Locally Compact Groups
134
Moreover, we have: Uk(A(w))U* = UA(w o rc)U* _ 9(w o K) = Ica(gw)
by 3.3.1 by 4.2.1(i)
by density, for all x in M, we get: Uk(x)U* = rCa(UXU*)
Finally, by 4.2.2, we have, for all w in I.: V(A(w)*A(w)) = pa(IgwI2) = Va(UA(w)*A(w)U*)
Since x -+ UxU* is an 1EE-isomorphism between two Kac algebras, cp and Spa two Haar weights, the result comes from 2.7.7.
4.2.5 Corollaries. (i) Let K = (M, F, r.., Sp) be a symmetric Kac algebra and G(K) its intrinsic group. There exist a Haar measure on G(K) and an isomorphism U from HW to L2(G(K)) such that the mapping x --> UxU* is an isomorphism from the Kac algebra K to K9(G(K)) ([170]). (ii) Let K _ (M, I', rC, cp) be an abelian Kac algebra. There exist a Haar measure on G(K) and an isomorphism U from HW on L2(G(K)) such that the mapping x --+ UxU* is an isomorphism from the Kac algebra K to Ka(G(K)) ([157]).
Proof. The assertion (i) results from 4.2.4, 3.7.6 and 3.7.5, and (ii) from (i), 4.1.4 (ii), 3.7.6 and 4.1.2. 4.2.6 Corollary (Weil's Theorem [197], [94]). Let G be a standard Borel group, and let there be a a-finite left-invariant measure m on G. Then G is a locally compact group and m is a left Haar measure on G.
Proof. The hypothesis allows us, using Fubini's theorem to see that the product measure m ® m is invariant under (x, y) --+ (y-1 x, y), and then, successively, by (x, y) -> (x,xy) and (x, y) (y-1, xy).
So, if the function f belongs to the abelian von Neumann algebra L' (G, m), the function (s, t) -+ f (st) on G x G belongs to L°O(G x G, m®m) (identified to LOO (G, m) ® L°° (G, m) ), and the function s -+ f (s -1) on G belongs to L' (G, m). We have then defined a coproduct P and a co-involution ic on L°O(G, m). Moreover, the measure m defines a normal semi-finite faithful trace on L' (G, m)+, which is left-invariant with respect to the coprod-
4.2 Takesaki's Theorem on Symmetric Kac Algebras
135
uct F. The fundamental operator associated by 2.4.2 is defined, for all f in L2 (G x G, m ®m ), x, y in G, by: (Wf)(x, y) = f (x, xy)
Let us now consider, for all s in G, the unitary operator µ(s) on L2(G, m) defined, for all f in L2 (G, m), t in G, by: 1 WS)f )(t) = f W t)
By ([94], lemma 7.4), y is injective. A straightforward calculation gives that the function s -+ µ(s)* on G, if it is considered as an element of the tensor product LOO (G, m) 0 ,C (L2 (G, m) ), is equal to W, and therefore, we get that
W is unitary, and that (here x is an automorphism because L°O(G, m) is abelian):
(K ®i)(W) = W* which, by 2.4.6 (v), proves that (L°°(G, m), F, ic, m) is an abelian Kac algebra. Moreover, for all s in G, we have:
WRWs) 0 1)W = µ(s) & N(s) and, by 3.6.10 and 3.2.2, µ(s) belongs to the intrinsic group G' of L°O(G, m)".
In fact, by 4.2.3 (i) and 4.2.5 (iii), there exists an isomorphism U from L2 (G, m) to L2(G') such that, for all v in G': UvU* = AGI (v)
and moreover, such that x -- UxU* is an isomorphism from the Kac algebra (L°°(G, m), 1 ', i , m) to K, ,(G') and, therefore, from the dual Kac algebra (L' (G, m), F, K, m) "* to K, (G').
So, (U 0 U)W(U* 0 U*) is the fundamental operator of Ka(G'), and, by 2.4.7 (i), (U ® U)W*(U* 0 U*) is the function v -+ AGI (v) on G', and (U ® I) W * (U* ® I) is the identity function on G', considered as an element of L°O(G') ® L°O(G, m)". So, for all w in (L°°(G, m)")*, we clearly see that U(i ® w)(W)U* is the function v -+ (v, w) on G' and, as (i 0 w)(W) is the function s --> (µ(s), w) on G, we infer that the isomorphism f -> U* f U from L°O(G') to L°O(G, m) is just the composition by y. Using f = XG'-µ(G), we see then that G'- µ(G) is of Haar measure 0. So, if v belongs to G' - µ(G), v j(G) C G' - p(G) is of Haar measure 0, and so is p(G), which is impossible; so ji is surjective, and the theorem is proved.
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4. Duality Theorems for Kac Algebras and Locally Compact Groups
4.3 Eymard's Duality Theorem for Locally Compact Groups In that paragraph G will denote a locally compact group. We shall apply the preceeding paragraph to K, (G). 4.3.1 Lemma ([44]). The left regular representation AG is a one-to-one homeomorphism from G to the intrinsic group of K,(G). Proof. Let s be in G. Then, by 3.3.6, AG (s) belongs to G(K, (G) ); the mapping AG is also clearly one-to-one.
Now let {AG(s)} denote a net converging to the identity 1 = AG(e), where e is the unit of the group G. We wish to show that s converges to e. Proceeding by way of contradiction, we suppose it does not. Then there
exists a subnet, say sn of sv and a compact neighbourhood V of e such that sn V, for all n. Choose a compact neighbourhood U of e such that UU 1 C V. Then, for each sn, we have (xu denotes the characteristic function of U and µ the Haar left measure on G): (AG(sn)Xu I Xu) =
Xs1zuxudf = y(Snu nu) = 0 G
Thus :
I((AG(Sn) -1)Xu I_ (Xu I XU) _ 11(u) > 0
Thus AG (sn) does not converge weakly to 1, which contradicts our first assumption and completes the proof. 4.3.2 Theorem. The left regular representation AG is a bicontinuous bijection from G onto the intrinsic group of K,(G).
Proof. Let denote Go the intrinsic group G(K,(G)). By 3.7.5, the Fourier representation A of K, (G) is the mapping, defined, for all w in .C(G)*, by: S -+ (AG(S)*,w)
which belongs to L' (G). By 4.2.4 applied to K, (G), there exists a unitary U from L2(G) on L2(Go) such that UA(w)U* be equal to gw, which is the mapping defined, for all t in Go, by: t -+ (t*, w}
which belongs to L°°(G0). Thus, we have:
U*gwU=gwoAG
4.3 Eymard's Duality Theorem for Locally Compact Groups
137
and, by continuity, for all f in LO° (Go) :
U*fU=foAG By selecting f =
(*)
XGO_AG(G), we see that f = 0, and Go - AG(G) is a zero-
measure set. Let so E Go and so AG(G); we have soAG(G) C Go - AG(G); therefore s oA G (G) is a zero-measure set as well as )tG (G) by left-invariance, but this is impossible by (*). Therefore AG(G) = Go, which completes the proof.
4.3.3 Corollary (Eymard's Theorem [46]). Let G be a locally compact group.
Let us recall (cf. 3.4.6 and 3.5.6 that the Fourier algebra A(G) has been defined as the set f f * 9, f , g E L (G)} (where f (s) = 1(s - 1) for all s in G, f in L2(G)), equipped with the norm: fG °(s)f(s)ds Then, the spectrum of A(G) is G; so every character on A(G) is involutive. Proof. By 4.3.2, the spectrum of L(G)* is equal to AG(G). By 3.4.6 and 3.5.6,
there is an isomorphism between C(G)* and A(G), which, to each w f,g in ,C(G)*, associates the function f * g, linked by:
(f*)(s) = (AG(s),fg) So, we see that the spectrum of A(G) is the set {s', s E G}, that is G. 4.3.4 Corollary. Let G1, G2 two locally compact groups, u an H-morphism from Hs (G1) to H9 (G2 ); then there exists a continuous group homomorphism a from G1 to G2 such that u(AG1(s)) = -G2 (a(s)), for all s in G1. The image a(G1) is a closed subgroup of G2. If u is injective, then a is injective; if u is surjective, then a is surjective too.
Proof. It is clear that u sends the intrinsic group of Hs(G1) into the intrinsic group of 1HIs (G2), and that the restriction of u is a continuous homomorphism
of groups. So the existence of a comes directly from 4.3.2. Then, the subset {x E £(G1); I's(x) = x 0 x} is a closed subset of the unit-ball of ,C(G1), and therefore, is compact for weak topology. Its image par u, that is: {AG2 (a(s)), s E G2} U {0}
by 3.6.10 and 4.3.2, is then also compact for the weak topology of C(G2). So {AG2 (a(s)), s E G1 } is locally compact, and by 4.3.1, a(G1) is a locally compact subgroup of G2, and so is a closed subgroup of G2.
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4. Duality Theorems for Kac Algebras and Locally Compact Groups
For every w in C(G2)*, we have: gw(a(s)) = (AG2 (a(s)*), w) = (UAG1(s),w) = (AG1(s)*,WoU) = g(w o u)(s)
by 4.2.1 by definition of a by 4.2.1
And:
gwoa=g(w0u) Therefore, gw o a = 0 implies w o u = 0, by 4.2.1 (v), and, so, if u is surjective,
it implies w = 0. Using 4.3.3, we get then that a(G1) is dense in G2, and so that a(G1) = G2. If u is injective, a is trivially injective.
4.3.5 Corollary. Let G1 and G2 be two locally compact groups; then, the following assertions are equivalent: (i) There exists a bicontinuous isomorphism u : G1 --- G2 (ii) There exists an ]H[-isomorphism from Ha(G2) onto 1H,, (G1). (iii) There exists an H-isomorphism from 1H1B(G1) onto 1HI8(G2).
Proof. We have (i) = (ii) because the application f -+ f o u is an I8Iisomorphism from Ha(G2) onto 1HIa (G1), (ii) = (iii) by 3.7.6, and (iii) = (i) by 4.3.2. 4.3.6 Corollary ([163]). Let G be a locally compact group.
(i) Let (M, I', i) be a sub co-involutive Hopf-von Neumann algebra of 18(G), such that there exists a Haar weight So on (M, F, K). Then, there exists a closed subgroup G' of G such that M is generated by {AG(3), S E G'} and is isomorphic to ,C(G').
(ii) Let K be a sub-Kac algebra of K8(G); then there exists G', open subgroup of G, and an H-isomorphism from K onto K8(G'). (iii) Let Kp be a reduced Kac algebra of Ka(G); then there exist G', open subgroup of G, and an H-isomorphism from Kp onto Ka(G'); moreover, we have:
P = XG' Proof. Let K = (M, I', K, cp); as K is a symmetric Kac algebra, by 4.2.5 (i), it is isomorphic to K8 (G(K) ). But it is clear, by 4.3.2, that: G(am) = M n {aG(s), s E G}
Let us put:
G'={sEG; AG(s)EM}
4.3 Eymard's Duality Theorem for Locally Compact Groups
139
Then, G' is a closed subgroup of G, and M is generated by {AG(s), s E G'}. Moreover, by 4.3.2, AG G' is a bicontinuous isomorphism from G' onto G(K), and by 4.3.5, (M, 1', i) is H-isomorphic to 1HI3 (G'), and we have (i). With the hypothesis of (ii), let E be the conditional expectation from ,C(G) to M obtained by 3.7.9 (i). For any s in G, we have:
l(Eac(s)) =
EAG(e)OEAG(s)
So, by 3.6.10, either EAG(s) = 0, or EAG(s) is in the intrinsic group of K, and is therefore unitary, which implies EAG(s) = AG(s).
So, the subset Is E G; AG (s) 0 M} is therefore equal to the subset Is E G; E AG (s) = 0} and is closed. Then G' = Is E G; AG (s) E M} is an open subgroup of G. So (ii) is proved and (iii) is proved using (ii) and 3.7.9 (ii).
4.3.7 Proposition. Let G be an abelian locally compact group, G the abelian locally compact group of all continuous characters of G (cf. 3.6.12), which will be called the dual group of G. Then: (i) The group G is the spectrum of L1(G), and the Gelfand representation of L1(G) is given by the Fourier transform: .f(X) =
VX E G, b'f E L1(G)
JQH()-ds
(ii) There exists a Haar measure dX on G. and an isomorphism U from L2(G) to L2(G) defined, for all f in L1(G) n L2(G), by:
Uf=f called the Fourier-Plancherel transform, such that the mapping x --> UxU* is
an isomorphism from the dual Kac algebra K3(G) to La(G), such that, for all f in L1(G): UAG(f)U* = J -
Proof. By 3.6.12, the set of continuous characters of G is the intrinsic group of Ka(G); as G is abelian, Ka(G) is symmetric, and, so, G' is the spectrum of the abelian Banach algebra L1(G), and the Gelfand transform, taking a coherent definition with 4.2.1, will be given by:
9f (x) = (x*, f ) = for all f in L' (G), X in G, which is (i).
fG
f (s) (X, s) - ds
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4. Duality Theorems for Kac Algebras and Locally Compact Groups
Using Takesaki's theorem (4.2.3 and 4.2.4), we see that there are a Haar measure dx on G, and an isomorphism U from L2(G) to L2(G), defined by U f = f for all f in L1(G) fl L2 (G), such that the mapping x -+ UxU* is an isomorphism of the dual Kac algebra 11(3 (G) to 10 (G). By 4.2.3 (ii), we get UAG(f )U* = f, for all f in L1(G), which ends the proof. 4.3.8 Pontrjagin's Theorem ([121]). Let G an abelian locally compact group, G its dual group, as defined in 4.3.7. Then the group G^^ is isomorphic to G. Proof. The isomorphism defined in 4.3.7 (ii) sends the intrinsic group of K8(G)
onto the intrinsic group of Ka(G), that is onto G^^. For all f in L1(G), we have, using 4.3.7 (ii): .f(s)(X's-l)ds
IG
=AX) = JGf(s)UAG(s)U*ds
from which we can deduce that UAG(s)U* is the function on G: x --+ (x, S-1)
Using Eymard's theorem (4.3.2), we get that these functions are all the characters on G, and so that the group G^^ is isomorphic to the group G.
4.4 The Kac Algebra li(3(G) It is now possible to describe the various objects associated to the Kac algebra K8 (G) by the general theory.
4.4.1 Proposition. (i) The enveloping C*-algebra C*(K8(G)) is the algebra Co(G) of continuous functions on G, vanishing at infinity; the canonical representation of C(G)* into C*(K8(G)) is then the Gelfand transform 9w(s) = (AG(8)*,w) for all w in £(G)*, s in G. (ii) The Fourier-Stieltjes algebra B(K8(G)) is the algebra M1(G) of bounded measures on G, and its Fourier-Stieltjes representation is the left regular representation of M1(G); an element x of £(G) is positive definite representable (in the sense of 1.3.6) if and only if there exists a (unique) positive bounded measure m on G such that: IG AG(s)dm(s) (iii) The enveloping W*-algebra W*(K3(G)) is the dual M1(G)* of M1(G). This Banach space, which is a W* -algebra, being equal to the bidual of C0(G),
4.4 The Kac Algebra K8 (G)
141
has then a structure of co-involutive Hopf-von Neumann algebra, given by:
(r(e), m1 0 m2)
m1 * m2) (is(9),m) _ (e, m°)
(0 E M1(G)*, m1, m2 E M1(G)) ME M1 (G)) (0 E M1(G)*)
where * is the multiplication of M1(G), and ° its involution, and where 8 is
defined by (0, m) = (9,rYi), with m(f) = (ffdm)- for all f in C0(G). (iv) The canonical imbedding (sA)* from L1(G) to B(K8(G)) = M1(G) is the usual imbedding from L1(G) to M1(G). Proof. By 4.3.3, the enveloping C*-algebra of C(G),, is the algebra of continuous functions on G, vanishing at infinity, and the canonical representation of L(G)* is its Gelfand transform. So (i) results from 4.3.3. As B(K8(G)) is the dual of C*(K8(G)), we deduce, from (i), that B(K8(G)) is equal, as a Banach space, to M'(G). Let us compute its Fourier-Stieltjes representation ice7r*; if m is in M1(G),
f, g in L2 (G), we shall have:
(lcslr*(m),wfg) = (lr(wf,g o tc), m) g(wf,g o tc)dm IG
IG
(KG(s)>wf,s)dm(s)
by 3.7.5 and (i)
And so, we have: AG(s)dm(s) JG We can deduce from it that the multiplication and the involution of B(K8 (G)) ,c87r*(m)
=
are the usual ones on M1(G), which gives (ii), with the help of 1.6.10. Then (iii) is a straightforward application of 3.1.5 (ii)).
Let f be in L' (G), and m = (sA)*(f). As A(f) is JGAG(s)f(s)ds, and, which is, by (ii), equal to by 1.6.1 (ii) and 3.7.3, equal to fG )G(s)dm(s), we see that m is the measure f (s)ds, which gives (iv).
4.4.2 Proposition. (i) Every non-degenerate representation y of ,C(G)* is given by a spectral measure Pµ on G, with values in Hµ, as defined, for example in ([105], IV, §17.4), such that, for all w in £(G)*:
µ(w)- Jc(ac(s)*iw)dPN(s) (ii) Let IL be a non-degenerate representation of ,C(G)*, Pµ its associated spectral measure on gµ, an element of (Aµ)*; then K8µ*(Sl) is the image by the left regular representation of the bounded measure d(P/(s), (1) on G.
4. Duality Theorems for Kac Algebras and Locally Compact Groups
142
(iii) Let µ1, 112 be two non-degenerate representations of ,C(G)*, Pµ17 Pµ2
their associated spectral measures. The spectral measure associated to the Kronecker product µ1 xµ2 is the convolution product of Pµ1 and Pµ2 defined by:
IC
.f(s)d(Pµl * Paz)(s) =
IGxG
.f(st)d(Pµz(t)(9 pill (s)).
Proof. By 4.4.1 (i) and 1.6.1 (i), there is a representation µ of C0(G) on ?-lµ such that µ o c = p; by ([105], IV, §17.4), the representation µ is given by a spectral measure on G with values in ?.lµ, which gives (i). We have then: (K. JL* A, W)
(W 0 rt..), fl)
=
J (aG (s)moo K)d(Pr(9), R) fc(ac(9),w)d(P. (s), n)
and therefore:
k8µ*(,fl) = J
Si)
G CG
which gives (ii). Let now f1 be in (Aµ1)*, Sf22 in (Aµ2)*; we have: ((µi x µ2)(w), J21 (9 02)
by 1.4.3
(111* (J?1)112* 022), W)
_ (IC8Y2*(.(l2)KSµ1s(fl1),w o rca)
s)d((P,42, n)* (P1, n))(s)
= IG(G = IG(
by (ii)
GH*
)w)d((t'NZ,.fl) * (F,41, f2))(s)
therefore, the measure d (Pµ1 x µ2 , D10 522) is the convolution product of the measures d(Pµ2 , ,f22) and d(Pµ1, Si); which gives (iii).
4.4.3 Theorem. Let P be a spectral measure on G with values in R. Then, there exists a unitary U in ,C(G) ®,C(f) such that, for all w in ,C(G)*, , is in H: (UI W 0 WC17) = fG (AG (S) *, w) d(P(s)
We shall write:
U=
AG(s)* ®dP(s) G
.
177)
4.4 The Kac Algebra K8 (G)
143
Proof. Let p be the non-degenerate representation of C0(G) associated to the spectral measure P, i.e. such that, for any f in C0(G):
y(f) = IG'(s)dP(e) Let us put v = p o 9; then v is a non-degenerate representation of L(G)* such that, for any w in £(G)*, we have: v(w) = frG. (AG (S) *, w) dP (s)
By 3.1.4, there exists a unitary U in ,C(1-() ®,C(G) which is the generator of v, and is such that, for all , 77 in 1-h, w in £(G): (UV I WC17 0
W) = Mw) 177)
So U = cUvcr satisfies the theorem. 4.4.4 Proposition. Let G be an abelian locally compact group, G its dual group, in the sense of 4.3.7. For any m in M1(G), let us define the Fourier transform
ofmby:
- IG(X,s)-dm(s)
m(X)
(x E G)
Then:
(i) For any m in M1(G), we have: AG(s)dm(s)) U* = m
U G
(ii) (Bochner's theorem) Every positive definite function on G is the Fourier transform of a unique positive bounded measure on G. (iii) (Stone's theorem) Every unitary representation p of G is given by a spectral measure Pµ on G, with values in 1-(µ, such that, for all s in G, we have:
µ(s) =
f(xs)-dF(x).
Proof. For any f in L1(G) (which is an ideal of M1(G)), (i) has been proved in 4.3.7 (ii). So, using the non-degeneracy of the representations, (i) is proved
for anyminM1(G). As L' (G^
has a bounded approximate unit, we see, using 4.4.1 (ii), that
every positive definite element in L°°(G) is of the form U(fG AG(s)dm(s))U*, with m in M1(G)+. So (ii) is proved, using (i) and 1.3.11.
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4. Duality Theorems for Kac Algebras and Locally Compact Groups
Let us consider the non-degenerate representation of L1 (G) obtained from p by 1.1.4. By 4.3.7 (ii), L' (G) is isomorphic to £(G)*, and, by 4.4.2 (i) and 4.3.7 (i), there is a spectral measure Pµ on G, with values in ?-(j,, such that, for all f in L1(G), we have:
µM =
jfG
f(x)dP/A(x)
from which we get (iii).
4.5 Characterisation of the Representations and Wendel's Theorem Let K = (M, I', re, co) be a Kac algebra. Let A be a von Neumann algebra. In this paragraph, Q will denote a normal, one-to-one morphism from M to A ® M such that
(a 0 Or - (2 (& r)a Q(1M) = lA 0 1M By 3.2.6, for any non-degenerate representation p of M*, the algebra All and the morphism yµ fulfill these conditions.
4.5.1 Proposition. We have, for all x in M^ + and t in R: (i (9 5P)/3(x) = SP(x)l A A
(i 0
IP)#
t
Proof. Let x in M+. We have: (i 0 5P)/9(x) ®1M = (i 0 i 0 V^)(i 0 I')/3(x) = (i ®i ® w)(/3 ®i)I'(x)
by 3.7.2
by hypothesis
_ /3((i ®5P)I'(x))
= = S (x)(1A 0 1M)
which brings (i). Let t in R. We have:
((i ® at
®t
by 3.7.2 by hypothesis
4.5 Characterisation of the Representations and Wendel's Theorem
t
145
by hypothesis
® i ® ot)(i ® F"')# by 2.7.6 (ii) applied to K
_
by hypothesis
_ (13®i)(iOoT)I' _ (P ® i)(ot ® i)I'
by 2.7.6 (ii) applied to K
From what follows that (i ®Ut )P ® i and 3 ° 0 i coincide on F"' (M*"* ); as it is obvious that they coincide also on C 0 M, thanks to 2.7.6 (iv) applied to K, they will coincide on M 0 M, which completes the proof.
4.5.2 Proposition. Let ?k be a faithful, semi-finite, normal weight on A (to simplify, we shall suppose A C £(H p )). Then: (i) for all x in'L,, y in 91, the operator fl(y)(x 0 1) belongs to 910®o, and there is an isometry U in HO 0 H such that: U(A,(x) 0 AW(y)) = A, ,®c(f3(y)(x ®1))
(ii) U belongs to A 0 M. (iii) for all z in M, we have: 13(z)U = U(1 ®z) Proof. We have: (0 0 A)((X* 0 J)P(Y*Y)(X 0 1)) = O(X*((i 0 ^)O(Y*Y))X) =+G(%*T)W(Y Y)
by 4.5.1 (i)
and so fl(y)(x 0 1) belongs to,®.. Let X1, X2 in 910, y1, Y2 in 91o ; polarizing the preceding equality, we find: (0 ® SP)((x2 0 1)@(y2)/3(yl)(xl (9 1)) =
O(x2x1)SP(y2y1)
which can also be written as follows: (AO®cp(la(yl)(xl 0 1)) I A pOV(9(y2)(x2 ®1))) = (A,,j(xi) 0 Al o(yl) I Ao(x2) 0 A (y2))
which completes the proof of (i). Let be in 2L , 77 in at', x in TO and y in 9l,. We have: 0 7**r"(77)) U (AO (x) 0 Ao (y))
0 7**r'(?7))AOOo(8(y)(x 0 1))
3(Y)(x 0 1)( 0 77) 0 1)(Ao(x) (9 77)
by (i)
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4. Duality Theorems for Kac Algebras and Locally Compact Groups
For all vector ( in H, by density, we shall have:
(ir'()
0 A,(y)) =
0 1)(C ®rl)
And, by having 7r' (e) converging to 1, we get: (10 7"r"(?7))U(( 0
0(y)(C 0 t7)
Let z in A'. We have:
(1 0 '(rl))U(z( 0 A,(y)) = /3(y)(z( ®rl)
= (z 01Ay)(( 0 i)
by hypothesis
= (z ®1)(1 (D ir'(r))U(( (9 AO (y))
by the same calculation as above So, by density and linearity, we get: 7'(77))U(z 0 1)
By having lr'(q) converging to 1, we get:
U(z ®1) = (z ®1)U and so U belongs to A 0 £(H). Let x in y in k n gi -, i in 2t0, such that 11rill = 1. We have: U(do(x) 0 J1\(wo_114n)JdO(y)) = U(Aw(x) 0 AW((z 0 wn)r(y)) = A+GOW(Q((z 0 wn)r(y))(x
1))
= nO(&O(((z 0 i Own0 0 i)I'(y))(x 0 1)) A00w(((z 0 i 0
-
AO(O
W((('
by 4.1.3 (iv) by (1)
by hypothesis
0 17)(i & P^)#(Y))(x 0
P (Y)) (X Jwo_,1,n)J)AOOW(,Q(y)(x (2) 1))
by 4.1.3 (v)
J/\(wo-1/4,,)J)U(Ao(x) 0 da(y))
by (i)
By linearity and density, we get: U(1 0 JA(wo_1/4,7)J) _ (1 ® J\(wa-1/4,7)J)U
4.5 Characterisation of the Representations and Wendel's Theorem
147
By density of A-1 42to in H, and by polarization, we get, for all w in M* :
U(1® JA(w)J) = (10 JA(w)J)U and, by density of A(M*) in M, we can conclude that U belongs to C(HO)®M, which completes the proof of (ii). Let x in Tto y in 910. We have: /3(z)U(A,p(x) 0 Ao(y)) = /3(z)A,p®,(/3(y)(x 0 1)) = A,p®,(,Q(zy)(x 01))
by (i) by (1)
= U(A,,(x) 0 A-(zy)) = U(1® z)(A,p(x) ® A,(y)) By density, continuity and linearity we can complete the proof of (iii).
4.5.3 Lemma. With the notations of 4.5.2, let X be in 910®0, Y in 910®o, x in MO, and y in 910. Then: and we have: (i) The operator (/3 0 i)(X)(x 010 1) belongs to
AO®W®o((fl 0 i)(X)(x 0101)) = (U 01)(Ao(x) 0 Ac®lp(X )) (ii) The operator (,l3 0 i)I'(y)(Y 0 1) belongs to 91V,®oO0 and we have: A
(1®cr)(10W*)(U®1)(1®c,)(A ®0(Y)®A0(y))
Proof. We have:
(000,O(X*(i 0 Po 0)(# 0 000 i)(X*X)X) =+V(=`((w0 V)(X* XXV 0 V)(X*X) < +00
by 4.5.1 (i)
which leads to the first part of (i). By polarization, for any X1, X2 in '10(g0 and X1, x2 in 91,x we shall find:
i)(Xi)(x 0101)) I Ap® _ (A,(x1) 0
((fl 0 i)(X2)(x 0101))) A (x2) 0 A (X2))
From what we can deduce the existence of an isometry of H,1,0 H ®H which
sends A,,(x) 0 A ®,-(X) on 110®,®0((/3 0 i)(X)(x 01 01)). It comes from 4.5.2 (i) that this isometry coincides with U ®1 on the elements of the form
148
4. Duality Theorems for Kac Algebras and Locally Compact Groups
AV) (x) ® AV (y1) 0 A, (y2) (where Y1, y2 are in 91.); therefore, by linearity and
continuity it is equal to U ®1, which completes the proof of (i). We have:
)((Y*
j) F (Y*Y)(Y 0 1))
0 0 i 0 0"')(# 0 0 F (Y*Y)Y) 0 )(Y*#(i 0 V) F (Y*Y)Y) W 0 )(Y*Y) O(Y*Y) A,
A
by 3.7.2
W
< + 00 which gives the first part of the proof of (ii). Using the same technique, through polarization, we get an isometry of 0 i)1,(y)(Y ®1)). Let H, ®H ® H sending A,,®,(Y) 0 A,(y) on x in N, y1, y2 in 9l ,. We have: A,,®
Sp((l3 0 i)(I'(yl))(x 0 Y2 ®1)) ((i 0 P)(/Q(y1))(x 0 Y2 ®1))
by hypothesis
= 0 r')(,a(yl)(x 0 1))(x 0 Y2 ®1)) (10 i)(1 0 W *)A,®c(f3(yl)(x 0 1)) 0 AO(y2)) = by 2.4.9 applied to cp and 3.7.3
(1 ® o)(1 ® W*)(U 01)(A,,(x) 0 A,(yl) ® A,(y2)) by (i)
This isometry does therefore coincide with (10 a)(1 0 W *) (U 0 1)(10 a) on those vectors of the form A,, (x) 0 A,, (y2) 0 A (yl ), which, by linearity, density and continuity, completes the proof of (ii).
4.5.4 Proposition. With the notations of .4.5.2, the isometry U satisfies:
(i 0 -r)(U) = (10 a)(U 0 1)(10 a)(U 0 1) . Proof. By 2.6.4, we have:
(i 0 4;-r)(U) = (10 a)(1 0 W)(1 0 a)(U 0 1)(10 a)(1 0 W*)(i 0 cr) Let x in 910 yl, y2 in
We have:
(10 W)(1 0 a)(U 0 1)(10 a)(1 0 W*)(Ao(x) (9 A - (yi) 0 A - (Y2)) (10 W)(1 0 cr)(U 0 1)(Ap(x) 0 AWOW(F(Y1)(Y2 0 1)))
by 2.4.2 (i) applied to K and 3.7.3 (10 W)(1 0 a)Atko -W(gW - (((# 0 i)(^ F(Y1)(Y2 0 1))(x 0 10 1)) by 4.5.3 (i)
4.5 Characterisation of the Representations and Wendel's Theorem
149
= (10 W)(1 0 a)AOOVOV((# 0 i)F(Yl)(fl(Y2)(x (9 1) (9 1)) = (U (9 1)(1 (9 0 1)) 0 by 4.5.3 (ii) (y1)) by 4.52(i) = (U 0 1)(1 0 a)(U(n+G(x) 0
= (U 0 1)(10 cr)(U 0 1)(10 a)(Ap(x) 0
0 AW(Y2))
therefore, we have:
(10 W)(1 0 Or)(U 0 1)(10 U)(1 0 W*) = (U 0 1)(1 (D q)(U 0 1)(10 a) which completes the proof.
4.5.5 Proposition. With the notations of 4.5.2, the isometry U is unitary. Then, by 2.6.5 and 4.5.4, it is the generator of a non-degenerate representation of M* which shall be denoted by it. Moreover, we have:
A=Aµ and
/9==yµ.
Proof. Let P the projection UU*. We have:
(i 0 F)(P) = (U 0 1)(i 0 c)(P 0 1)(U* 0 1)
by 4.5.4
P=Q®1
Let z in M. We have: ,Q(z)(Q ® 1) = fi(z)UU*
= U(1 0 z)U* = UU*U(1 0 z)U* = (Q ®1)U(1®z)U* = (Q 0 1),8(z)(Q ® 1)
by 4.5.2 (iii)
by the same calculation
Passing to the adjoint operators, we get: f3(z)(Q ®1) = (Q ® 1),@(z)
Now let x in
91,p y
91,.
We have:
U(Q ® 1)(A ,p(x) 0 Ac(y))
= U(A,(Qx) 0 Ac(y)) = A,,®^ (,Q(y)(Qx ®1)) = A o(&O((Q 0 1),8(y)(x 0 1)) = (Q 0 1)A,,®O(/3(y)(x ®1))
by the result above
_ (Q 01)U(A,,(x) ® A,(y))
by 4.5.2 (i)
by 4.5.2 (i)
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4. Duality Theorems for Kac Algebras and Locally Compact Groups
By linearity and density, we get:
UP= U(Q®1)=(Q®1)U=PU=UU*U=U and:
UU* = P = U* UP = 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 µ. Then, we have, for z in M: P(Z)
U(1 0 Z)U* 714 (Z)
by 4.5.2 (iii) by 3.2.2 (i)
which completes the proof. 4.5.6 Theorem. Let K = (M, F, ,c, gyp) be a Kac algebra, K = (M, I', ik, Sp) the dual Kac algebra, A a von Neumann algebra, /3 an injective normal morphism
from M to A 0 M. Then, the following assertions are equivalent: (i) We have: (l3 ® OF= (i ®I')/3 and /3(1) = 1
(ii) There exists a non-degenerate representation µ from M* to A such that:
/3oA=cA x 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, l3 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 ® I'e)/3(AG(S)) = /S(AG(S)) 0 AG(S) /3(1) = 1
(ii) There exists a unitary representation µG of G, such that A is the weak closure of PG(L1(G)), and, for all s in G: 9(ac(s)) = Nc(s) 0,\G(S)
Proof. It is just an application of 4.5.6 to K. (G). 4.5.8 Corollary. Let K be a Kac algebra. Then an element u of £(H) belongs to the intrinsic group G(K) if and only if it is the canonical implementation of an automorphism /3 of M such that (/30 i)1' = F/3 ([18]).
4.5 Characterisation of the Representations and Wendel's Theorem
151
Proof. Let /3 an automorphism of M such that (f3® i)1' = r13. 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 /3 = %yu. Moreover, by 4.5.2, we shall have: uAO(y) = Ac (/3(y)) Therefore u is the canonical implementation of P. 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:
(/oi)f'=t$u Using the first part of this proof, we see that the canonical implementation of /3u is an element v of G(K), which satisfies flu = Nv. Then uv* belongs to M' 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 Jal = 1. As F(u) = u ®u, and
r(v)=v0v,weget a = a2,soa=1 andu=v.
4.5.9 Wendel's Theorem ([199] ). Let G be a locally compact group; every automorphism of L°O(G) which commutes with the right translations is a left translation.
Proof. Let /3 be an automorphism of L' (G) commuting with right translations; we have Tap = (/3 0 i)ra, from which, applying 4.5.8 to K3(G), we get that there is u in G(K8 (G)) such that /3(f) = u f u* for all f in L°O(G). So, by Eymard's theorem (4.3.2), there exists s in G such that /9(f) = )tG(s) f AG(s)*, and so Q is then the left translation by s-1. 4.5.10 Theorem ([163]). Let G be a locally compact group.
(i) Let (M,1', ic) be a sub co-involutive Hopf-von Neumann algebra of Ha(G), such that there exists a Haar weight yo on (M, F, rc). Then, there exists
a normal subgroup H of G such that M is the subalgebra of the functions in LOO(G) invariant by H. Then (M, 1', re) is isomorphic to Ha(G/H).
(ii) Let K be a sub Kac algebra of Ka(G); then there exists a normal compact subgroup K of G such that K is isomorphic to Ka(G/K). (iii) Let K8(G)p be a reduced Kac algebra of K8(G); then, there exists
a normal compact subgroup K of G such that K8(G)p is isomorphic to K8(G/K). Proof. (i) Let K = (M,1'', x, So), K ic, Sp) the dual Kac algebra. By 4.2.5 (ii), K is isomorphic to Ka (G(K)). As (M, r, ic) is a sub co-involutive
Hopf-von Neumann algebra of (L°°(G), ra,'ca), it is easy to see that M is globally invariant under the automorphisms implemented by AG(s), for all s in G. Then x -- AG(s)xAG(s)* (x E M) is an automorphism /3s of M satisfying:
152
4. Duality Theorems for Kac Algebras and Locally Compact Groups
Fos = (Ps ®i)f We have then a continuous morphism from G to the group of automorphisms
f of M satisfying Ffl = (f3 ® i )f , which, by 4.5.9, is isomorphic to the intrinsic group of K. Let H be the kernel of this morphism; we then get a continuous one-to-one morphism u : G/H -+ G(K). By definition of H, M is included in the subalgebra If E L°O(G); AdAG(s) f = f, Vs E G} which is isomorphic to L°°(G/H), and this inclusion L°O(G(K)) --> L°°(G/H) is given by f -- f o u (f E L°°(G(K))). Then, taking f = XG(j)-u(G/H)l we see that G(K) - u(G/H) is of measure 0. Let now s be in G(K) - u(G/H); as the set su(G/H) is included in the set G(K) - u(G/H), it is of measure 0, and so is u(G/H), which is impossible. So, u is surjective, and G(K) is isomorphic to G/H, and (i) comes then from 4.3.5. If K is a sub-Kac algebra of K, ,,(G), its dual K is a reduced Kac algebra of KS(G), by 3.7.9 (ii). Let r be the reduction ,C(G) -+ M, and let K be the kernel of the morphism r o AG from G to the unitaries of M. Let us consider the set: {x E ,C(G); F, (x) = x ®x} which is compact and equal to {0}UAG(C), by 3.6.10 and 4.3.2. Then )G(K) is the intersection of this subset with the kernel of r, and so is compact. So, by 4.3.2, K is a compact normal subgroup of G. Using 4.3.4, we see that the application r o AG from G to G(K) is surjective. As r is open, we see that
G/K is then isomorphic and homeomorphic to G(K). So (ii) and (iii) are proved using (i) and 3.7.9 (ii).
4.6 Heisenberg's Pairing Operator 4.6.1 Lemma. Let K = (M, f, re, cp) be a Kac algebra. Let y (resp. v) be a nondegenerate representation of M. (resp. M*) and U/A (resp. U,,) its generator (we thus have Ut, E Aµ ® M and U E A,, 0 M where A/A (resp. A,,) is the von Neumann algebra generated by y (resp. v)); with the notations of 3.2.2, applied to K and K. we define yµ for all w in M*, and 'y, for all w in M* by: 7Y (A(w)) = S(A x M)(W ) = UM(1 0 a(w))UM 7v (A(w)) = s(a x P)(w) = vv (10 A(w))uv*
Then, we have: (i 07`*A)(UV)(i 0 4;)(UV* 0 1) = (10 U14k; 0 0(i 0 -tV)(U,*,)
-
4.6 Heisenberg's Pairing Operator
153
Proof. Let x in M, y in M; as, by 3.2.2 (i) and by definition, we have: yµ(x) = Uµ(1 0 x)Uµ and: UV(1 0 y)Uv
we can deduce: 7",M)WO(i 0 4;)(UV* & 1)
(10 U14k; 0 0(10 UV)(1 0 UID(i 0 4;)(UV* 0 1)
_ (10 vN)(c 0 :)((1 0 uV)(c 0 0 (10 uN)(1 0 vVN _ (10 Up)0r 0 e)(i 0 7v)WN) which completes the proof.
4.6.2 Corollary and Definition. With the above hypothesis, there exists a unitary element of Av ®Aµ, denoted V, (z
such that:
µ) (UM) (Z 0 40 (UV* 01) = Vµ)
It will be said that Vµ,,, is the Heisenberg's pairing operator of the represen-
tations i and v. ®c)(U,* ®1) belongs Proof. By the definition of yµ, the unitary (i to A 0 Aµ ® M; it results from 4.6.1 that it belongs as well to A 0 Aµ 0 M.
By 4.1.5, we then get that it belongs to A 0 Aµ 0 C, which completes the proof.
4.6.3 Proposition. With the above notations, let /11 and 92 be two nondegenerate representations of M* and v a non-degenerate representation of M*. We have: Vµ1 X 112Y = (i ®c)(Vµ2,v ®1)(Vµ ,v ®1)
Proof. By definition, we have: Vµ Xµ2,v ®1 = (z ® 7µ1 Xµ2)(U,,)(i ®(z ®s)(c 0 i))(UY 0 10 1) = (Z ®s ®z)((i ®i ®yµl)(i ®y/A2)\U1/)) by 3.2.5 (i®z®c)(i®s®i)(Uv ®1®1)
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4. Duality Theorems for Kac Algebras and Locally Compact Groups
Now, we have also:
(ioio51)(i O 7u2)(UV) _ (i o2
S)(Uv
by 4.6.2
_ (Vµ2, 0 10 1)(s 0'0')(1 & ('0 7µl)(UV)) 0 V141 IV 0 1) = (VP2 0 10 1k; (1 = (VP2
(Z 0 s)(vv 0 1))) 0 10 1k; 0'0')(10 VIA1 IV 0 1)
by 4.6.2
(cOiOi)(iOiOc)(10U1,01) From what we can deduce that the operator Vµ1 X/12,1 01 is equal to:
('0 4; 0')((V1J2,V 0 1k; 0'0')(10 VA1)V 0 1k; 0'00 (i 0 i 0 0(10 UV 0 1))(i 0 i 0 4;)(i 0 4; 0 i)(Uv* 0 10 1)
which may be written: ((i 0 S)(VP2,V 0 1) 0 1)(vµi,V 0 10 1)
which completes the proof.
4.6.4 Proposition. With the notations of 4.6.2, let V,,,µ be the Heisenberg's pairing operator of the representations v and p (the background Kac algebra being then K). We have: V1,,µ = 0Vµ,1, .
Proof. By definition, we have: (i 0 70(um)(i 0 (;)(U.U* 0 1)
(U*)) o 0 (up _ (c 0 0((1 0 uN)(c 0 i)(i 0 7o)(Uu ))' i)(Vp;v
by 4.6.2
which completes the proof. 4.6.5 Proposition. Let 111, µ2 be non-degenerate representations of M* and v a non-degenerate representation of M*. (i) Let tin Hom(µ1, 92); we have: (10 t)Vµ1,v = Vµ2i1,(10 t)
4.6 Heisenberg's Pairing Operator
155
(ii) Let 4$ be a normal morphism from Aµ1 to AJA2 such that 4$(1) = 1 and 4$ o
/11 = 92; we have: Vµ2,1,
(iii) In particular, with the notations of 1.6.1 (ii), we have: (i ® sµ)(V,.,v) = Vµ11, .
Proof. The assertion (i) results immediately of 3.2.3 (i), (ii) results of 3.2.3 (ii) and (ii) implies (iii).
4.6.6 Corollaries. (i) Let u, u1, u2 be elements of G(K), i.e. one-dimensional representations of M*. Let v a representation of M*, by definition, the unitary Vu,,, belongs to A and moreover, by 2.6.6 (ii) and 4.6.3, we have: Vul u2,v = Vul ,zVu2,v
Therefore the mapping u Vu,,, is a unitary representation of the intrinsic group G(K) in A. We have, in particular, V1,,, = 1. Let us remark that: Vv,u = vu ,v =
vu*,,,
(ii) Let u in G(K) and v in G(K). The unitary Vu,v is then a complex number of modulus one which shall be denoted X(u, v). By 3.2.2 (i), we get: X(u, v) = 7u(V)v* = uvu*v*
Therefore, we have: UV = X(u, v)vu
and by (i), x is a bicharacter on the product G(K) x G(Jf() which will be called the Heisenberg's bicharacter associated to K. (iii) Let A be the Fourier representation of M*, v a representation of M*. The unitary VA,v thus belongs to A 0 M; moreover, by definition, we have: VA,v ®1 = (i 0 7A)(UV)(i 0 (;) (UV ®1)
0 I')(U1)(i 0 c)(U,*, ®1)
=U1,®1
by 3.2.2 (iv) by 1.5.1 (i)
Then, VA,, , is equal to the generator U of v. In particular, for v in G(K), VA,v is equal to v. We see also that VA is nothing but the fundamental operator W. Finally, by the results above, we have:
(i0sir)(V,v)= U1,
.
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4. Duality Theorems for Kac Algebras and Locally Compact Groups
4.6.7 Examples. Let K = lira (G), K = K8 (G). To every representation v of ,C(G)*, we can associate, by 4.4.2 (i), a spectral measure P on G with values in fv such that for all w in ,C(G)*, we have: v(w) = JG(A(s)*,w)dPv(s)
It is then easy to check that, for all unitary representations y of G: y(s)dPP(s)
V1,v = G
and in particular
W=
IGA(s)dPA(s)
Now let y be a continuous character on G, i.e., by 3.6.12, y in G(Ka(G)); for s in G, f in L1(G), we have: (AG(S)-YAG(S)*,f) ='T(S_1)(-f1f)
Therefore:
x(AG(s), 7) = 7(s-1) and:
X(7, G(s)) = 7(s) when G is abelian, for s in G and y in G, it can be expressed as follows: AG(S)'YAG(3)* = (7,s)7
and therefore: AG(s)7 = (7,S)7AG(S)
which is the classical Heisenberg's commutation relation.
4.6.8 Proposition. Let v be a non-degenerate representation of M*. For all 9 in B(K), let us put (where ir denotes the universal representation of M*): v(9) _ (i ®9)(V7.,v)
Then v is the unique non-degenerate representation of B(K) on fl,, such that: V -(SA)* = V
If, using 3.3.4, one identifies M* with A(K) = (sa)*(M*), then v extends v. The representation v will be called the extention of v to B(K).
4.6 Heisenberg's Pairing Operator
157
Proof. Let us recall that (W* (K), cs,i x,r, s;r) is a co-involutive Hopf-von Neumann algebra, the predual of which is B(K) (3.1.5 (ii)). We have: by 4.6.5 (iii) = (Vir,v 01)(1 0 o)(V,r,v 01)(1 0 a) by 4.6.3
(i ®ss7cx7r)(V7r,v) = (i ®S)(V7rx7r,v)
And using 1.5.1 (i), we see that v is a multiplicative mapping from B(K) to A . Let W^ in M*. We have: v(sa)*(w) _ (i 0 (SA)*(.Z'))(Vr,v) )(i 0 sa)(Vir,u) by 4.6.5 (iii) and 4.6.6 (iii) _ (i ® w)(UU) by the definition of Uv = v(w)
By 3.4.4, we know that A(K) is a bilateral ideal of B(K). The restriction to A(K) of v being a non-degenerate representation, it follows easily that v is non-degenerate and involutive. As A(K) is an ideal of B(K), we easily get the unicity of v: let v' be another extension of v, for all 9 in B(K) and W^ in A(K), we have:
v)v(w) '(9 = v'(9 * (sa)*(w = v(9 * (sA)*(Z)) = v(9)v(w)
by hypothesis
which completes the proof, because v is non-degenerate.
4.6.9 Remarks. (i) The extension of the Fourier representation A to B(K) is the Fourier-Stieltjes representation x7r* defined in 1.6.9. (ii) In the group case, the algebra B(Ka(G)) identifies itself with the classical Fourier-Stieltjes algebra B(G) (1.6.3 (iii)). Moreover, with the notations of 4.4.2, it is clear that, for every f in B(G), and any non-degenerate representation v of A(G) (or of C0(G)), we get:
v(.f) =
IG
f(s)dPP(s)
(iii) Still in the group case, if K = K3(G), Af is identified with L1 (G) and B(KS(G)) with the involutive Banach algebra M1(G). To each nondegenerate representation v of L1(G), one associates a representation v of M1(G). By the unicity of 4.6.8, it is clear that it is just the usual procedure for locally compact groups, as described in 1.1.4.
4.6.10 Proposition. Let 7r (resp. *r) be the universal representation of M* (resp. M*), * (resp. 7r) be its extension to B(K) (resp. B(K)) as defined in 4.6.8, and 7r* (resp. *r*) the transposed mapping of the latter. We have:
4. Duality Theorems for Kac Algebras and Locally Compact Groups
158
(i)sSr*=5. and * are faithful
(ii) the representations (iii) =K7r*.
Proof. Let 9 in B(K). By definition of fr, for every 0 in B(K), we have: 9)
_ (e
0 Sam)) = (e, ^(9° 0 Sfr)*)
= (9, (i ® 90 o Sfr)(v* 0}) =
by 4.6.8 by 1.2.5
(e, (i 0 0)(v7*r,7-r))
by 4.6.4
_(e0 e1 v7r,fr)
by 4.6.8
(70), 0)
which completes the proof of (i). By transposing the equality in 4.6.8, we find, using (i): 7r* = Sa7r)* = SAS;r7r
as, thanks to 1.6.9 applied to K, 7r is faithful, it yields (ii). By using the above equality, we get: by 3.3.3 by (i)
sAsfr7r
= sA7r
which completes the proof.
4.
7 A Tatsuuma Type Theorem for Kac Algebras
4.7.1 Lemma. Let K be a Kac algebra. Let x in W*(K); x is a character on B(K) = (W*(K))* (i.e. s,rx,r(x) = x 0 x) if and only if, for every nondegenerate representations it and v of M*, we have:
Sp(x) 0 sv(x)
.
Proof. Let w in M*; we have: (/1 x v)(w)
= (sµ ® = (sp (9
x 7r)(w)
by 1.6.1 (ii) by 1.6.4 (iii) by 1.6.1 (ii)
4.7 A Tatsuuma Type Theorem for Kac Algebras
159
So, by continuity, for all x in W * (K), we have:
sµxv(x) = (sµ 0 Sv)sirx7r(x)
Let us assume that x defines a character on B(K). We have then: Sv)(x 0 x) = sµ(x) 0 sv(x) spxv(x) = (Sµ 0 Conversely, let us assume that this last equality is satisfied for every pair
(µ, v) of non-degenerate representations. In the case of (r-, ir), we get:
s,rx,r(x) = s,r(x) 0 s,.(x) = x 0 x which completes the proof.
4.7.2 Theorem. Let K be a Kac algebra. (i) An element x of W * (K) belongs to G(W * (K)) if and only if s. \(x) 54 0 and, for any non-degenerate representations y and v of M*, we have: sµxv(x) = sµ(x) 0 sv(x)
(ii) The restriction of s.\ to G(W*(K)) is an isomorphism and a homeomorphism from this group to G(K). Proof. Let x be a character on B(K) such that sA(x) 0 0. We have: i'(sA(x)) = (sA (9 sA)(cs,rxn(x)) = (sA 0 sA)(x 0 x) = sA(x) 0 sA(x)
by 3.3.3
Thus, by 3.6.10, sA(x) belongs to G(K), and is therefore unitary. Also, we find, with the definitions of 3.2.2:
y,rsax) = cs.\x,r(x) = c(sa 0 i)s,rx,r(x)
by 3.2.2 (iii) by 1.6.4 (i)
(i ®sa)(x ®x) = x ® sA(x)
As %y,. (1) = 101, it is easily deduced from this that x is unitary and therefore belongs to G(W * (K) ). As we have s,\(1) = 1, the converse is immediate and (i) follows.
Let xi and x2 be in G(W*(K)) such that sA(xl) = sA(x2). It results from the above computation that: xl 0 SA(xi) _ 77r(Sa(xi)) = y,r(s,\ (x2)) = X2 0 S.\(x2)
by hypothesis
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4. Duality Theorems for Kac Algebras and Locally Compact Groups
which implies xl = x2 and the injectivity of the restriction of sa to G(W*(K)). Let y in G(K). Then, with the definitions of 4.6.2, the Heisenberg's pairing operator Vy,, belongs to W * (lid) and satisfies the following: by 4.6.5 (iii) by 4.6.3
s7rx7r(Vy,7r) = Vy,rxir
=Vy,7r®Vy,ir Therefore, Vy,,1. belongs to G(W * (K)). Finally, we have: sA(Vy,,r) = Vy,\
=y
by 4.6.5 (iii) by 4.6.6 (iii) and 2.6.6 (i)
which proves that the restriction of sA to G(W * (K)) is a bijective application from G(W*(K)) to G(K); it is clearly a group homomorphism. The continuity of sA is trivial. Conversely, as, by 4.6.2, we have: Vy,ir ®1 = (i ®'Yy)(U7r)U7r
we get the continuity of the application y -+ Vy,,r.
4.7.3 Corollary (Ernest's Theorem [44]). Let G be a locally compact group. The mapping 7rG : G --+ W*(G) implements an isomorphism and a homeomorphism from G onto G(W*(G)). Proof. By 4.7.2, sA implements an isomorphism and a homeomorphism from G(W*(G)) onto G(K,(G)) and, by Eymard's theorem (4.3.2), AG implements an isomorphism and a homeomorphism from G onto G(K3(G)). The results come from the formula AG = s A 7rG .
4.7.4 Corollary (Tatsunma's Theorem [168]). Let G be a locally compact group. Let x in W * (G). The two following assertions are equivalent: (i) there exists some s in G such that x = 7rG(s) (ii) for every continuous unitary representations /, v of G, we have: (a) sµ(x) ®sv(x) (b) sAG(x) 0 0.
Proof. By 4.7.2 and 4.7.1, the second assertion is equivalent to x belonging to G(W * (G)), the corollary then results from 4.7.3.
Chapter 5 The Category of Kac Algebras
In what follows, K1 = (M1,r1,k1,cp1) and K2 = (M2, 1'2, K2, p2) are two Kac algebras, K1 = (M1, I'1, ki, SPi) and K2 = (312, E2, K2, p2) their duals. In this chapter, we put on the class of Kac algebras a structure of category, by defining convenient morphisms (called K-morphims), as it was done by the
authors in [40]. The definition is not straightforward and involves the von Neumann algebras generated by the preduals of dual Kac algebras (5.1.1). Thanks to the Heisenberg's pairing operator, it is then a routine to define a dual K-morphism, and we have now a duality functor in the category of Kac algebras (5.1.3 ). Moreover, the full subcategory of symmetric Kac algebras, will be equivalent, thanks to Takesaki's theorem as seen in Chap. 4, to the category of locally compact groups (5.1.4 ). Let now u be an 1H[-morphism from (M1, fl, i1) to (M2, F2, K2); we can define a K-morphism u from K1 to K2, called the extension of u (5.2.3 ), and give a characterization of those K-morphisms which are extensions (5.2.4 ). If the dual K-morphism u^ is an extension too, the 1HI-morphism u will be called strict (5.3.1), and we have then got another class of arrows, stable by duality (5.3.2 ), which is the category introduced by the authors in [36]. Strict IHI-morphisms are characterized (5.3.1); in both categories, isomorphisms are the same (5.6.8 ). The end of the chapter is devoted to other results about isomorphisms, due to de Canniere and the authors ([22] ). Let us suppose now that the preduals (M1)* and (M2)* are isomorphic Banach algebras (nothing is assumed about
the involutions); then the Kac algebra K1 is isomorphic either to K2 or to K2 (5.5.5 ), and the dual Kac algebra K1 is isomorphic either to K2 or to (K2)s (5.5.9). In the case of two locally compact groups G1 and G2, we recover Wendel's theorem ([198]) about isomorphisms between the Banach algebras L1(G1) and L1(G2) (which implies isomorphisms between G1 and G2) (5.1.11), and Walter's theorem ([194]) about isomorphisms between the Fourier algebras A(G1) and A(G2) (which implies isomorphisms between G1 and either G2 or G2' ) (5.5.12 ). Moreover, from that result, we can deduce the unicity of the co-involution of a Kac algebra (5.5.7).
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5. The Category of Kac Algebras
As a corollary, similar results occur for isomorphisms of the FourierStieltjes algebras B(Ki) and B(K2) (5.6.6), and, in the case of two locally compact groups, we recover Johnson's theorem ([65]) about isomorphisms between the Banach algebras M1(Gi) and M1(G2) (5.6.9 ), and Walter's theorem ([194]) about isomorphisms between the Fourier-Stieltjes algebras B(G1) and B(G2). Thus, these four different results, which used different types of proof, are there shown, as M. Walter guessed (see the introduction of [194] ), to be actually the same property.
5.1 Kac Algebra Morphisms 5.1.1 Definitions. We shall call K-morphism from K1 to K2 an H-morphism a from W * (Ki) to W * (K2). By transposition, we get an involutive Banach algebras a* from B(K2) to B(Kl ). The class of the Kac algebras, equipped with these morphisms, thereby becomes a category. It shall be denoted by 1C. A
5.1.2 Theorem. With the above notations and those of 4.6.8, there is a unique normal morphism & from W * (K2) to W * (]iii) such that: N
A -
a7r2 = 7rl a*
or also, and equivalently, such that: (a ® i)(Virl 7rl)
a)(V1r2 i*2 ) A
A
Moreover, a is a K-morphism from K2 to 1k1. By iterating the process, we find:
a=a AA
We shall say that a is the dual morphism of a. Proof. Let W2 in M2*. By 4.6.8, we have: 7rla*(SA2)*(w2) = (i ®a*(sA2)*(w2))(V,rl,*rl )
= (i 0 w2)(i 0 sA2)(i 0 a)(V,rl,frl )
is a unitary As a(1) = 1 and SA2 (1) = 1, the operator (iOsA )(i of W * (Kl) 0 M2, and by 1.5.3, it implies that Sri a* (sae )* is non-degenerate. Thus, if we put:
a= S*1a*(, 2)* A
1\
5.1 Kac Algebra Morphisms
163
we have: &(1) = 1
In another way, a is defined as to make the following diagram commute: ir1
«*
B(K1)
B(K2)
M2
W*(Ki)
i2
We shall have: a7r2(sA2)*(w2) = a7r2(w2)
_ *1a*(SA2)*(w2)
by 4.6.8
by definition of a
Therefore ar2 and 7rl a* coincide over A(K2 ); as A(K2) is a an ideal of B(K2) and as the restriction of 7r1 a* to A(K2) is non-degenerate, we easily get: a7r2 = 7rl a*
By definition of *7'l and 1r2, this equality can be equivalently written as follows, for all B in B (K2) : a(i ® 9)(V,r2,W2)
= (i ®e o a)(V,rl ,*1 )
which is also equivalent to: (a ®i) (V7r2 ,,r2) = (i (9 a) (V7r1,fr1)
or, thanks to 4.6.4, to: (i®&)(V*2,.2) _ (a®i)(*1,,.1)
(*)
Let be a normal morphism from W * (K2) to W * (Kl ), such that,8 72 - = 7rl a* For all w2 in M2*, we have: IQ7r2(w2) _ /31r2 (s
by 4.6.8
by assumption a7r2(sA2 )*(w2)
= a72 (w2 )
therefore /3 = a, which yields the unicity of a.
by definition by 4.6.8
5. The Category of Kac Algebras
164
Let a* be the transposed of a and 01 in B(K1). We have: 7r1(a*(81)) = (i 0 a*el)(V*2,7r2 )
_
by 4.6.8
01)('o a)(Vfr2,7r2 )
= (i ® 01)(a ® i)(Vir1 7r, )
by (*)
= a(i 0 01)(Vfrl ,7r, ) by 4.6.8
= a7ri (e1)
Therefore, we get :
N
N
7r1 a* = a7r1
and, as, by 4.6.10 (ii), 7r2 is faithful, we obtain that, as a7rl, a* is an involutive algebra morphism.
By transposing, recalling that a(1) = 1, we get the fact that a is a morphism of Kac algebras. Finally, it is clear, by (*), that a^^ = a, which completes the proof.
5.1.3 Theorem. The correspondance which associates to any Kac algebra its
dual Kac algebra (as defined in 3.7.4), and to any morphism the dual Kmorphism (as defined in 5.1.2), is a duality functor of 1C into itself. It shall be denoted by D.
Proof. Let K1, K2 and K3 be three Kac algebras, a be a morphism from K1 to K2 and 0 be a morphism from K2 to K3. Let us consider the morphism /9a from K1 to K3. By using 5.1.2 repeatedly, we get: (/3a 0 i)(V*1 i7r1) = (i3 0 i)(i ® a)(V*2,7r2 )
= ('0 a)(/3 0 ')(V*2,7r2 ) (i 0 aN) (V7r3 ,7r3
Therefore, we have (/9a) = a#, which, because of the already known results, completes the proof.
5.1.4 Theorem. Let G1 and G2 be two locally compact groups and m be a continuous morphism from G1 to G2. Then: (i) There exists a unique K-morphism, denoted by Ka (m), from Ka(G2 )
to Ka(G1) (i.e., here, an H-morphism from Ml(G2)* to Ml(G1)*), the transposed of which (it is an involutive Banach algebras morphism) is the mapping from M1(G1) to M1(G2) that sends every measure of M'(Gi) on its image by in. (ii) There exists a unique K-morphism, denoted by K3 (m), from K3 (G1) to K3(G2) (i.e. an H-morphism from W*(G1) to W*(G2)) such that, for all
ginG1
Ks(m)lrl(9) = 1r2(m(9))
5.1 Kac Algebra Morphisms
165
where 7rl and 7r2 stand respectively for the universal representations of G1 and G2. (iii) Let us denote 1Ca (resp. 1C8) the full sub-category of IC made up of the abelian (resp. symmetric) Kac algebras. The mapping which associates to a locally compact group G the Kac algebra Ka(G) (resp. Ks(G)), and to a continuous morphism of groups the morphism Ka (m) (resp. K8 (m)) as above defined, is a duality (resp. an equivalence) functor between the category of locally compact groups equipped with the continuous morphisms and the category 1Ca (resp. JC8); it shall be denoted by Ia (resp. K8). (iv) We have:
K8=DoKa. Proof. The mapping g -+ 7r2(m(g)) is a continuous representation of G1 in W * (G2 ); thus there exists a normal morphism, denoted by K8 (m), from W * (Gi) to W * (G2) such that, for all g in G1: K,(m)7ri(g) = 72(m(g))
In particular, we have: K8(m)(1) = 1
by using 1.6.8, we immediately check that K8 (m) is an H-morphism; the unicity is trivial, which completes the proof of (ii). The transposed of the dual morphism, i.e. (K3(m)"), is an involutive Banach algebra morphism from M1(G1) to M1(G2) such that, for ally in M1(Gi), we have: 7r2((K8(m)^)*(y)) _ Ks(m)(*i(y)) G1
by 5.1.2
72 (m(g))dµ(g)
From where we immediately get that (K8 (m)")* (p) is the image measure m(µ); starting from this equality, by transposing and dualizing, the unicity in (ii) implies the unicity of this morphism Ka(m), which yields (i). Let 9 be a morphism from K8 (G1) to K8 (G2 ). It is clear that /3 will map the intrinsic group of W * (Gi) in the intrinsic group of W * (G2). As, by 4.7.3, these groups are isomorphic both algebraically and topologically to G1 and G2 respectively, there is a continuous morphism m from G1 to G2 such that, for every g in G1, we have:
i1(g) = 1r2(m(g))
which is nothing but to say that:
Q=K8(m)
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5. The Category of Kac Algebras
As each symmetric Kac algebra is of the form K8(G) (4.2.5 (i)), it follows that Ka is an equivalence between the category of locally compact groups and K3. The ends of (iii) and (iv) are straightforward.
5.1.5 Remark. With the above notations, and identifying B(lE(Gi)) to B(Gi ), for i = 1, 2, it is easily checked that for all f in B(G2), we have:
K8(m).(f) = f o m
-
5.2 H-Morphisms of Kac Algebras 5.2.1 Lemma. Let K1 and K2 be two Kac algebras, b be an H-morphism from (M1, .l'1i ril) to (M2, .l'2i K2) and p be a non-degenerate representation of Ml*. Then p o S* is a non-degenerate representation of M2* on ?i, , the generator of which is equal to (i ®S)(Uµ ). Proof. Let w2 in M2*. By definition of the generator Uµ, we have: µ o d.(wz) = (* (9 u2b)(Uv) = (' (9 W2)(' 0 b)(Uu ) As we have S(1) = 1, the operator (i ®S)(Uµ) is a unitary of A®®M2; by 1.5.2, it is the generator of the representation p o S*, which is then non-degenerate.
5.2.2 Theorem. Let K1 and K2 be two Kac algebras, S be an H-morphism from (Ml , I'1, ki) to (M2, I'2, X2 )y S* be the involutive Banach algebras morphism from M2* to Ml* obtained by transposing b, W*(S*) be the homomorphism
from W*(K2) to W*(Kl) obtained by applying the functor W* to S* (i.e. such that W * (S* )7r2 = ir1 S*). Then, we have:
(i) W * (S*) is a K-morphism from K2 to K1;
(ii) the involutive algebra morphism W*(S*)*, from B(K1) to B(K2), obtained by transposing W*(S*), is the unique Banach space morphism which makes the following diagram commute: W* (1* )*
B(K1)
B(K2) K27r2
M17r1*
Mi
M2
where it stands for the universal representation of Mi*, and xi7ri for the Fourier-Stieltjes representation of B(Ki) (i = 1, 2);
5.2 H-Morphisms of Kac Algebras
167
(iii) The mapping W*() ^ is the unique morphism from K1 to K2 which makes the following diagram commute: W*(5 )_
W*(1[(2)
W*(1I(1)
812
8A1 tS
M2
M1
Proof. By definition, W * (S*) is a normal morphism from W * (K2) to W * (Kl )
such that: W*(S*)7r2 = 7r16*
As, by 5.2.1, 7rl S* is non-degenerate, we get:
W*(S*)(1) = 1
and by transposing the above equality, we find: 7r2*W*(S*)* = 67r1*
and, by hypothesis: K27r2*W*(6*)* = 8K17r1*
As, by 1.6.9, K27r2* is faithful, we get the unicity of W * (S*)*; we may show, in
the same way, that W* (S*) * is an involutive Banach algebras morphism; by transposing, we get that W* (S*) is a Kac algebra morphism, which completes the proof of (1) and (ii). Let wl in Ml *. We have: by 4.6.8
sA_2W*(S*)^7rl(wl) =
s - 7r2 W*(S*)*(sa1)*(w1)
by 5.1.2
= rc27r2*W*(S*)*(SA1)*(wl)
by 4.6.10 (iii) by (ii)
= 6K17r1* (Sa1)*(LZ'l)
=
by 4.6.9 (i) 6SA1 1(W1)
Therefore, we have: S_
1\ 2
W*(S*)- =
Conversely, let a be a K-morphism from K1 to K2 such that:
168
5. The Category of Kac Algebras
by transposing, it comes: a* (s,\ 2 )* _ (SAl )*S* _ (W*(S*))*(sA)*
And thus, a* and (W*(6))* coincide over A(K2 ); for A(K2) being an ideal of B(1K2) and for the restriction of (W* (S* )^) * being non-degenerate, we get
the unicity and therefore (iii). 5.2.3 Definition. With the above notations, we shall say that the K-morphism W*(S*)^ from K1 to K2 is the extension of the IHI-morphism b; it will be denoted by S. The dual K-morphism W * (S*) from K2 to K1 will be called the coextension of S.
The H-morphism b being given, the K-morphisms S and S^ are respectively characterized by the equalities: S-
A2
= Ss
Al
S^72 = lr1 S*
.
5.2.4 Proposition. Let a be a K-morphism from K1 to K2. The following assertions are equivalent: (i) The representation of (M1)* in M2 defined by 3A2arr1 is quasi-equivalent to a sub-representation of the Fourier representation Al. (ii) There exists an H-morphism b from (Mi, 1'1, il) to (M2, T'2, K2) such that a is the extension of S. Then the H-morphism b is unique.
Proof. Let us assume (i). It means that there is a normal morphism b from M1 to M2 such that b(l) = 1 and: sae
a7rl = SAl = Ss-1 *r1
which gives: s- a = SsAl A2
Thanks to 3.3.3 it can easily be checked that 8 is an H-morphism; with 5.2.2 (ii) we then get (ii). Let us assume (ii). By 5.2.2 (iii) and 5.2.3, it implies:
s a= Ss-2 Al therefore: a7r1 = SAl
which is nothing but (i).
5.2 IHI Morphisms of Kac Algebras
169
5.2.5 Proposition. Let a be a K-morphism from K1 to K2. The following assertions are equivalent: (i) The representation of M2* in M1 defined by kiwi*a*(sA2 )* is quasiequivalent to a sub-representation of the Fourier representation A2. (ii) There exists an H-morphism 8 from (M2,1'2,k2) to (M1,r1, Rill) such that a is the co-extension of 6. Then, the H-morphism 6 is unique.
Proof. Let us assume (i). It means that there is a normal morphism 6 from M2 to M1, such that 8(1) = 1, and that we have: 6A2 = i^i7ri*a*(SA2 )*
by 4.6.10 (iii)
=SA
= SAl a7r2(SA2 )*
by 5.1.2
=
by 4.6.8
Thanks to 5.2.4, we then see that 6 is an ]H[-morphism from (M2, r'2,' 2) to (Mi, l'i, k1) such that a = 8, which yields (ii). Let us assume (ii). By 5.2.3 and 5.2.2 (ii), we have: K17rl*a* _ 6k27r2*
and therefore: l 7rl*a*(sA2 )* = 6^27r
= 6A2
by 4.6.9 (i) applied to K2
which completes the proof.
5.2.6 Proposition. Let G1 and G2 be two locally compact groups, u be a continuous morphism from G1 to G2 and Ka(u) and K3(u) be the morphisms defined in 5.1.4 (i) and (ii). Then, we have: (a) The following assertions are equivalent: (i) K, ,,(u) is an extension. (ii) Ks (u) is a co-extension.
(iii) There exists an H-morphism Ha(u) : Ha(G2) -; Ha(Gi) such that, for all f in L°O(G2): IH[a (u) (f) = f o u
(iv) The image of the left Saar measure on G1 by u is absolutely (v)
continuous with respect to the left Saar measure on G2. The morphism u is strict, and has an open range.
170
5. The Category of Kac Algebras
(b) The following assertions are equivalent: (i) K8 (u) is an extension. (ii) Ka(u) is a co-extension.
(iii) There exists an H-morphism H8(u) : H8(G1) -> H8(G2) such that, for all s in G2: Hs(u)(AG1(s)) = AG2 (u(s))
(iv) The representation AG2 o u of G1 is quasi sub-equivalent to the representation AGi (v)
The morphism u is strict, and has a compact kernel.
Proof. It is clear that (a)(i) and (ii) (resp. (b)(i) and (ii)) are equivalent. Let us assume (a)(i). From 5.2.5, it follows that there is an H-morphism S from Lm (G2) to LOO(G1) such that b(f) = f o u for all f in B(G2). This equality can be extended, by norm continuity, to all continuous bounded functions f on G2, and then, by ultraweak continuity, to all f in L°O (G2 ). So, we have (a)(iii). Let us suppose now (a)(iii) and call Wi the Haar weight on L°O (Gi) (for i = 1, 2). It is immediate that W1 o Ha(u) is a semi-finite, normal trace on L°O (G2 ); therefore, there exists a positive element g, affiliated to L°°(G2) such that SP1(f o u) = co (f g) for all f in L°O(G2), which implies (a)(iv). Let us suppose now (a)(iv); it is then clear that the application f -+ f o u defined from L' (G2) to L°O (G1) is a normal morphism; it is easy to check that it is an H-morphism, whose extension is ][ea(u). So we have proved that (a)(i)-a(iv) are equivalent. Let us assume these properties. Coming back to the H-morphism Ha(u), we see that this morphism may be decomposed into a reduction (which, by 4.3.6 (ii), is the restriction L°O (G2) -+ L°O (G'), where G' is an open subgroup of G2), and an H-isomorphism from L' (G') to a sub co-involutive Hopf-von
Neumann algebra of L°O (G1). By 4.5.10 (i), there is a normal subgroup H of G1 such that this sub von Neumann algebra of L°O (G1) is isomorphic to L°O(G1 /H), and, using 4.3.5, we get that G' is isomorphic to G1 /H. It is easy to see that we have just got the canonical decomposition of u, which is then strict and with an open range, which is (a)(v). Let us now assume (a)(v). Let w be in £(G2)*, f, g in K (G1). Then, we have (where, for sl in G1, s1 means its class in Gi /Ker u): fGi
Cjw o u(sl)f(sl)9(31)dsl
_ (w, f aGZ(ucsl>*)f(s1)9(s1)ds1} G1
= (W7
f
d1 l /Ker u
f
AG2 (u(t)*) f (t)g(t)dt 1 Ker u
5.2 H Morphisms of Kac Algebras
(w,J 1 / Keru
A G2 (u(s1)* )dsl
J
1 Ker u
171
f (t)(t)dt
(wJAG2 (s2 )* h(s2 )ds2 2
by changing the variable u(s1) = s2 and defining the function h by:
h(u(si)) =
J
f (t)q(t)dt 1 Ker u
h(s2) = 0 if s2 does not belong to u(G1)
It is then clear that h is continuous on a compact, and is null outside it. So fG2 AG2 (s2) h(s2)ds2 belongs to the definition ideal of the Haar weight on £(G2) (3.6.11). By 3.1.3, the representation w --> 9w o u of ,C(G2)* is quasiequivalent to a sub-representation of the Fourier representation of Ks (G2 ), that is of the Gelfand representation of ,C(G2)*. So we get (a)(i), by 5.2.5. The equivalence of (b)(i), (b)(iii) and (b)(iv) is a corollary of 5.2.4. Let R = support %(u); from 1.2.7 and 4.5.10 (iii), we see that Ks (G1)R is isomorphic to Ks(G1/K), where K is a compact normal subgroup of G1, and, from 4.3.6 (i), we see that 1H[e(u)(,C(G1)) is generated by all AG2 (s), where s runs into a closed subgroup of G2. It is then easy to see that K = Keru, and that G' = Im u; as Hs (G1 /K) is isomorphic to H. (G'), we see, by 4.3.5, that u is strict, which is (v). Let us now assume (b)(v). Let f be in L1(Gl ), g, h continuous functions on G2, with compact supports. Then, we have: AG2(u(si))f(si)dsi) 9(32)h(s2)ds2 f1Gr2 (fCG1
9(u(s11)32)h(32)d32J f(sl)dsl
JG1 (fG2 f(sl)k(si)ds
JG1
wit h: k(si) = fG2 9( u(31 )32)h(s2)d32
It is clear that k is continuous with compact support; so k belongs to the definition ideal of the Haar weight on L°O (G1). So, using 3.1.3, we see that the representation f -+ fG1 AG2 (u(sl)) f (sl )dsl of L1(G1) is quasi-equivalent to a subrepresentation of the Fourier representation of Ka (Gl ), that is the left regular representation AGl. So we get (b)(iv).
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5. The Category of Kac Algebras
5.3 Strict IHI-Morphisms 5.3.1 Theorem. Let K1 and K2 be two Kac algebras, u be an H-morphism from (M1, Ti, rq) to (M2, .l'2, K2) and u the extension of u. The following assertions are equivalent: (i) The morphism u is the co-extension of an H-morphism u from the Hopf-von Neumann algebra (M2, I'2) iK2) to (M1,T1,k1); (ii) There exists a von Neumann algebra morphism v from M2 to M1 such that, for every w in M2*, we have: v(A2(w)) = )11(w o u)
Then, the morphisms v and u are equal. Proof. By definition, the mapping u : W * (1f 1) ---+ W * (1f 2) satifies:
s-A2 ou=uos-Al therefore, we have: (A1 )* 0 u* _ (u)* 0 (s - )* 1\ 2
and, also: K17r1*(u)*(sa2 )* _ lc17r1*(sAl)*u*
_ Al u*
by 4.6.9 (i)
the theorem is then a direct consequence of 5.2.5. 5.3.2 Definition. Every H-morphism verifying the conditions of 5.3.1 will be
said to be strict. Given a pair of Kac algebras, the extension operation is clearly putting strict H-morphisms, and K-morphisms which are both an extension and a co-extension, into a bijective correspondance. Equipped with the class of strict H-morphisms, the Kac algebras form then a category with
a duality, the dual of a strict H-morphism u being then defined as the Hmorphism u which verify, for all w in (M2): u(A2 (w))= Al(w o u)
We have already met some strict 1HI-morphisms, namely the H-isomorphisms
(3.7.6), the reduction of a Kac algebra (3.7.10) and the injection of a Kac sub-algebra into a Kac algebra (3.3.8). 5.3.3 Proposition. Let G1 and G2 be two locally compact groups and m be a continuous morphism from G1 to G2. The following assertions are equivalent:
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173
(i) The morphism Ha (m) is both an extension and a co-extension. (ii) The morphism Ks(m) is both an extension and a co-extension. (iii) The morphism m is strict, has an open range and a compact kernel. If they are satisfied, it is possible to define the following two strict Hmorphisms: Ha(m) : L°O(G2) -+ L°O(G1) such that Ha(m) f = f o m for all f in L°O(G2) such that Hs(m)AG1(s) = \G2 (m(s)) H3(m) : £(G1) -- £(G2) for all s in G1 Proof. It is a straightforward consequence of 5.2.6 (a) and (b) and 5.3.2. 5.3.4 Theorem. Let K1 and K2 be two Kac algebras and u be an H-morphism from (Mi, F1, K1) to (M2, F2, K2). The following two assertions are equivalent: (i) The H-morphism u is a strict H-morphism. (ii) The subalgebra u(M1) is at 2 -invariant and the restriction of V2 to u(M1) is semi-finite.
Proof. Let us assume (ii). Let Ru be the support of u. The morphism u can be decomposed into three components, u = i o a o r, where i the injection of u(Ml) in M2, a is an H-isomorphism of M1Ru onto u(M1) and r is the reduction M1 --i M1Ru . We know that M1Ru and u(Ml) can be equipped with Kac algebra structures and that r, i and a are strict H-morphisms; therefore, by composition, so is u, which brings (i). Conversely, let us assume that u is strict. With the same definitions as above, let us decompose u into the product j o r where j = i o a, i.e. j is the H-morphism from M1Ru to M2 such that j (xRu) = u(x) for all x in M1. The dual strict H-morphism r is injective and allows the identification of it is defined, by 5.3.2, by rA1R = Al r*, where AiR (M1R) and r((M1Ru is the Fourier representation of M1Ru . We have: rA1 R j* = A1r* j* = )11u* = u^A2
I(*fi)
by definition of the dual strict H-morphism U^. As r is an injective homomorphism, we get, for every w2 in M2*:
Let w1 in M. Thanks to Kaplansky's theorem, we have:
IIu*(2'i)II = aua{s.(cai)) as(ws))6 W2 E Mz., Ilax(wz)II c 1}
= suP{I(Ua2(w2),W1)I, W2 E m2.7 I< 1}
(*)
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5. The Category of Kac Algebras
= = 9uP{(aln
^06 W2 e mt2., IS 1}
by (*)
)7rr '1 )), W2 E MYr) I< 1} W2 E M2.. Ilaz(wz)II < 1}
<_
by
:5 11r*(w1)11
The mapping r* being surjective, we can define a continuous linear mapping e from (M1R)* to (M2)* such that er* = u*.
It is a simple exercise to prove that a is an involutive Banach algebras morphism. By transposing, one gets that the normal linear mapping ?* from M2 to (M1R ) satisfies rE = u. By r being invective, we get that E* is mutiplicative and that E*(1) = 1, therefore, ?* is an H-morphism. Let us assume that u* (W^ 1) = 0; by the above calculation, it implies:
(A1j*(w1),*(21)) = 0 for all w2 in M2* such that IIA2(w2)11 > 1, and thus for all w2 in M2*. As j is injective, j* is surjective and then we have, for all w in (M1R )*:
(AlR(w),r*(wl)) = 0
which, by density, implies i(Z1) = 0. It ensures that e is injective and therefore that e* is surjective, we then have: u(M2) = r(e* (M2 )) = r((M1R,, )^)
By 3.7.10, we know that r((M1Ru )^) is a sub-Kac algebra of M1, therefore u(M2) is of 1-invariant and the restriction W1 to u(M2) is semi-finite. The H-morphism u being strict, its dual u is strict as well, and we can apply this proof to u, and, as u = u, (ii) can be deduced, which completes the proof.
5.3.5 Corollary. Let K1 and K2 be two Kac algebras and u be a surjective H-morphism from (Mi, I'1, i1) to (M2, F2, K2); then u is strict.
5.4 Preliminaries Abort Jordan Homomorphisms 5.4.1 Definition. Let M1 and M2 be two von Neumann algebras and £ be a linear bijective isometry from M1 to M2. For any projection P in the centre of M2, we define a linear mapping £p : M1 -> M2 by writing, for all x in Ml:
Cp(x) = t(x)t(1)"P
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175
In particular, we shall have ii (x) = i(x)i(1)*. We define the two following sets:
Ph = {P; P projection in the centre of M2 such that Lp is a homomorphism of algebras from Mi to M2 }.
Pa = {P; P projection in the centre of M2 such that i p is an anti-homomorphism of algebras from Mi to M2}.
Let us remark that for any P in Ph (resp. Pa ), Lp is an involutive homomorphism (resp. anti-homomorphism), for all Jordan homomorphisms are involutive, by definition. 5.4.2 Theorem ([76], [149] ). With the above notations, we have: (i) L(1) is unitary;
(ii) Li is a Jordan homomorphism from Mi to M2;
(iii) There exists a projection R such that R belongs to Ph and 1 - R belongs to Pa.
Proof. By ([76], th. 7), we have (i) and the mapping £ from M1 to M2 defined by t(x) = L(1)*t(x), for all x in M1, is a Jordan isomorphism. Using (i), we get (ii). The last assertion then results from ([149], th. 3.3).
5.4.3 Lemma. With the above definitions, let P in Ph (resp. Pa) and Q a central projection of M2 such that Q < P; then Q belongs to Ph (resp. Pa). Proof. It is a trivial consequence of the following equality:
LQ(x) ='tp(x)Q
for all x in Mi
.
5.4.4 Lemma. With the above definitions, the set Ph (resp. Pa) is a lattice. Proof. It is enough to check that, if P and Q belong to Ph, so does P+Q-PQ. Let x, y be in M1. We have:
VPW + QQ-PQ(x)VP(y) + tQ-Pq(y)) = tP(x)QP(y) + tQ-PQ(x)tQ-PQ(y) = tP(xy) + QQ-PQ(xy) because P belongs to Ph, as well as Q - PQ, thanks to 5.4.3. Therefore: LP+Q-PQ(x)LP+Q-PQ(y) = LP+Q-PQ(xy)
and P + Q - PQ belongs to Ph. The proof for Pa is identical.
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5. The Category of Kac Algebras
5.4.5 Lemma. With the above notations, the set Ph (resp. Pa) has a greatest element.
Proof. Thanks to 5.4.4, the set Ph, which is bounded, does have an upperbound Sh which belongs to the weak closure of Ph. Let us prove that Sh belongs to Ph Let x, y be in M1. We have: £Sh (xy) = £(xy)t(1)*Sh = lira Q(xy).e(1)*P PEPh
= lim £(x)t(1)*Pt(y)i(1)*P PEPh
= lim £(x).e(1)*t(y)t(1)*P PEPh
= t(xX1)*t(y)t(1)*Sh = .2Sh (x)tSh (y)
which completes the proof for Ph. The proof for Pa is identical. 5.4.6 Definition. With the above definitions, let us put: Sh = max Ph and Sa = max Pa .
Let us remark that 5.4.2 (iii) implies Sh + Sa > 1.
5.5 Isometries of the Preduals of Kac Algebras In what follows, we consider two Kac algebras K1 = (Mi, rl , tq, V1) and K2 = (M2, I'2, K2, SP2) and T, a linear multiplicative, bijective isometry from M2* to Ml*. We shall note £ its transposed, which is then linear, bijective, ultraweakly continuous isometry from M1 to M2.
5.5.1 Lemma. (i) The operator £(1) belongs to the intrinsic group of K2, and so:
(e(1)) _ £(1)*
(ii) Let w in M2* . We have: £(1)* .wo = (e(1)* .w)°
(iii) The mapping £p being ultraweakly continuous, we can consider its M1*. We have, then, for all projections P in the transposed (ep)* : M2* centre of M2 and for all w in M2*:
(tp)*(w) = T(t(l)*P - w)
.
5.5 Isometries of the Preduals of Kac Algebras
177
Proof. Let w, wl in M2*. We have:
(r2(e(i)),w w = (1, T(w wl)) = (1, T(w) T(i.i ))
by hypothesis
= (r1(1), T(w) 0 T(w )) (10 1 T(w) 0 T(wl)) _ (1,T(w))(1,T(w')) 1
therefore, by linearity, density and continuity, we have:
T2(t(l)) = 41) 0 Al) since, by 5.4.2 (i), we have £(1) 0 0, it completes the proof of (i), using 2.6.6 (i). Let now x be in M2. We have: (x) i(1)* w°) = (xE(1)*,wO) (K2(x*)K2V(1)),W)(IC2(x*W1)*,W)(rl-2 (x *), i(l) * - W)
(x, (AW - W)O)
which completes the proof of (ii). Let now y be in M1. We have: (y,(p)*(w))
=
(ep(y),w) - (t(y)e(1)*P,w)
(t(y), t(l) * P - w) = (y, T(t(l) * P
which completes the proof.
5.5.2 Proposition. With the notations of 5.4, for all x in Mi, we have: <=J
(ash 0 Qsh)(rl W) = F2 V1 (X))(Sh (9 SO
(ii)
Vsa 0 esa)(riW) = F2 V1 (X))(Sa 0 Sa)
(iii) Ov)
T2 (Sh) T2(Sa)
Sh & Sh Sa 0 Sa
-
by (i)
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5. The Category of Kac Algebras
Proof. Let w, w' in M2*. We have:
WSh 0 tSh)(rl(x)), w ®w') = (-Pl(X)l (tSll)*(W) (& ('Sh)*(W'))
_ (I'1(x), T(2(1)*Sh w) OT(2(1)*Sh w )) by 5.5.1 (iii) = (x, T(Q(1)*Sh w) * T(t(1)*Sh wl)) = (x, T((t(1)*Sh w) * (t(1)*Sh w'))) by hypothesis = (t(x), (t(1)*Sh w) * (t(1)*Sh 'w')) ((1)*Sh ' w) 0 (t(1)*Sh ' w')) = (1'2(t(x)), = (1'2(t(x)), (t(1)*Sh 0 £(1)*Sh) . (w 0 w')) ®t(1)*Sh),w (12(t(x))(t(1)*Sh _ ®w')
_ (r2(t(x)t(1)*)(Sh 0 Sh), w ®w')
by 5.5.1 (i)
_ (f2(ti(X))(Sh 0 Sh),w ®w') by linearity, continuity and density, we get (i). The proof of (ii) is identical. Let Q be the projection in the centre of M2 such that r2(Q) is the central support of Sh 0 Sh in the commutant of r2 (M2) (Sh 0 Sh belongs indeed to M2 0 M2 which is included in (I'2(M2))'). Let us call 4$ the isomorphism between the two involutive algebras r2 (M2)(Sh 0 Sh) and F2 (M2 Q). Let x in M1; we have:
r2(tQ(x)) = r2(t(x)t(1)*Q) = r2(4(x)Q) = r2(4(x))r2(Q) = 4$(1'2(4(x))(Sh 0 Sh)) = 4$((tsh 0 tsh)(r'i (x))) by (i). Therefore the mapping r2tQ is multiplicative, r2 being injective, QQ is multiplicative, hence Q < Sh, by 5.4.6. Therefore, we have:
r2(sh)>r2(Q)?sh0 sh which brings (iii). The proof of (iv) is identical. 5.5.3 Lemma. Let R be a projection satisfying the condition 5.4.2 (iii). Then the operator: U = WR 0 E)(wl ) + ($1-RIC1 0 Z)(Wi))(Q(1) (9 1)
is a unitary of M2 0 M1, and, for all w in M2* and 0 in (Mi), we have: (U,w 0 0) = (Ai(Tw),0)
Let us remark that iR and £i_RrKi are, both, normal homomorphisms from M1 to M2-
5.5 Isometries of the Preduals of Kac Algebras
179
Proof. We have:
U* = (AW 0 1)((tR 0 O(WI) + V1-RK1 0 i)(Wl*)) hence:
u*v = (e(1)* (9 1)(YR 0 i)(Wiwi) + (9 i)Ri Wi)V(1) 0 1)) (t(1)* = 0 1)((PR 0 i)(1 0 1) + (ti_RKi 0 0(10 1)(41) 0 1)) = (Q(1)* 0 1)(R 0 1 + (1 - R) 0 1)(Q(1) 0 1)
=101 because t(1) is unitary by 5.5.1 (i). The same kind of calculation would yield
UU* = 10 1. Since W1 belongs to M1 ® M1, it is clear that U belongs to M2 0 M1. Now, we have:
(U,w09) MR (9 0 (Wl*) 41) - W 0 0) + V1 - R IC 10 0 (WO 41) - LO 0 0) (Wl*i (tR)*W1) - W) 0 0) + (Wli (4-RM41) - LO) 0 K1 0 0) 1
_
(QR)*(e(1) w)
1
e) + (w; ,
_ (We, T(R w) 0 B + T((1 - R) w) 0 B) (Wj*
7
W) e
by z.s.i (i)
by 5.5.1 (iii) and 5.4.2
T(w) 0 0)
_ (Ai(T()), 8)
by 2.6.1 (i)
which completes the proof.
5.5.4 Proposition. With the above notations: (i) the mapping T is involutive. We have also: ii
Id2ti = L1'c1
(iii)
'd2 (Sh) = Sh ic2 (Sa) = Sa .
(iv)
Proof. We have, with the notations of 5.5.3:
A1(T(w)) = (w 0 i)(U) = (i 0 w)(SU)
The operator cU being unitary in Ml 0 M2 and the mapping A1T being multiplicative, it follows from 1.5.1 (i) and 2.6.5 that T is an involutive representation of M2*; as Al is involutive and injective, the proof of (i) is completed.
5. The Category of Kac Algebras
180
Let now x be in M1, w in M2*. We have:
(ic2ti(x),w) _ (K2tl(x*)*,w) V1(x*)'W0)-
by 5.4.2 (ii)
(X 7V1)*(W0))
(x*,T(e(1)* .w°))
by 5.5.1 (iii)
=(x*,T((t(1)* .w)°))
by 5.5.1 (ii)
=(x*,T(t(1)* 'w)0) (i.ci(x),T(L(1)* . W))
by (i)
_
(rci W, Vi)* (w))
by 5.5.1 (iii)
which completes the proof of (ii).
Let Pin Ph, x, y in M1. We have: Q-z(P)(xy) = Axy)AWK2W)
= 4(xy)K2(P) = IC2V1tq(xY)P) _ K2VPK1(xy)) = N2(QP(K1(y)k1(x))) = K2VPK1(yYPK1(x)) = k2QPK1(x)k2tPK1(y)
by (ii)
by hypothesis
= K2V1(tC1(x))P)NW1(k1(y))P) _ K2Q1tc1(x)tc2(P)tc2t1tc1(y)tc2(P)
= 4 (x)K2(PYl(y)K2(P)
by (ii)
= Qecz(p)(xYr1z(P)(y)
therefore K2(P) belongs to Ph. So, K2(Sh) belongs to Ph and 'c2 (Sh) < Sh by definition; since K2 is involutive, we get (iii). The proof of (iv) is identical.
5.5.5 Theorem. Let K1 = (Mi, r1, t1, (P1) and K2 = (M2, r2, IC2, (P2) be two Kac algebras. Let us assume that there exists a linear, multiplicative, bijective isometry T from the Banach algebra M2* to the Banach algebra Ml*. Then
K1 is isomorphic either to K2 or K. More precisely, if £ stands for the transposed mapping of T, we have: (i) The operator t(l) belongs to the intrinsic group of K2. (ii) At least one of the two following assertions is true: (a) The mapping £1 : M1 -+ M2 defined for all x in M1 by:
Q (y) = e(sy(l) .
5.5 Isometries of the Preduals of Kac Algebras
181
is an IHI-isomorphism from K1 to K2 (it implies that 4i is a (b)
homomorphism of von Neumann algebras). The mapping 41 is an antihomomorphism of von Neumann algebras and the mapping Li : M1 - M2 defined, for all x in M1, by:
L1(x) = J2e1(x)*J2
is an H-isomorphism from K1 to K. Proof. By 5.4.6, 5.5.2 (iii) and (iv), 5.5.4 (iii) and (iv) the couple of projections (Sh, Sa) as defined in 5.4.6 satisfies the hypothesis of the lemma 2.6.8 applied to K2. One of the two projections is therefore equal to 1.
Let us assume that Sh = 1. Then, the mapping 4i is involutive and multiplicative by 5.4.2 (ii), normal and bijective by construction; we have: X1(1) = 1
by 5.4.2 (1)
r2e1= Yl 0 4>r2 rc24 = 4kl
by 5.5.2 (i) by 5.5.4 (ii)
So, Li is an IHI-isomorphism from Kl to K2.
Let us assume that Sa = 1. Then the mapping L1 is involutive, multiplicative and linear by 5.4.2 (ii), normal and bijective from Ml to M2 by construction. Finally, it verifies: L1(1) = 1
r24 = (L (9 Li)I'1 4t'1 = 1 ic1
by 5.4.2 (i) by 5.5.2 (ii) and 12.10 (i) by 5.5.4 (ii) and 1.2.10 (i)
So, Li is an IH[-isomorphism from K1 to K. Thus, (ii) is proved and (i) was proved in 5.5.1 (i). 5.5.6 Theorem. Let K1 = (M1,1'1,ic1,cp1) and K2 = (M2,1'2,ic2,p2) be two Kac algebras. Let u be a normal isomorphism from Ml to M2 such that: U(1)
1
I'2u = (U 0 u)rl Then u is an IHI-isomorphism from (Mi, I'1, m1) to (M2, I'2, r2).
Proof. Let us apply 5.5.5 (ii) to the transposed mapping u* : M2* -+ Ml,,; with u being multiplicative, we are in the first case of 5.5.5 (ii); as u(1) = 1, we get that u is an i-isomorphism.
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5. The Category of Kac Algebras
5.5.7 Corollary. Let K = (M, I', x, co) be a Kac algebra. If the quadruple x and Spo is pro(M, r, n,,, apt,) is also a Kac algebra, then we have n,, portional to W.
Proof. We just apply 5.5.6 to the identity map of M. 5.5.8 Lemma. With the hypothesis and notations of the second case of theorem 5.5.5 (ii), we have, for all w in M2*: VI1)*(W') = T(t(l)* - w) .
Proof. Let x in Ml. We have:
(x,(t'1)(w')) = (t'1(x),w') = (L1(x),w) = (x,(ei)(w)) = (x,T(t(1)*
. w))
by 5.5.1 (iii), which completes the proof. 5.5.9 Corollary. Let K1 = (Mi, I'1, i1, Spi) and K2 = (M2, I'2, r.2, (P2) be two
Kac algebras. Let us assume that there is a linear multiplicative, bijective isometry T, from the Banach algebra M2* on the Banach algebra M1*. Then K2 is El-isomorphic to either K1 or to K1. More precisely, if £ denotes the transposed mapping of T, we have: (i) The operator £(1) belongs to the intrinsic group of K2; (ii) There is an H-isomorphism 15 from K2 onto K1 or to Ki (in the first case Li is a von Neumann algebras homomorphism from M1 to M2, in the second case it is an anti-homomorphism) such that, for all w in M2*, we have: q5(*7^1(J)(A2(W))) = Ai(Tw)
where 741) denotes the automorphism of M2 implemented by £(1) (cf. 3.2.2).
Proof. Let us assume that we are in the first situation of 5.5.5 (ii). Then £1 is 1H[-isomorphism from K1 on K2. The dual ]H[-isomorphism (cf. 5.3.2) .Li from K2 to K1 is defined, for all w in M2*, by:
i02M)
A1W1)*(W))
= ai(T(2(1)* w))
by 5.5.1 (iii)
Now, we have:
ii
(A2
il (A2 (41)
= Ai(T(w))
The theorem stands for
_ Li .
by 3.2.2 (v)
by what is above
5.5 Isometries of the Preduals of Kac Algebras
183
Let us assume now that we are in the second situation of 5.5.5 (ii). Then is is an H-isomorphism from K1 on K. The dual il-isomorphism L1 from K2 on K1 will also be, straightforwardly, an il-isomorphism from K2 on Ki K. By 5.3.2, it is defined for all w' in (M2)*, by:
tl(A2(w
Al W,1 )* Pl))
with the help of lemma 5.5.8, it can as well be written: i'l (A2 (w)) = A1(T(t(1)* . w))
As before, we can deduce from it that the theorem stands with '1 = Li .
5.5.10 Corollary. Let K1 = (M1,r1,iq,so1) and K2 = (M2,f2,ic2,p2) be two Kac algebras. Let us assume that there is a linear, multiplicative, bijective isometry T from the Fourier algebra A(K1) to the Fourier algebra A(K2). Then K1 is IEI-isomorphic to either K2 or K2. More precisely, up to the canonical isomorphims (cf. 3.3.4) between (Mi)* and A(Ki) (for i = 1, 2), if £ denotes the linear mapping from M2 to MI transposed of T, we have:
(i) The operator £(1) belongs to the instrinsic group of K1 (ii) There is an H-isomorphism lfi from K1 onto K2 or KZ (in the first case £1 is a von Neumann homomorphism from M2 to M1, in the sec-
ond case it is an anti-homomorphism), such that for all 9 in A(K1), we have: (7t(i) (ic1ir1*(9))) = ,c2ir2*(T (e))
Proof. It is nothing but corollary 5.5.9 applied to the pair (K2, K1). 5.5.11 Wendel's Theorem ([198]). Let G1, G2 be two locally compact groups, T a linear, multiplicative isometry from L1(G1) to L1(G2). Then, there exist: (a) a character X on G2
(b) a bicontinuous isomorphism a from G2 to G1 such that, for all f in L1(G1) and almost all s in G2:
(Tf)(s) = x(s)f(a(s)) Proof. Let us apply 5.5.5 to Ka (G1) and Ka (G2 ); as Ka (v2) is commutative, there is only one case; the intrinsic group of Ka (G2) is the set of characters on G2 (3.6.12), and the H-isomorphism between Ka(G2) and Ka (G1) comes from an isomorphism of G1 and G2 (4.3.5).
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5. The Category of Kac Algebras
5.5.12 Walter's Theorem ([194]). Let G1i G2 be two locally compact groups, T a linear, multiplicative isometry from A(Gi) to A(G2). Then there exist:
(a) an elements in G1 (b) a bicontinuous isomorphism a from G2 on G1 or G1PP such that, for all t in G2 and f in A(Gi) :
(Tf)(t) = f(s Ia(t)) Proof. Let us apply 5.5.10 to Ka (Gi) and Ka (G2 ); the intrinsic group of K8 (G1) is the set {)tG(s), s E Gi } (4.3.2), and the H-isomorphism between Ka(Gi) and Ka(G2) comes from an isomorphism of G1 and G2 (4.3.5). The result comes then from the fact that K (G2 )' = Ka (G2PP) (2.2.5).
5.6 Isometries of Fourier-Stieltjes Algebras Let K1 and K2 be two Kac algebras. In what follows, we consider a multiplicative, isometric, linear bijection T from B(K2) to B(K1) (as defined in 1.6). Its transposed mapping £ which sends W * (Ki) into W * (K2) is then an ultraweakly continuous, isometric, linear bijection. 5.6.1 Lemma. (i) The operator £(1) belongs to the intrinsic group of W*(K2). (ii) The operator 5A2 (L(1)) belongs to the intrinsic group of K2. (iii) As £1 is ultraweakly continuous, it is legitimate to consider its transposed (t1)* : B(K2) -- B(Ki). Then, with the notations of 5.4, we have, for all 9 in B(K2):
(L1)(9) = T(t(1)* 9) Moreover, if 9 is positive, so will be T(.e(1)* 9).
Proof. The proof of (i) is strictly analogous to 5.5.1 (i), because it results from 5.4.2 that £(1) is invertible; (ii) is just a corollary of (i). The proof of the first part of (iii) is absolutely identical to the one of 5.5.1(iii). Let us assume 9 to be positive. Let x be a positive element of W * (Ki ). We have
(x) (Li)(9)) = (L1(x), 9) > 0 because, by 5.4.2 (ii), Li is a Jordan isomorphism, which completes the proof.
5.6.2 Lemma. (i) Let K be a Kac algebra. We define a set Q by:
Q = {Q E W*(K); Q projection and Q 0 1, s,x,(Q) < Q ® Q}
5.6 Isometries of Fourier-Stieltjes Algebras
185
Then Q has a greatest element and:
max Q = 1 - supp s,\ (ii) With the constructions and notations of (i) associated to the two Kac algebras K1 and K2, we have:
fl(Ql) = Q2
-
Proof. Because A 54 0, it is clear that the projection 1 - suppsA is different from 1. Moreover, we have:
(sA 0 i)s,rx,r(1 - supp sA) = s'\x,r(1 - supp sA) = 4; y7rsA(1 - supp sA) = 0
by 1. 6.4 (i)
by 3.2.2 (iii)
Therefore s,x,r(1 - supp sA) belongs to Ker(sa 0 i) and: s,rx,r(1 - supp sA) < (1 - supp sA) 0 1 By, 3.3.3, we also have: s a s;r (1 - supps,\) = i n s A (1 - supp s a) = 0
Therefore:
s* (1 - supp S,\) < 1 - supps '\ by s;r being involutive, we get in fact: s;r (1 - supp S,\) = 1 - supps A
We then can write down: s,r x ,r (1 - supp S,\) = s,r x ,r s * (1 - supp s A )
= S (s* ® s*) s,r x,r (1 - supp s,\) < c(s;r(1 supp S,\) 1)
= 1®(1 - supp S,\) Finally, by using again (*), it comes:
s,rx,r(1 - suppsA) < (1 - suppsA) ®(1 - supps,\) therefore 1 - supp sA belongs to Q.
by 1.6.6 by by (*
186
5. The Category of Kac Algebras
Let Q be in Q. We have:
rsa(Q) = ryasa(Q) = ssaXa(Q) = s(sa sa)s7rx7r(Q) < 4;(Sa
=
S,\(Q)
< S,\(Q)
sa)(Q 0 Q) sa(4l)
by 3.2.2 (iv) by 3.2.2 (iii) by 1.6.4 (iii) by hypothesis
1
It then results from 2.7.3 (ii) that s.\(Q) is either equal to 0 or 1. Let us assume s.\ (Q) = 1; it is equivalent to Q > supp sA, which implies: Q + (1 - supp S,\) > 1 and:
sirxir(Q) +sirxir(1 - suppsa) > 1 ® 1 and then:
Q0Q+(1-suppsa)®(1-suppsa) > 1®1
And:
((1 - Q) 0 suppsa)(Q0 Q + (1 -
supps,\))((1 - Q) 0 suppsa) > (1 - Q) 0 suppsa
which leads to:
(1-Q)0supps,\ =0 which is impossible, Q being different from 1, and supp s,\ different from 0. Therefore we must have s,\ (Q) = 0, which is Q < 1- supp sA and it completes the proof of (i). Let now Q be in Q1. As L1 is a Jordan isomorphism, L1(Q) is a projector of W* (K2 ). Moreover it is not equal to 1, because L1(Q) = 1 would obviously be equivalent to Q = 1. Now, let 9 and 9' be two positive elements of B(K2). We have:
(s.2x.2(Li(Q)),9 0 9') (sir2xir2(L(Q)L(1)*),909) = = (S1r2 X7r2 (L(Q))(L(1)* 0 L(1)*), 9 0 9') = (Sir2xir2(L(Q)),L(1)* . 9 0 L(1)* . 9') = (L(Q), (L(1)* .9) * (L(1)* .9'))
= (Q, T((L(1)* . 9 * (L(1)* . 9'))) = (Q) T(L(1)* 9) * T(L(1)* 9'))
by 5.4.1 by 5.6.1 (i)
by hypothesis
5.6 Isometries of Fourier- Stieltjes Algebras
187
(S7r2 X 72 (Q), T(t(l)* - 0) 0 T(t(l) * -0 1 ) ) < (Q 0 Q, a(e(i) 0) 0 T(e(i)' 0')) by assumption
(QI=A (L)(0))(Q,Vi ).(01)) V, (Q) 0) V, (Q) 1
V1 (Q) (& 4 (Q)
7
1
by 5.6.1 (iii)
01)
0 (D 01
So, we get: sW2 x V2
(L1(Q)) < L1 (Q) ®L1(Q)
and therefore fl (Q) belongs to Q2-
Therefore we get Li (Qi) C Q2. As £1 is bijective, we could prove ti 1(Q2) C Q1 the same way, which completes the proof of (ii). 5.6.3 Proposition. With the above notations, we have:
(ii) (iii)
supp s 1\1) = 1 - supp S.\2 . (Ker s,\1) = Ker sA2
T(A(K2)) = A(K1)
Proof. Let us apply 5.6.2 to prove (i), considering that Li preserves the Let order. The ideal Ker 3A1 is generated by the projection 1 - supp sal.
x in W * (Ki ). As L1 is a Jordan isomorphism (5.4.2), we have:
Ll(x(1 - supp sa1) = (ei(x) i(1 - supp sat) + Li(1 - suPP sa1)L1(x)) = L1(x)(1 - suPP sae)
by (1)
As L1 is bijective, we get L1(Ker sa1) = Ker 8A2 . And L(1) being unitary and Ker sae a bilateral ideal, it completes the proof of (ii). Let 0 in B(K2); by (ii), T(6) vanishes over Ker 31\2 if and only if 0 vanishes over .(Ker sat) = Ker sae , which gives the result, thanks to 3.3.4.
5.6.4 Notations. The restriction of T to A(K2) satisfies the hypothesis of 5.5.10. There is thus an element u of the intrinsic group of K2 and an ]isomorphism
from K2 to K1 or Wi such that, for all 0 in A(K2), we have: 4i^ruK27r2*(0) = x1ir1*(T9)
Let us determine u more accurately; the mapping (s,\,),-,'T(S,\2 )* is an isometric linear bijection from (M2)* to (M1)* which shall be denoted by T.
5. The Category of Kac Algebras
188
we get u =N 1(1); and by Let £ : M1 -4 M2 its transposed. By 5.5.9, N transposing the relation T(sa2 )* _ (sA1 )T which defines T, we get that N N s,2. _ £s, , by definition of £ and £. Then, we have: U = Al) = 3A2 (t(1)) .
5.6.5 Lemma. With the above notations, we have, for all 9 in B(K2): (l(1))'121r2*(9) = tq7r1*T(9)
Proof. To simplify, we shall put y = Ysa2
In 5.6.4, the above relation has
been proved for 9 in A(K2). Now let w in M2*. Let us recall that (sA2 )*(w) is the generic element of A(K2) and that A(K2) is a bilateral ideal of B(K2) (3.4.4). By applying 5.6.4, we then find: (7(1c21r2*(9+(saz)*(W)))) = wtni.(T(B * (sa2 )* (w)))
or:
(7'c27r2*(O))(7'c2ir2*(sA2)*(w)) _ (lciiri*T(9))(lcilri*T(sA2 )*(w))
and, by using 5.6.4 again: (7!c27r2*(9))(7/c2ir2*(Sa2)*(w)) _ ('c17r1*T(9))(4i'Y'c21r2*(Sa2 )*(w))
which, by 4.6.9 (ii), can also be written: (4i,ytC272*(O)kP'YA2(w))
= (iciiri*T(9))(i'$t2(w))
by having A2(w) converging to 1, we complete the proof. 5.6.6 Theorem. Let K1 and K2 be two Kac algebras. We assume that there exists a multiplicative, isometric, linear, bijective mapping T from the FourierStieltjes algebra B(K2) on B(1K1). Then, there exists an 1H[-isomorphism from
K2 onto K1 or K . More precisely, if £ stands for the transposed of T, we have:
(i) The operator 3A2 (L(1)) belongs to the intrinsic group of K2. (ii) There is an IH[-isomorphism 45 from K2 onto K1 or Ki (in the first case £1 is a von Neumann algebra homomorphism from W*(Ki) to W*(K2), in the second case it is an anti-homomorphism) such that, for all 9 in B(K2), we have:
y8a2
(j(1))1C21'r2*(9) = rcl7ri*T(9)
5.6 Isometries of Fourier-Stieltjes Algebras
189
Proof. It is enough to put 5.6.1 (i) and 5.6.5 together.
5.6.7 Corollary. Let K1 and K2 be two Kac algebras. Let isomorphism from W*(Ki) onto W*(K2) such that:
be a normal
-9 72 X 72 W = (W 0 W) s,,.l x Il
(i.e. such that respects the canonical coproduct of W*(K1) and W*(K2)). Then, there exists an H-isomorphism 4$ from K2 onto K1 such that, for all w in M1*, we have: T/(,7ri (w)) = 7r2 (W 0 q5)
We have also: S,n2 W _ WS;nl
.
Proof. Let us apply 5.6.6 to the transposed mapping W* = B(K2) --- B(K1). As W is multiplicative, we are in the first case, furthermore, as W(1) = 1, there is an H-isomorphism 4$ from K2 onto K1 such that, for all 0 in B(K2), we have: 4$(!27r2*(O)) = ic1ir1*(W*(O))
Because !PK2 = 114$, it can also be written: -7rl*(Tl*(O))
Therefore, for all w in M1*, we have: (W,7rl*o*(O))
(T/(7rl (W)), 0)
(w, 4i7r2*(0)) (w o 43, 7r2*(0))
_
(7r2(c4.o4$),O)
which gives the first result. We can see that: s;r2W7ri(w) = s*27r2(w o 4$) = 7r2(w o 4$) _ 7r2(w o 0 o'2) = 7'2(w o K1 00)
_Wir1(woK1) _ W*1(w) _ Ws*l 7r1 (w)
which gives the second result, by the ultraweak density of 7r1(M1*) in W*(K1).
190
5. The Category of Kac Algebras
5.6.8 Corollary. Let K1 and K2 be two Kac-algebras, W a K-isomorphism from K1 to K2 (i.e. an IH[-isomorphism from W * (K1) to W * (K2 )). Then there exists an IHI-isomorphism u from (M1, I'1, rc1) to (M2, F2, 1C2) such that W is the extension of u. Therefore, K-isomorphisms are H-isomorphisms.
Proof. It is a particular case of 5.6.7. 5.6.9 Corollary (Johnson's Theorem [65] ). Let G1 and G2 be two locally com-
pact groups. Let T a multiplicative, linear, bijective mapping from M1(G1) to M1(G2 ). Then there exists: (i) a character x on G2 (ii) a bi-continuous isomorphism a from G2 to G1 such that for all measure p of M1(G1) we have:
Tp = xa-'(P) Proof. By 4.4.1 (ii), the algebra M1(G1) is the Fourier-Stieltjes algebra associated to the Kac algebra K8 (G1). Let us recall that, by 3.6.12, the intrinsic group of Ka (G1) is composed of the characters on G1. Therefore, by using 5.6.6, we see that there is a character xr on G1 and an IHI-isomorphism 4$ from K8 (G1) to K8 (G2) (because K3(G2) = K3 (G2)) such that, for all p in M1(G1), we have: AG2 (T y) = 4(OXI AG1(12))
We easily compute that for all p in M1(G1) we have: aX-cAG1 W» = aG1 WY)
On the other hand, by 4.3.5, there is a bicontinuous isomorphism a' from G1 to G2 such that, for all s in G1: 4i(AG1(s)) = AG2(ar(s))
By integrating, we find, for all µ in M1(G1): P(AG1(µ)) = AG2 (a'(p))
Going back to (*), we have: by (**) by (***)
KGs (Tµ) = oP(AGi Wµ)) = AGZ
and therefore:
T p= a/(re'IL) = (le
o
(aI
(X µ)
al-1)(aI
W)
We finally reach the result by writing X = x' o a r -1 and a = ar -1
5.6 Isometries of Fourier-Stieltjes Algebras
191
5.6.10 Corollary (Walter's Theorem [194]). Let Gi and G2 be two locally compact groups. Let T be a multiplicative, isometric, linear, bijective mapping
from B(G1) to B(G2). Then there exists: (i) an elements in G1 (ii) a bicontinuous isomorphism a from G2 to G1 or to G1 PP such that, for all t in G2 and f in B(G1), we have:
(Tf)(t) = f(s 'mo(t))
Proof. By 1.6.3 (iii), up to the Fourier-Stieltjes representations, we have B(G2) = B(1[ (Gi)) (i = 1, 2). Therefore, applying 5.6.6, we get the existence of an element u in G(K3 (G1)) and an IHI-isomorphism from Ha (G1) to Ha (G2) or ]HIa (G2 )' = 1HIa (G2PP ), such that for all f in B (G1), we have: Tf = P (flu (f ))
By 4.3.2, there exists s in G1 such that u = AG1(s). Then, we have for all f in L°O(G1) and almost all t in G1: 1 (fl,\,;1 (S) Y )) M = f (s - t)
on the other hand, by 4.3.5, it exists a bicontinuous isomorphism a from G2 to Gi or G1PP such that: (***) lp(f) = f 0 a
Going back to (*), we finally find, for all t in G1 and f in B(G1), that: (Tf ) (t)
(PAG 1 (,q) (f )) (a
f (s-la(t)) which completes the proof.
Chapter 6 Special Cases: Unimodular, Compact, Discrete and Finite-Dimensional Kac Algebras
Let K = (M, T, re, cp) be a Kac algebra, K = (Ar,I, k, c) the dual Kac algebra. We have seen that the modular operator A = aV is the RadonNikodym derivative of the weight Sp with respect to the weight SO o i (3.6.7 ).
So, it is just a straightforward remark to notice that W is invariant under if and only if is a trace. Moreover, the class of Kac algebras whose Haar weight is a trace invariant under x is closed under duality (6.1.4 ). These Kac algebras are called "unimodular" because, for any locally compact group G, the Kac algebra ]I (G) is unimodular if and only if the group G is unimodular. Unimodular Kac algebras are the objects studied by Kac in 1961 ([66], [70] ).
We show later another analogy with the group case, namely that if Sp a finite weight, then (M, T, rs, cp) is a unimodular Kac algebra (6.2.1); it is called "of compact type", because Ka(G) is of compact type if and only if G is compact. We prove then, after Kac ([67] ), that every representation of the involutive Banach algebra M, is the sum of irreducible representations (this leads, for compact groups, to the Peter-Weyl theorem) and that the Fourier representation is the sum of all (equivalent classes) of irreducible representations of M. (6.2.5 ). With the help of Eymard's theorem, this leads to Tannaka's duality theorem for compact groups (6.2.6 ). If K is such that the Banach algebra M. has a unit, then K is a unimodular algebra and K is of compact type (6.3.3 ). So, such Kac algebras will be called "of discrete type". Moreover ([67] ), the von Neumann algebra M is then the sum of finite-dimensional matrix algebras:
M=eC(Hi)
withdi =dim Hi
i
and the trace Sp is then given by: p
ED xi i
i
diTr (xi )
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, I', rc) be a co-involutive Hopf-von Neumann algebra, such that M = ®i,C(Hi) with di = dim Hz < oo and some Hi,, equal to C; let p be the one-dimensional projector associated to Hzo; if p gives a unity of the Banach algebra M*, and if F(p) satisfies a certain (quite natural) condition involving !K, then, there is a Haar trace Sp and (M, I', x, Sp) is a Kac algebra of discrete type (6.8.5 ). This result appears, then, to be, in the non-commutative case, the analog of Krein's matrix block algebras (6.4.5 ), and, so leads to Krein'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 UnimodularKac Algebras 6.1.1 Lemma. Let (M, I', ic) be a co-involutive Hopf-von Neumann algebra, cp be a faithful, semi-finite, normal trace on M (we consider elements of M as operators on Hip). For all x in 91w, we shall consider the element wx of M* defined for all y of M by: WX(Y)
= W(xy) = W(YX)
Then:
(i) For all x1, x2 in 9, we have: wx1 x2 = W AW (x1),Ap (x2 )
Therefore the set {wx, x E %p} is dense in M*. (ii) For all x in 9ZW, the element wx belongs to IV and we have, with the notations of 2.1.6: a(w.,) = AWW
194
6. Special Cases
(iii) For all x in 9l,, we have with the notations of 1.1.1 (ii): wx = wx *
(iv) For all w in M*, we have, with the notations of 1.1.1 (ii) and 2.1.6: II'IlW = Ik'lIW
(v) If Sp = W o i, then, for all x in, we have: 0
Wx = WK(x* )
and wx belongs then to I , fl I. (vi) Let us suppose that M = ®iEIC(Hi ), with di = dim Hi < oo, and cp = Ej diTri, where Tri is the canonical trace on £(H1); then ( = cp on, and we have, for all i in I, and 1 < j, k < di: Wei
=
>>k
ilk
1
k
k
i
where the {e}1<
Proof. We have, for all y in M: (yA<,(x1) I AW(x2)) = W (x2yx1) = co(x1x2y)
which gives (i). If z is in 91., we have: Wx(z*)
= Sp(z*x) = (Av(x) I AV(z))
which immediately yields (ii), by 2.1.6 (i) and (ii). By definition, we have: wx (y) = wx
(y*)
= SP(x y*)
= (x*y) = wx* (y)
which is (iii). We have: II'II V = sup{ I (x*, w) I , x E 4V, V(x*x) < 1 } = sup{ 1(x, w) 1, x E gyp, Sp(x*x) < 1} = sup{ I (x*, w)1, x E 91w, W(x*x) < 1} = IIwIIW
by 2.1.6 (i)
because V is a trace
6.1 Unimodular Kac Algebras
195
which gives (iv). Let us suppose that Sp = Sp o rc; we have:
by 1.2.5
by hypothesis
which gives the first part of (v), the second part being trivial then. For x = ®ixi in M+ (with xi in ,C(Hi)+), we have, with the notations of 1.2.11(ii): d:Tri, (VixiVi*) _
SO 0 rc(x) _
diTrix: = cp(x)
Moreover, we have then: weij,k (x) = SO(xej,k)
= di E(xiej,kSl I Qh
i = di(xiC =diS?
,
I Ck) k(x)
which completes the proof, thanks to (v). 6.1.2 Theorem. Let K = (M, I ' , i ,5P) be a Kac algebra and K = (M, F, k, (3) be the dual Kac algebra. The following assertions are equivalent: (i) The Haar weight cp is a trace. (ii) The dual Haar weight cP satisfies: S0^ 01%^= .
Proof. It is a straightforward consequence of 3.6.7 and 4.1.1. 6.1.3 Definition. A Kac algebra K = (M, I', rv, Sp) will be called unimodular if W is a K-invariant trace.
6.1.4 Proposition. A Kac algebra is unimodular if and only if its dual is unimodular.
196
6. Special Cases
Proof. It is an immediate corollary of 6.1.2.
6.1.5 Proposition. Let K = (M, I', K, Sp) be a unimodular Kac algebra and K = (M, I', k, Sp) be the dual Kac algebra. Then, for any x, y in asp the element wA(p(x),A (y) belongs to 4 and we have: a(wn40(x),Aw(y)) = a(wx * w,,(y)* )
Proof. By 6.1.1 (ii), wx and wy belong to IV, and by 3.5.2 (i), A(wx) and A(wy) to 91W. By 6.1.4, cp is a trace and therefore A(wx) A(wy )* belongs to So,
by 6.1.1(ii) applied to SP7 wbelongs to I0. We have also:
wa(w,,) ,a(wy )
= wnW(x),AW (y)
Therefore we have: a(wAw (x),Aw (y)) = 4^LoA(wT),\(wy)*
= A,(A(wx)A(wy)*) = a(wx*wy) = a (wx *wK(Y)*)
by 6.1.1 (ii) applied to cp 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 Saar weight Spa of Ka(G) is !Ca -invariant. (iii) The Kac algebra Ka(G) is unimodular. (iv) The Kac algebra K3(G) is unimodular. (v) The Saar weight cp9 of K8(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 K = (M, I', rc, Sp) be a Kac algebra. The following assertions are equivalent: (i) The weight cp is strictly semi-finite (in the sense 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 of K which is an invariant weight Kac algebra.
(iv) The sub-algebra M' is a sub-Kac algebra of K.
Proof. For every Kac algebra, it is clear from (HWiii) that r,.(M') is equal to M" and from 2.7.6 (ii) that I'(M'P) is included in M° ® M'. Moreover, we know, by [15], that cp is strictly semi-finite if and only if the restriction of cp to M° is a semi-finite trace. Therefore, we see, by using 2.7.7, that cp is strictly semi-finite if and only if M'P is a trace sub-Kac algebra of K. Thus, (i) implies (iv) which implies (ii). N N Conversely, let us assume (ii) and denote by (M, I', k, gyp) the trace sub-Kac
algebra of K. We have, for any x in M and t in R, o (x) = 01'(x) = x and therefore M is included in MP, which implies that the restriction of Sp to M' is semi-finite, so, 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. 6.1.8 Corollary. Let G be a locally compact group. The weight W. 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 pp, will be strictly semi-finite if and only if there exists a reduced Kac algebra of K3(G)" (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 K ,,(G) being unimodular. By 4.3.6 (ii), it is equivalent to the existence of an open subgroup H of G such that 1[,(H) is unimodular which is, in turn, by 6.1.6 equivalent to H being unimodular.
6.2 Compact Type Kac Algebras 6.2.1 Theorem. Let (M, I', rc) be a co-involutive Hopf von Neumann algebra. Let cp be a finite faithful normal weight on M such that, for all x, y in M, we have: (i ®S0)((1 ® y*)r(x)) = K(i 0 W)(I'(y*)(1 0 x))
Then, (M, r, rc, Sp) is a unimodular Kac algebra. Such a Kac algebra will be called of compact type. We have then: w*Sp=Sp*w=w(1)cp
6. Special Cases
198
Proof. Putting y = 1 in the above formula, it comes: (i ® SP)(r(x)) = Sp(x)l
applied to K(x), it gives: Sp o IC(x)1 = (i 0 W)r(K(x)) = (i 0 p)c(K 0 rc)r(x) = (SP o Ic 0 i)r(x)
and: (p o lc(x)SP(i) = ((p o re ®()r(x)
oK(i)
by (*)
= p(x)5p(i)
therefore Sp o is = W and y° = v. Let t in R, we have:
roT = (i 0 o )r rat = (o° 0 i)r therefore, we get:
by 2.5.6 by 2.7.5 (i) because cp o K = SP
ra2 = (at ®ot )r
On the other hand, it results from (*) that ((p ®cp)r(x) = in M, and as r(M) is of®" invariant, by 2.7.6 (iii), we have:
for all x
o®®'r = .raw we finally get raw = rat, and, r being injective, it implies 4 t = at and then, for all t in R, of = id. Therefore Sp is a trace, the axiom (HWiii) obviously holds, and (M, r, rc, cp) is a Kac algebra, it is unimodular because cp is a rc-invariant trace.
The formula w*y = w(1)y is given by (*); using the involution and the fact that W = Sp°, we get (*w = w(1)cp.
6.2.2 Theorem. Let G be a locally compact group. The following assertions are equivalent: (i) The group G is compact. (ii) The Kac algebra Ka(G) is of compact type. Proof. It is trivial.
6.2.3 Lemma. Let K = (M, r, re, () a compact type Kac algebra such that ((1) = 1. Then there exists an isometry I from Hp to Hw 0 Hcp such that, for all x, y, z in M, all w in Iw, all , 17 in H, we have: (i)
(ii) (iii)
IAW(x) = A ®w(r(x)) I*(Ao(y) (9 (z)) = a(wy*wz)
(\(w)Iri)=(Ia(w)Iri®Je).
6.2 Compact Type Kac Algebras
199
Proof. As, by 6.2.1 (*), we have (®p)f(x) = p(x) for all x in M, by density and polarization, we can define a unique isometry I from Hp to H. ® Hop verifying (i). Then, we get: (I*(A<(y) 0 A,(z)) I AV(x)) = (A,(y) 0 AV(z) I IAV (x)) = (AW(y) ® A,(z) I AVOV(r(x)))
_ (Sp (9 ,)(r(x*)(y 0 z)) _ (r(x*),w y ®wz)
_
by 6.1.1
(X*,Wy*WZ)
_ (a(wy*wz) I A,(x))
by 2.1.6 (ii)
which gives (ii). We have: (X(w)A(y) I AW(z)) =
(A(w)*
7 WAp (z),AW (y))
= (A,(A(w)) I a(wA4,(z),A60(y)))
by 2.1.6 (ii)
by 3.5.4 (ii) and 6.1.5 by (ii) = (a(w) I*(Av(y) 0 AW(rv(z)*)) = (a(w) I a(wy*w.(z)* ))
= (a(w) I*(AV(y) 0 JAAV(z))
by 3.5.2 (i)
= (Ia(w) I AW(J) 0 JAp(z)) and (iii) follows, by density of A,p(M) in Hip.
6.2.4 Proposition. Let K = (M, r, ,, Sp) be a compact type Kac algebra such that V(1) = 1. The Hilbert algebra A,(910) is then equal to H; moreover, it exists a continuous algebra morphism b from H to M* such that, for all w
in V: Ma(w)) = w .
Proof. By 6.2.3 (iii), for all w in I,, and
in H, we have:
II,\(w)eII < IIIII therefore, every vector 6 in H belongs to the achieved Hilbert algebra of a(4, r,1 ), i.e. to A,(SYt,). Also, we have, for all w in IV: IIa(')II = IIWIIP
= sup{I(x*,w)I, x E M, cp(x*x) < 1} > sup{I(x*,w)I, x E M, lIxil < 1} because (1) = 1 = IIwII
Thus, it exists a linear mapping b from a(IV) to IV such that Ilbil C 1 and b(a(w)) = w for all w in IV. By density, it is possible to extend b to the whole
200
6. Special Cases
of H; moreover b is clearly an involutive algebra morphism from a(Iw fl to I. fl 1, which, by continuity, completes the proof. 6.2.5 Theorem. Let K = (M, I', ,c, cp) be a compact type Kac algebra, such that W(1) = 1. Then we have:
(i) The Banach algebra M* admits wA, (1) as unit. Therefore we have A(K) = B(K), and s,\ is a 1H[-isomorphism from W*(K) to (M, I', k). (ii) The element A(W) is a one-dimensional projection p of the centre of M. and we have, for all x in M: px = XP = WA,(1)(x)p
I'(x)(p 1) =pox P(x)(1 ®p) = x ®p Moreover, the unit wA,(1) is a homomorphism from M to C. (iii) For all w in M*, A(w) is a compact operator on H, and, if w belongs to IW, A(w) is a Hilbert-Schmidt operator on H. The Fourier representation A can be decomposed into a direct sum of irreducible finite-dimensional representations Therefore, for all i in I, there exist Hilbert spaces
Hi, with di = dim Hi < -boo, such that the algebra M is isomorphic to ®iEI,C(Hi).
(iv) We have, with the above notations:
F(p) = > di 1 iEI
j,k
k(ej,k) ®ek,j
is a system of matrix units of ,C(Hi). (v) The Haar weight 5P is equal to Ei diTri, where Tri is the canonical trace on ,C(Hi). It is also equal to the restriction to M of the canonical trace on ,C(H), and we have, for all x in M: where
WA,(1)(x) = 0(xp)
(vi) For all i in I and integers 1, m such that 0 < 1, m < d1, we have: di
L
k=1
k
'o
k
+n
where 61 m is the Kronecker symbol and {L}k=1,...,d1 is the orthogonal basis
of Hi corresponding to the matrix units es
jlk.
6.2 Compact Type Kac Algebras
201
Proof. For all w in M*, we have: (A P
(,\*(w)AW(1) I AW(1) _ (,1(w o r%,)11,(1) I AW(1))
by 3.7.3
_ (Aw((w 0 z)-P(1)) I
by 2.3.5
= w(1) therefore A(CA,(l)) = 1, and, thanks to A being injective, we see that wAW(1) is the unit of M*, which gives (i). The relations p *5o = cp and V° = co (6.2.1) imply that A (V) is a projection. As we have w*Sp = Sp*w = w(l)So for all w in M*, we get that this projection lies in the centre of M. Moreover, for all x in M, we have: o rt. 0 Z)r(x))
by 2.3.5
= P(x)Aw(1) = (A,(x) I AW(l))AW(l)
therefore A(V) = p is the one-dimensional projection on CA,,(1). Now, for x in M, e, ,q in H, we have: (xp 177) = ( I A,(1))(z7 I A,(1)) (xA,(1) I A W(1))
= (PI)(2?n(1)(x) therefore pxp = wA, (1) (x) p, and, as p is central, we get the first formula of (ii), from which it is easy to get that wAp (1) is multiplicative. Then, for all w in M. we have: c<'(xp) = w(px) = wAW(1)(x)w(p)
Then for all 1, w2 in M* and x in M, we have: ((2)1 (&w2)(r(=)(1 fir)) = ((2'i 0 wAbci>)(r(=))wz(n)
therefore we get the second formula of (ii). The last one can be proved the same way, therefore we have (ii). As, by 6.2.4, H is a complete Hilbert algebra, we know by ([24], 1 §8.5) that H = ®i E I pi H, where the family {pj} i E I is the set of the minimal projections
of the centre of M. We know that pi H, up to a constant c?, is isomorphic to the Hilbert-Schmidt operators algebra on a Hilbert space Hi, that M is isomorphic to ®iEI,C(Hi) and then that EiEI ciTri. By using the standard representation of M on ®i EI (H1 0 Hi ), we get that Sp is equal to
202
6. Special Cases
the restriction of the canonical trace on L(®i (Hi ® Hi )). The same holds for every standard representation of M, and in particular on H. The representation A is therefore the sum of irreducible representations w -+ A(w)pi, equivalent to representations Ai of M* on spaces Hi such that £(Hi) = Ai(M*)".
If {j}jEJ is an orthonormal basis of H, for all w in I., we can compute the Hilbert-Schmidt norm of A(w):
IIA(w)IIis=
_ j,k
I(A(W)ejlek)12
IJ)t2
j,kE
by 6.2.3 (iii)
= IIIa(w)112
=
IIa(w)112
=IIw!I,<+00. Therefore, A(w) is a Hilbert-Schmidt operator. By density it implies that for all w in M*, A(w) is a compact operator on H. And, for all i, pi A(w) is also a compact operator on pi H. Now, the isomorphism which maps (up to ci ) piH onto the Hilbert-Schmidt operators on Hi (i.e. onto Hi ® Hi) is actually mapping pi A(w) on Ai (w) ®1; thus, the compactness of these operators imply:
di = dim Hi < 00 which completes the proof of (iii). Let now be an orthonormal basis of Hi and the associated matrix units. Let Y1, Y2 in which are decomposed into yl = Ei yi and Y2 = Ei y2 with y1, y2 in C(Hi ). We have: (wyl 0 iC ®wy2)(1(P)) = (wy2 ®wyl o
X A) ((p)
= (A*(W^ Y2) A* (wyl o k ), SP)
= SPA(WY2 ° k)A(wyj )) = co(A(wy2)*A(wyl ))
= (A,(A(1)) IAV(A(wy2))) _ (A,(yi) I Ac(y2 )) = c (y2Y1)
=
ciTri (y2y1) di
_
ci > (y2 yl C I C ) i
j=1
by 3.2.2 (iv) by 1.4.3
by 3.7.3
by 1.2.5 and 6.1.1 (iii) by 6.1.1 (ii)
6.2 Compact Type Kac Algebras
203
di
ci
(Yi6j
dxy2j
6
i
j,k=1
i
di
ci L (YS!6j' I 6k)(y26k I 6j ilk=1
i
di
>c1 L Tti(ejkYDTri(e"jy' ) j,k=1
i
di
ci L (Tr (9 Tr )(ej,kyl ®ek)3y2) j,k=1
i
di
i
cz
=(SP®SP)
j ej,kyl
i
i
ek,jy2
j,k=1
di
_ (c ® c)
Ie,k ® ek, j (Yi ®y2)
ci 1
j,k=1
di
ci
_ ('2'yi 0wy2) i
1
j k=1
e j,ky1 ®ek, j y2
,
Thanks to 6.1.1 (i), by linearity and density, this implies: di
F(P) E ci L rv(e j j,k=1
i
®ekJ
So, the operator ci 1 ',k=1 k(ezilk) ® ek . is equal to P(p)(x(pi) ® pi) and, therefore, is a projection; by 1.2.11 (ii), it implies ci = di, which completes the proof of (iv). For all x in M, we have:
(xp) = wAV (1) (x)SP(p)
by (ii)
= WA, (1) (x)
by (iv)
which completes the proof of (v). We also have, for 0 < 1, m < di: i'(p)(l ®e12,,z) = di 1 > i (ej,k) 0 ek 'elm j,k
= di 1 E k(el,k) ®ek,m k
by (iv) (*)
204
6. Special Cases
Now, we can write down, thanks to 6.1.1 (vi):
i
k
k1
`%'v km, x
= di
2
('k(e1)* *W k
k,m
/\Wkc(ek 1)* ® Wet
= dz
x) 1
k,
k
,
m
I'(x))
2
= dz E (Sp ® SP)(r(x)(k(ek) l)* ® ek,m)) k
= di
®V)(F(x)I'(p)(1 0 e21)M))
by (***)
= di 1(cp 0 )(i'(xp)(1 0el,m)) = W A,(1) (x) di
®SP)(I'(p)(1 ®el,m ))
by (ii)
= di 1wA,(1)(x)SP((i 0 W )(I'(p)(1 0 el,m)))
= d-1
(1)(° rx, )((Z ®SP)((1 0 p)I'(el,m))) by (HWii) SO
= di 12A,(1)S ((i 0 S)((1®p)I'(el,m))) by 6.2.1 and 6.1.4 = di 1wA,(1) (x) IP^ ((i 0 5?)(el,m ®p))
by (ii)
= di 1wA,(1)(x)c (el,m)
by (v) by (v)
= 6l,rn AW A,(1)(x)
which completes the proof. 6.2.6 Theorem. Let K = (M, T, ,c, Sp) be a compact type Kac algebra such that
p(1) = 1. Then we have: (i) Every non-degenerate representation of M* can be decomposed in a direct sum of finite-dimensional irreducible representations. (ii) Every irreducible representation of M* is finite-dimensional and is equivalent to a component of the Fourier representation A. So, the Fourier representation is the sum of all the (equivalence classes of) irreducible representations of M. Let us note Irr this set. (iii) For y in Irr, let Hµ be a Hilbert space such that µ(M*) = L(Hp ), dP = dim Hp < +oo. The Hilbert space H is then isomorphic to the Hilbert sum ®,EIrrH, 0 Hµ, where the norm on H. is multiplied by dµ. be an orthonormal basis of Hµ. The vectors (iv) Let {di2A,(,i*(f1is,)*)} (E Irr, 0 < j, k < dµ) form an orthonormal basis k
6.2 Compact Type Kac Algebras
205
of H and for all a in H, we have: dµ
dM E pEIrr
,u)*)
I
-
j,k=1
Proof. Let µ be a non-degenerate representation of M* on a Hilbert space Hµ . With the notations of 6.2.5, we put: K; = {ii(b(e))ri, C E p;H, 77 E Hµ}- C Hµ
If i 54 j, it is easy to check that Ki is orthogonal to Kj. As H = ®iElpiH and b(H) is dense in M*, we get that Hµ = ®iEIKi. Let now 77 be in Ki. It is clear that the subspace /2(M*)77 contains µ(b(piH))i which is finite-dimensional, as dense subspace. Therefore the space K = {µ(M* )r7} - is finite-dimensional; it is also p(M*) invariant; therefore K has a subspace K' which is 1t(M*) invariant and such that the corresponding representation of M* is irreducible.
If we consider a maximal family Ka of two by two orthogonal, finitedimensional and p(M* )-invariant subspaces of H. such that the corresponding representations of M* are irreducible, we have necessarily Hµ = ®aKa which completes the proof of (i). Let us assume that /2 is irreducible and let us use the above arguments. There exists i in I such that H,, = Ki. Let q, 77' be in Hµ . We have, for all w in IV:
(/A(w)s' 1 ij') =
By continuity, we have, for all
(a(w) I1V(y*(S2n,n`))*)
in H, thanks to 6.2.4:
(,u(b(e))ri I ij') = ( I AV(L*(flf7,T71)*))
We then have: (/2(b(p))rl 177')
by (*) _ (p(b())r1 I ri') because q belongs to K=
Therefore, we get that: ASP (' * (S?17 j'0 *) = pi AV (p * (QQ7,,7,) *) E piH
and then ) (wA* (0 , )*) (which is equal to 6.1.1(ii) and 3.5.4 (i)) belongs to C(Hi).
,., Hi 0 Hi
(A,(p*(Q,,,,)*)), thanks to
6. Special Cases
206
For w in Ie fl n., we have: (7^r(A,(y*(fl,7)*)a(w) I a(w))
(AV(y*(flj7)*) I a(uj*w")) Ily(w,)77112 > 0
7r(Aw(µ* (Q, )* )) belongs to L(Hi)+, and therefore there exists such that fr(e) = ir(Ap(µ*(sl7)*))1/2 and for any w in IV, we have:
So,
(A()e I
)
_ (a(w)l
in p1H
(a(w) I -7^r'(6)*6)
_ (a(w) I A ,(µ*(Qn)*)) = (p(w)q 177)
As A, and µ are irreducible, we see that µ is unitarily equivalent to AZ, which completes the proof of (ii).
Thanks to (ii) it is possible to put a bijection between Irr and the set I defined in 6.2.5 (i); then (iii) can be deduced from 6.2.5 (iii). For µ E Irr, let eJ k be the matrix units associated to the basis }j=1,...,dµ . We have: 3)
where 'I
3
®'k
is the canonical trace on ,C(H,,). Therefore, by (iii), an orthonormal
basis of H is made of the following elements {d1"2A,(e), µ E Irr, 1 < i l k < dµ } .
For w in I,,, we then have: (a(w)
k
_ (p(w)e I 6k)
= Trµ(e3µ(w))
,
= dµ 1SP(e%A(w))
through the identification of µ to a component of A, by (iii) = d; 1(a(w) I A-(e' ,j)) An orthonormal basis of H is thus made of:
{di2A(p*(Q,)*), pEIrr, 1<j,k
(IA
(p(b())' I %k)
which completes the proof of (iv) and of the theorem.
6.2 Compact Type Kac Algebras
207
6.2.7 Corollary. Let G be a compact group, equipped with a normalized left Haar measure. We have: (i) The Hilbert space L2(G) is a subalgebra of L1(G) (ii) The Fourier algebra A(G) has a unit, A(G) is equal to B(G), and sAG is an 1H[-isomorphism from W*(G) to K3(G).
(iii) Every representation of G can be decomposed into a direct sum of finite - dimensional irreducible representations. Moreover, every irreducible representation of G is finite-dimensional and is equivalent to a component of
the left regular representation AG. So, the left regular representation AG is the sum of all (classes of) irreducible representations of G. Let us note this set Irr G. j
d
di, E µEIrrG
f (s)737-
) yj)k
j,k=1 (ICG
Proof. Through the use of 6.2.2, it is the translation of 6.2.4, 6.2.5 (i) and 6.2.6 applied to K, ,,(G).
6.2.8 Corollary (Tannaka's Theorem [166] ). Let G a compact group, Irr G the set of all (classes of) irreducibles representations of G, and, for each
v in Irr G, H a Hilbert space such that v(L1(G)) = ,C(H1,); let us now choose x = ®IrrGxv an element Of ®IrrG(Hv) different from 0. Then, the two following assertions are equivalent:
(i) There exists a unique s in G such that, for all v in IrrG:
xv = v(s) (ii) For any pair y, v in Irr G, and 7rl, ... , 7rn in Irr G, if V is a unitary in Hµ 0 Hv, and mk integers such that: µ(s) ® v(s) = V *
®(1G(Cmk) 0 irk(s)) V k
then: xµ ® xv = V *
((1(Cmk) (9 xirk )
V.
k
Proof. For any y, v in Irr G, . .there exist 7rl, U, ,,, in . , in, in Irr G, a unitary ,C(Hµ 0 Hv), and integers mµ,v,,I.k such that, for all s in G, we have: Uµ,v(P(s) 0 v(s))U*,,, =
®(1G(Cmµ,v,rk) 0 irk(s))
k
208
6. Special Cases
By hypothesis, this implies: UµIv(xµ (9 xv)Uµ,v
@(1L(CM1A1V1Wk) 0 x7rk ) k
By 6.2.7, we have G(G) = E)IrrGG(Hv), and, for any s in G:
r8(ac(3)) _ ,Xc(3) 0 ac(s) = E)(µ(s) 0 v(3)) µ,v
_ ®Uµ,v ED(1,C(C'A1v1Tk) 0 Irk(s)) µ,v
k
and therefore, for any y = ®v yv in ,C(G), we have: ra(y) = ED Uu*,v µ,v
®(1L(CmA,v,'rk) (9 Yk) k
and then, thanks to (*): F. (x) _
µ,v
(x0x)
=x®x As, by hypothesis, x is not equal to 0, x belongs then to the intrinsic group of K.(G), i.e. to the set {AG(s), s E G} by 4.3.2. As AG = ®IrrGv by 6.2.7 (iii), we get (i). As the implication (i) = (ii) is obvious, the result is proved.
6.3 Discrete Tyke Kac Algebras 6.3.1 Definition. Let K = (M, I', rc, Sp) 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 L'(G) has a unit (1.1.3). 6.3.3 Theorem. Let K be a Kac algebra. The following assertions are equivalent: (i) The Kac algebra K is of discrete type. (ii) The Kac algebra K is of compact type.
6.3 Discrete Type Kac Algebras
209
Proof. Let us assume (i). Because of M. being unital, so is )(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 I, fl I,,,, is dense in M* and A norm-continuous, we get that 1 belongs to the norm closure of A(ip fl II) and therefore to the norm closure of So, there is an invertible element in nc fl g . Therefore the left ideal 91.- is equal to all M and the weight SP is finite, which is (ii). The converse implication has been proved in 6.2.5 (i). 91!
6.3.4 Corollary. Let K be a discrete type Kac algebra. Then we have: (i) The Kac algebra K is unimodular. (ii) There exist Hilbert spaces Hi, with di = dim Hi < -boo, such that the algebra M is isomorphic to ®2ETC(Hi). (iii) There exists p, one-dimensional projection of the centre of M such that, for all x in M, we have:
T(x)(p O 1) =pox 1'(x)(1 OP) = x OP F(p)
di 1
iEI
rv(e.7,k)
j,k
are matrix units of C(Hi). (iv) We have:
where
E diTri iEI
where Tri is the canonical trace on C(Hi), and the unit c of M* is defined, for all x in M, by: E(x) = co(px)
The Haar weight 5o is also equal to the restriction to M of the canonical trace on C (H).
(v) We have, for all i in I: di
L ( S? i e )° * 1
k=1
km
is the Kronecker symbol and {}k=i,...,d1 is the orthogonal basis of Hi corresponding to the matrix units es where
ilk.
Proof. The assertion (i) results from 6.2.1 and 6.1.4, (ii) from 6.2.5 (iii), (iii) from 6.2.5 (ii) and (iv), (iv) from 6.2.5 (v), and finally (v) from 6.2.5 (vi).
210
6. Special Cases
6.3.5 Theorem ([109]). Let (M, I', ,c) be a co-involutive Hopf-von Neumann algebra such that M = ®iEIC(Hi), with di = dim Hi < oo. Then the following assertions are equivalent: (i) There exists a weight Sp on M such that (M, r, x, W) is a discrete type Kac algebra. (ii) There exists a one-dimensional projection p in the centre of M such that, for all x in M, we have:
I'(x)(p ®1) = p ® x
I'(x)(10 p) = x ®p F(p) _
iEI
di 1
j,k
rc(e,k) ® ek1j
(*)
where the are matrix unit of £(H1). Moreover, Sp is then equal to >jEI diTri, where Tri is the canonical trace on ,C(H1). The unit of e of M* is a homomorphism and satisfies e(x) = 5P(xp) for
all x in M. If we use the standard representation of M on H = ®iEJ(Hi®Hi), then cp is equal to the restriction to M of the canonical trace on ,C(H). Proof. By 6.3.4 (iii), we know that (i) implies (ii). Let us assume (ii), and let us put co = >jE1 diTri. We have then cp(p) = 1. Let us put c(x) _ p(xp) for all x in M. Then, for all w in M*, we have:
(x,e*w) _ (r(x),e®w) _ (Sp ® w)(r(x)(p ®1))
_ (® w)(p 0 x)
by assumption
_ (x,w) Therefore e*w = w for all w in M*. We show that w*c = w the very same way.
Let
be the orthonormal basis of Hi associated to
We have:
i)=Skljdi 1
Therefore, using (* ), we find:
(i (9 p)I'(p) _
iEI
di
1
1:
ic(e
jj) = i (1) = 1
iEI
j,k
From (*) we can also get : J
k
o !. ® i)r(p) = di k
ekI.7
)F(p) =d, 1i(e' ,.7)
(**)
6.3 Discrete Type Kac Algebras
211
And then:
r(4 j)
k oK®i0i)(i0F)r(p) = di(fl
k o r.. ® i ® i)(r (9 i)r(P)
(***)
which yields:
(i 0 Sp)r(ek,j) = di(fl
_,
ik o r ®i)r(Ci ® W)r(p)) = dill
k
(1)1
= Sj,kdi = 5P(ek )1
By linearity, we get (i 0 Sp)r(x) = cp(x)1, for all i in I and all x in £(Hi), and by linearity and normality, for all x in M+. Therefore Sp is left-invariant,
and r(91Sp) c®p. Similarly, using (*), we find, for i1 in I, and 0 < m, I < dii: i di 1 1] K(e4,q)®e4,q
.1
(10 e,1)r(p) =C1®eml) iEI
P,q
di,1E r(eP,1)®em,P P
and then:
C1®ei,m)*r(ek,;) = di(fl i i o
®i ®i)CC1®1 0 At* 2m)(r 0 i)T(p))
=di(fl ij) k oK®i®i)(r(3 didi, 1
E(%
i)((10eit
by (**)
l)I'(P))
i o IC 0 i)r(IC(eit 1)) ®em,p
k
by the above
P
This implies: (i 0 cp)((l 0 ei1)t m)*r(ek ')) = di(fl iti k o x (9 i)r(K(eit 1)) it = di K(i ®flq )F(e M91) k
and, using (***):
i;= didi,K(fl
(i0co)((10em)*r(4,j))
, oK®i(&fl
i )(1'Oi)f(p) C****)
Using once more (*), we get:
r(p)(1 ®ek,j)
eP,q) ®eq,P (10 e
dig
it
P.q
= di 1 E K(ej, q) 0 eqj q
212
6. Special Cases
and then:
r(e"11M* )(10 ek ') =
r(em 1)(1 (9 ek, ') I
= dj,(Q =, i, O KO i ® i)((r 0 i)r(p)(1 ®1® 4,)) by (***)
m
dii(Q i, ;, o rc 0 i 0 i)(r 0 i)(r(p)(1 0 ek,; ))
= di,d- 1
(Q1,1i o r 0 i)r(r(ek,q)) ®eqj by the above computation
This implies: (i 0 p)(r(et/>rn)(1 ®ek,.7 '))
= di/(,fl ;1 ;1 O X. 0 i)r(lc(e4,j))
= didi,(Q jl = didi,(S21,
o
M 41
0
,
by (**) k
(9 SP)((1(9 elVm)F(e' 1j))
by
By linearity, we get, for all xii in ,C(Hii) and yi in C(Hi ):
(i 0 pP)(r(x )(1 0 yi)) = ic((i 0 SP)((' 0 x )r(yi)) Let W be the fundamental operator constructed in 2.4.2 (i), thanks to Sp being left-invariant, and A the bounded linear application constructed in 2.3.5 and 2.4.6 (i). Thanks to 2.4.3, the above formula may be written: (AW(yi) I A(w°)Asp(xii )) = (AW(yi) I A(w)* AW(xi1))
which, by linearity, density, and thanks to 2.4.6 (iv), leads to (HWiii). As Sp
is a trace, we see that (M, r, ,c, 'p) is then a Kac algebra and, as M* has a unit, it is of discrete type, which completes the proof.
6.3.6 Corollary. Let I be a set, and let (M, r, n) be a co-involutive Hopfvon Neumann algebra, with M being abelian and isomorphic to £°°(I). The following assertions are equivalent: (i) The set I can be equipped with a structure of discrete group and then
(M,1',i) is equal to Ha(I). (ii) There exists an element e in I such that, for all x in M:
r(x)(6E®1)_be®x r(x)(1 ®SE) = x 0 &
r(8) _ 11 ic(8i) ®Si iEI
(where Si stands for the characteristic function of {i} over I).
6.4 KreYn's Duality Theorem
213
Proof. It is a consequence of 6.3.5, 4.2.5 (ii) and 6.3.2.
6.3.7 Corollary. Let (M, I', i) be a co-involutive Hopf-von Neumann algebra,
such that M = ®iEI.C(Hi) with di = dim Hi < -boo, and F is symmetric. Then, the following assertions are equivalent: (i) There exists a compact group G such that (M, I', ,) ^r ]HI8(G). (ii) There exists a one-dimensional projection p in the centre of M such
that for all x in M, we have:
f(x)(p®1) = p® x
F(p)
- iEI di 1 1: n(ej,k) ®ek1j j,k
where the ek, j are matrix units for £(Hi ). Proof. It is a consequence of 6.3.5, 4.2.5 (i), 6.3.3 and 6.2.2.
6.4 Krei'n's Duality Theorem 6.4.1 Preliminaries and Notations. Let I be a set, and, for all i in I, let di be in N, Hi be an Hilbert space of dimension di, {e}1<
O LII =Ek118k11 and, pi.(Ek8k)=8. ,C(Hik)*, we have, in D*, II Using the Hahn-Banach theorem, we see that A is then dense in D*.
6.4.2 Definition. With the notations of 6.4.1, we shall say that A is a Krein algebra if:
(i) there is a product * and an involution ° on A, such that A is then an involutive algebra and that there is io in I with di° = 1, such that the unit element of ,C(Hi.) (which is isomorphic to C), noted eZ°, is a unit in A.
(ii) for every i, j in I, there exist k1, ... , kn in I such that, for any yi in £(Hi) and yj in ,C(Hj), the product yi*yj belongs to ®p 1,C(Hk,,). More precisely, there exists min j kp in N such that did j = >p mi, j,kp dkP (so that we
6. Special Cases
214
®,C(Hkp ))), and a unitary may identify ,C(Hi) ®,C(Hj) with Ui, j in C(Hi) ®,C(Hj) such that, for any yi in C(Hi) and yj in G(HQ) and ®p_1(C','iJ,k®
zkp in £ (Hkp) such as yi * Y j = ®p zkp , we have: n
Ui, j (yi (9 yj)U j = (1) (1G(Cmi,j,kp) 0 zkp) p=1
(iii) For every i in I, there exists i' in I such that, for any yi in C(Hi), yo belongs to £(Hii ). More precisely, we have di = dig (so that we may identify
,C(Hi) with C(Hii)) and that there exists a unitary V in C(Hi) such that, for any yi in £(Hi ), we have:
yio =
Vi*(yi)tVi *
where (y)t means the element of C(Hi) whose matrix in the basis {e} is the transposed matrix of yi. (iv) In the decomposition described in axiom (ii), the space Hi,, defined in (i) appears if and only if j is equal to the element i1 defined in (iii); moreover, we have then mi,ii,io = 1. (v) For all un I, we have:
(e,)°*e,,, = Sm,,peio
.
6.4.3 Theorem. With the notations of 6.4.1, let us suppose that there exist on D a coproduct TD and a co-involution XD, such that (D, I'D, KD, SP) is a discrete type Kac algebra. Then, A (considered as a subspace of D*) is a dense sub-involutive algebra of D*, which is a Krein algebra.
Proof. By 1.2.11 (i), for all i, j in I, there exist k1,. .. , kn in I, mi, j,kl , .. ., mi) j,kn
in N, such that didj = Epmi,j,kpdkp (so that we may identify
,C(Hi) ® £(Hj) with ®p_1(Cmi,1,kp ®,C(Hkp ))) and, and a unitary Ui, j in ,C(Hi) ®,C(H,) such that, for all x = ®kxk in D, we have: P
rD(x)(Pi ®pj) = Ui,j ((1r(cmi,j,kl)
(9 xkl )
1=1
Uz j
Therefore, for any (1i in C (Hi) * and (1i in £(H), we get :
(x,Q2*S1') =
(1nia,® xkl) 1=1
,
,j .
(Qi
fli)
which gives that A is a subalgebra of D*, which satisfies 6.4.2 (ii).
('k)
6.4 Krein's Duality Theorem
215
Using 1.2.11 (ii), we get that A is invariant under the involution of D*, and satisfies 6.4.2 (iii).
Let e be the unit of D*; as e is a homomorphism, its support pe is a dimension-one projection in the centre of D; therefore, there exists io in I
such that dio = 1, pE = pi0 and e = fko, where Co is a unit vector of the one-dimensional space Hio ; therefore A satisfies 6.4.2 (i). Moreover, in
the decomposition (*), the index io appears if and only if there exists a dimension-one projection p in C(Hi ® Hj) such that, for all x in D, we have I'(x)p = e(x)p. But then, we have:
P = E(PE)P = rD(pE)n = rn(nf)n(pi 0 P;) But, by 6.3.4, we have:
rD(PE)(Pi 0 pj) = 0 if j 0 i' rD (PE)P(Pil 0 Pi) = Pi where the dimension-one projection Pi has been defined in 1.2.11 (ii). So, such a projection p does not exist if j 54 i', and is equal to Pi (and therefore unique) if j = i'. Therefore A satifies 6.4.2 (iv); as 6.4.2 (v) is given by 6.3.4 (v), the result is proved.
6.4.4 Theorem [109]. With the notations of 6.4.1, let us suppose that A is a Krein algebra. Then, there exist on D a coproduct rD and a co-involution 'CD, such that (D, rD, r, D, SP) is a discrete type Kac algebra, and A (considered as a subspace of D*) is a dense sub-involutive algebra of D*. Proof. Let us consider that A is a subspace of D*; by 6.4.2 (ii) we have then,
for all x=®kxk in D,i,j in 1, 0
((1(m;,j,kp)
u) -
0 xkp )' Ui, j - (4,m 0 f1,8) . i j
P
® xk, ` 2i,m*or,8 k
or:
((D xk) l,rn*f2r,8 k
Uz . ®(l,c(Cmi,l,kp) 0 xkp) Ui,j, `?;,m 0 (1 ,s
1)
P
When taking for j the element io defined in 6.4.2 (i), we see that, for any i in I, we have mi,i,,,i = 1. Therefore, if we put:
ni = >mj,k,i j,k
we shall have 1 < ni < oo.
216
6. Special Cases
Let us put U = ®i,jUi,j; it belongs to C(H 0 H); for x = ®kxk in D, let us put:
TD(®xk) = U* ®(l,c(cnk) ®xk) U kEI
Then, clearly, TD is a normal one-to-one morphism, TD (1) = 1, and we have I
= I I x I I for all x in D. Moreover, using (*):
(rD(e Xk), !,m ®Q ,s) = ((E)xk)7 hhl,m*1r,s) So, by linearity, we get for all x in D and ,fl, 12', ,fln, fl' in A:
(i'D(S), fl ® ,fl') = (x, ,fl*1')
(i'D(S),E,fln ®,fln
(ii)
= x, n
n
and then: E ,Gn*,Gn n
EQn ®On n
(i'D(S),1: On 0 J?nl)
<_ II=II
1: On ®'2n
n
n
Therefore, by density of A in D*, we see that TD(x) belongs to D 0 D. Moreover, the product * being associative, we get that
(i'D 0 Z>rn = (ti 0 rn)rn and so TD is a coproduct over D.
By 6.4.2(iii), we have, for alliinI,0
'l,rn )O = V * . Jli,m Vi
or, for ails = EDxj, in )
®xk) (f2l,m)o = (Vi(x*V* k
Let us now put V = ®iVit; it belongs to ,C(H), and, for x in D, we have:
' D(x) = V(®xk)V*
We have I= 11x1j, and, using (***): (, D(x)*, !,mY _ (1/i(x)*Vi*, (2l,m) _ (x, (fl,m)°)
6.4 KreYn's Duality Theorem
217
and so, by linearity, for all S? in A, we have:
(icD(x)*, 0) - = (x, fl*) which leads to I< 11J?11, and, then, to I (lcD(x), Q)1 < lix li ,i 11; by density
of A in D*, we get that iD(x) belongs to D. Then, by transposing (ii) and (iv), we get that (D, I'D, 'CD) is a co-involutive Hopf-von Neumann algebra; by the same arguments, we see that A is a sub-involutive algebra of D*; let Rio be the unit of A; just by density of A in D*, we see that it is a unit for all D*, which by definition, satisfies, for all x in D: d2io (x)pi° = xpi0
(v)
We have, then: -PD (X) (10 Pio) = (i 0 J?io)(F(X))(1 0 Pio) = X 0 Pio
rn(=)(nio 0 1) =
(nio 0 ,)(r(=))(nio 0 1) = nio 0
In (i), the one-dimensional space Hi° appears if and only if there exists a one-dimensional projection pi, j in C(Hi 0 Hj) such that, for all x in D, we have: pi,j U*
®(1,C(Cmij,kp) 0 xkp) Ui . P
= Ui j ®(1G(Cmi,i,kp) ® xkp) Ui,jpi,j = xi°pi,j P
or, thanks to the definition of I'D: pi, j l'D (x) = I'D (x )pi, j = Rio (x )pi 2j
By 6.4.2 (iv), this happens only if j is equal to the index i1 defined in 6.4.2 (iii), and, moreover, this projection pi)ii is unique.
Let us now consider the projection I'D(pi°)(pi 0 pj ); we have, thanks to (v), and because pi 0 pj is in the centre of D 0 D:
rn(Pio)(ni 0 p;)rn(X) = rv(Xp,o)(p, 0 P;) = R'°(x)rn(pio)(pi 0 P;) rn(X)rD(Pio)(pi 0 P,) = rD(xp=o)(Pi 0 Pj) = R=°(X)rn(nao)(p= 0 n;)
So, it implies that rD (pi0) (pi 0 p j) = 0 if j is different from i', defined in 6.4.2 (iii), and that, for all i, rD(pi°)(pi 0 pi') is a one-dimensional projection in C(Hi 0 Hii ).
6. Special Cases
218
On the other hand, by 1.2.11 (ii), we get that the operator: Pi = dZ-1 E KD (eq)s) ®e8,q q,s
is a projection in £(Hii) 0 ,C(Hi ), which satisfies, by 6.1.1 (vi), for any i1, i2 in I:
(Pjl'D(x) J('Oil it
'p ,'q
''G i2
di11dZ21(P=fD(x), wr
l ,'mi2 =
di 3bi,il
= di
esi
q ,P)
bi,i2 rD(x),
0 wei2
1,m
W
®wei
16i)il bi,i2 (XI
m
m
by 6.4.2 (v)
= di 16i,il 6i,i2 6rn,gQio (x)
By putting x = 1 in the preceding calculation, we get:
, q1 m
(Pi l ,'Oil it ®D i2 i2) = dZ 16Z,Zl 8i,i2 V m,q p
and therefore: (PIrD(X), ,rloil ;l ® Q i2 i2) = (P., n'. it ®D i2 p 141 j , m p
i2
)Qio(x)
which, by linearity and continuity, leads to:
PzrD(x) = Qbo(x)Pi Therefore we get, by the unicity of the dimension-one projection pi.,is rD (pi o) (P F (9 pi) = Pi
And, as FD (pi o) (p j (9 pi) = 0 if j is different from i', we have:
d-1 E 1 D (eq,s) 0 eq
Pi =
rD (pio) _ Z
Z
(viii)
q,s
and, thanks to (vi), (vii), (viii), pio satisfies the conditions of 6.3.5, which completes the proof.
6.4.5 Corollary. With the notations of 6.4.1, the following propositions are equivalent:
(i) A is a Krein algebra.
6.5 Characterisation of Compact Type Kac Algebras
219
(ii) There exist on D a coproduct 1'D and a co-involution IUD, such that (D, TD, lD, Sp) is a Kac algebra of discrete type. Then, A (considered as a subspace of D*) is a dense sub-involutive algebra of D*.
Proof. We have proved in 6.4.3 that (ii) implies (i), and in 6.4.4 that (i) implies (ii).
6.4.6 Corollary (Krein's Theorem [83]). With the notations of 6.4.1, the following propositions are equivalent: (i) A is an abelian Krein algebra. (ii) The set of the characters on A which are continuous with respect to the norm of D*, is, for the weak topology of D, a compact group G. The set I may then be identified to the set of (classes of) irreducible representations of G. For all v in I, let H be the finite-dimensional Hilbert space such that is the algebra generated by v(G); then A may be identified to the algebra of
functions s --+ (v(s)e I ii), for all v in I, , i in Hv. Proof. Using 6.4.5, 4.2.4, 6.3.3 and 6.2.2, we see that (i) is equivalent to D being isomorphic to some ,C(G), with G compact; then, the involutive algebra D* is isomorphic to A(G), and, by 4.3.3, G is isomorphic and homeomorphic
to the spectrum of D*; as AG = ®Irr Gv by 6.2.7 (iii), we see that the set I is equal to Irr G, and, for all v in I, H is the finite-dimensional Hilbert space such that £(H,) is the algebra generated by v(G). Moreover, via the isomorphism between D* and A(G), A may be considered as a subspace of A(G), precisely the space generated by the functions s -* (u(s)er e ), for all v in Irr G, {r} being a basis for H,,, which completes the proof.
6.5 Characterisation of Compact Type Kac Algebras 6.5.1 Notations. Let (M, I', rc) be a co-involutive Hopf-von Neumann algebra. Let J be the set of (equivalence classes of) finite-dimensional representations
of M*, with a unitary generator in the sense of 1.5.2. Let v1 be in J and v2 C v1, then, by 1.5.4 (ii), v2 belongs to J. Let J' the subset of J formed by irreducible representations. As the trivial representation belongs to J', this set is not empty. Let v be in J, the representation v defined, for any w in M*, by v(w) = v(w o ,c)t (where t stands for the transposition), belongs to J, by 1.5.9. It is also clear that, if v is in J', so is v. Let us consider J' as equivalent classes, and let us pick up, in each class, a representation v which operates on a Hilbert space H such that £(H,,) is the von Neumann algebra generated by v. Let us write I for the set of such v's. We shall denote by p the representation ®EIv.
220
6. Special Cases
By 1.5.4, p has a unitary generator and the elements of I being two by two disjoint, the von Neumann algebra D generated by p is ®VEIC(Hv). We shall then use all the notations of 6.4.1. Each element O , of C(HY )* can be isometrically identified with an element of D* which shall still be denoted by 8Y . As we have, through this identification, for all w in M* :
(p(w),Ou) _ (V(W),O)
we get, in M, for all eY in .C(H,)*: v*
p*
Moreover, let us recall that, for all v in I, as v is non-degenerate, the mappings v* are one-to-one.
6.5.2 Theorem. With the notations of 6.5.1, we have: (i) the algebra D can be equipped with a coproduct 1'd and a co-involution lcd such that, for all w in M*: rd(P(UJ)) = SAP X P)(w)
kd(V(U')) = Pew o k)
and (D, rd, lcd) is a co-involutive Hopf-von Neumann algebra. (ii) the subspace A is a Krein algebra; if considered as a subspace of D*, it is a dense sub-involutive algebra of D*. (iii) the quadruple (D,rd,icd,p) is a discrete type Kac algebra.
Proof. Let fit, v be in I. By 1.5.5, y x v belongs to Y. Decomposing this representation into irreducible components, we find a unitary Uµ,,, belonging to £(HH 0 HY), integers mµ,Y,,.k and elements ik in I, such that, for any w in M*, we have: UP Y(I x v)(w)U,,Y
®(1G((Cmµ,v,rk) 0 k)(w) k
It implies, for any w in M,:
Ix v)(w)II <_k sup I< sup I= IIp(w)II vEJ As (p x p)(w) is a direct sum of elements of the form (y x v)(w), we have: I
P x p)(41 :5
I
As(vx1)(w)=v(w)forallwinM*andallvinI,wehave m,,i,v=1; therefore if we put n,r = Eµ,Y mµ,,,,,r, we have 1 < n,r < oo.
6.5 Characterisation of Compact Type Kac Algebras
221
Let us put U = ®µ,vEIUP,v in £(Hp 0 Hp), and, for x,r in £(H), let us define rd by:
rd ®xir = CrU* irEI
®(1I(Cn,r) 0 X7r)
UcY
irEI
For any w in M, we have: cid(P(w)) = ;I'd U*
ED 7r(w) 7rEJ
by
ED(1,C(Cn,) 0 -7r (w)) U AEI
= ®u;,v II,v
ED(jL(CinjuPPIT) 0 lr(w))
Utl,v
7rEI
= (D (xv)(w)
by (*)
µ,v
=(PxP)(w) Therefore, by continuity, we find that rd(x) belongs to D 0 D for all x in D.
Using the definition it is immediate to check that (rd 0 i)rd = (i 0 rd)rd. Finally it is clear by (**) that I'd(1) = 1 and that I'd is injective. Let v be in I. The representation v(w) = v(w o rc)t is in J'; so there exists an element M in I and there exists a unitary V in C(H,) such that, through the natural identification of Hv and Hµ, we have, for all w in M*:
(w) = Vvv(w)V' Let us put V =
(***)
in £(Hp) and, for x in ,C(H',), let us define:
Icd ® xv = V ® xy V*
(****)
vEI
vEI
For all w in M*, we have: Kd(P(w)) = Kd @ V(W) \v EI
= V ® v(w o rc) V * vEI ® Vvv(w o 10 Vv vEI
_ ® v(worc) vEI
= p(woIC)
by (****)
222
6. Special Cases
Therefore, by continuity rcd(x) belongs to D for all x in D and it is clearly an involution. By (****) we have r,,d(1) = 1. Let 8 be in D*, w in M*, we have: (p*(Oo!sd),w) = (p(w),90!'cd) = (Pcd(p(w)), 8)
= (p(woK),9) _ (p*(9),w o K)
_ (isp(O),w) and, thus, we have, for all 8 in D* : p*(8olCd)=rCp*(0) From this result we can get, for 81, 82 in D* : ((lCd (9 lcd)rd(P(w)), 81 (9 82) _ (c(lCd (9 lCd)(P X P)(w), e1® 82)
_ ((P X P)(w), 82 0 1d ®81 0
lCd)
_ (p*(92 ° PCd)P*(81 ° rud)7 w)
_ (,gyp*(82),cp*(81),w)
= (IC(p*(91)p*(92)),w) = (p*(91)p*(92),w o IC)
= ((P X P)(w), 010 02) _ (crd(p(w o n))' 81 0 82) _ (cl'd(Kd(p(w)), 010 82)
and so, we get: (xd 0 lCd)rd = crdlcd which completes the proof of (i). Moreover, for x = ®,1.x, in D, we get, using (*) and (* *) : µ (U;,P((1C(CmivTk) ®x1rk) UU,v)`1rs 0 Qrm P
(crd(xir)Q',s p
Qr rn
v
®X7 ,14 rn * `4r,s and so, A is a subalgebra of D* which satisfies 6.4.2 (ii). Moreover, the element
(l1, associated to the trivial representation 1, belongs to A, and is a unit for D*. Moreover, we have, for all v in I, using (****):
(Vv4Vj,Qr) = (VxiV* flr a)
_ K Icd
x71.
ED x71., (QT 8)O 7r
r,a
6.5 Characterisation of Compact Type Kac Algebras
223
and: Vv '
s,r Vv = (JGr)8)°
''
which shows that A is globally invariant under the involution of D*, and satisfies 6.4.2 (iii).
By 1.5.7,wehave, forallvinI, 0
and then, thanks to the injectivity of v*, we get that A satifies 6.4.2 (v). The trivial representation 1 appears in (*) if and only if it exists a onedimensional projection ppy in .C(HI2 0 Hv) such that: P,O,v(,u X v)(w) = (p X v)(w)Pµ,v = w(1)Pp,v
(*)
Let {4}iq,3 aq,86q 0 e ' in Hµ ®H,, such that Eq,8 laq,s 12 = 1 and, for all r, in Hµ 0 H,,: Pr' =(77 1
)6
As in 1.5.8, let us put xT,q = p* (Qµ ), xt$8 = v* Eq ,Er Then, for all w in M*, we have, using 1.4.3:
Evt ).
((Y X v)(w)(eq ®Es )1 er, ®et) = ((p X v)* (flEq )Cr ®JGE45 ,E ), W)
(x',qx',s,w)
and, thus: (p(p X v)(w)(eq ® ee) 16r ® Et )
_ ((µ X v)(w)(eq 0 6') I P(er ® Et )) (µ X v)(w)(eq
Es') qIE,s/ ar,tagIisle qI ®e8
= ar,t > agi,six ;gx8 81W q',s'
The same way, we get: ((p X v)(w)P(e®®c') I eµ ®Et ) X V) (W) E aq,8aq,,8,eq gi
g
,
8i
v
= cxq,B E aqI ,8i x r,q, x t,e, , W qI $a/
224
6. Special Cases
and: ('(l)p(eq ® Cs) I er (94t) = w(1)aq,sar,t
The relation (*) is therefore equivalent to the existence of mn complex numbers {aqua} 1
E kq,I2 =1 q,8
x9 is aqI ,8Ixq q )q
=a q,s 1
Vq Vs
, xt,ai
= art 1
Vr Vt
q1 8 agi'8IxT
q
gi )si
From (**) we deduce: aqua
(xI )* = 9 )q
agI,8i (XqII
)q
)* xµ xa q 2q
qI ,s1
and: aq)s(x11
q )9 )*
=
q
E aq')8I >(xµ, )*xµq Iq q N9
,
aqI
0qI qIIx e
aqI/ Is
xv
q',s1
_
xa ,8
q
q/ 31
by 1.5.7
8
a/
and also: CYr't (xr,qIX qI a
ar,t(xpr r
ay,s `xr,qIXxr,q'
xt,s'
v µ ) *E_ aq',s' II(xqii)*xi,q) t8 r r q' X81
aqI,,I bq',q" xt a, qI 81
_
a.Jll sl xt 81
therefore (**) implies: 1q)* _ q
agl,SIxsi,8 8'
E aqIxt a, r
at
by 1.5.7
6.5 Characterisation of Compact Type Kac Algebras
225
Conversely, let us assume (***). We have: Eaq/isXq
lx8 '9 _ 1: ar,8Xq l(xr l)* lq
,q
)q r
81
and: aqi 81 xµ /X
xµ , (x .i , )*
ar,8
q)q
g1,s1
= aq,81
)q
q$q
r
The same way, we have: agi,81 x
µ v II x81,8 = q q
µ
µ
aq,sx q qi1(x q
)* 1q
q
81
and: agi,8/ xA I -ixa ,8 = s1)g11
q )9
1: aq,s
q"
q
XP ; (x1; )* q Iq q )q
= aq11 X81
And then we see that (**) and (***) are equivalent. It implies that (*) is, in fact, equivalent to the existence of complex numbers {aq,s} J
=1
EI q,s
aq,s(xµ q )q
)* _
q
E
aq1
xs
s1
ar,t(xµ
l)*
,q
=
aq/ ,8Ixt 8,
r
The projection
of (*) being then associated to the vector >
q,8
aq,seq ®Es .
qq
Let then U be the mapping from Hµ to H defined by (U4 I es) = aq,s . We have: (p(w)U*ev
Eq) =
aq,s (A(w)eq 1'e' q) s (A
col q
q
Nis \(xq ,q)*' q
E agI,8I (xa ,8) w) 8' I
aq',s'(v(w)&v 14s )
81
v(w)e" E agI,8I E8 (v(w)e ' U41)
226
6. Special Cases
Therefore, we have '(w) U* = U* v(w) for all w in M*. Taking the adjoints, we get Uj(w) = v(w)U for all w in M*. Therefore: U*Up(w) = U*v(w)U = µ(w)U*U
And, j being irreducible, we get that there is c in C such that U*U = c1H. Then m = n. But, we have: aq,s E aq,BIE8, _
UU*Eg = q
s1
aq,saq,B'Eµi q J81
Therefore, for all s, we have Eq faq,s 1 2 = c and then:
1 = 1: laq,s 12 = nc q,s
which implies c = n-1 .
Let us put V = n-1/2U. Then V is a unitary which makes v and 9 equivalent. Conversely, let us assume that there exists a unitary V making v and µ equivalent. Let [aq,8] 1
E kq,sI2 =1 q,s
Furthemore, it is clear that UP(w) = v(w)U implies: ar,t (xr ,q1)* _ E (xgi,81 xt 81 r
81
and that j(w)U* = U*v(w) implies: iiq,'(XPI
q )q )*
q
_ E aqII xs, 81
and, therefore, we get (*).
Let us assume that the multiplicity mp,v is greater than 1. Then, there exists two orthogonal projections p and q satisfying (*), that is two orthogonal vectors C = Eq,8 aq,BEq ®Ee and rj = E q,. ,Qq,s44 ®Es which are of norm one and satisfying (***). Therefore there exists two matrices U1 and U2 satisfying, for all w in M* :
p(w)U1 = Ul v(w) ,(w)U2 = U2 v(w)
and U1 9(w) = v(w)Ul
and U2j(w) = v(w)U2
6.5 Characterisation of Compact Type Kac Algebras
227
It implies U2 U11(w) = U2 v(w)U1 = U2 Ulµ(w). And j being irreducible, there exists c' in C such that U2 U1 = c' 1. But, we have: U2*U16184
=
13q,s' E aq,e'e8 q
/3q,8aq,81 E8
q,8'
81
which implies, for all s:
1: 3q,saq,8 = c q
and:
0 = (6 177) = E aq,sPq,s = nc q,8
Therefore we have c' = 0, and then U2 U1 = 0, which is impossible because U2U2 = n1, by the above, which completes the proof of (ii). Then (iii) is just a corollary of 6.4.3, because we have, with the notations
of 6.4.3, ID =cI'd 6.5.3 Definition. Let H = (M, I', ,c) a co-involutive Hopf-von Neumann algebra. We shall denote by D(H) = (D, I'd) 'ed, Wd) the discrete type Kac algebra defined in 6.5.2. If K = (M, I', ,c, Sp) is a Kac algebra, the discrete type Kac algebra associated to the triple (M, T, ic) will be denoted by D(K). By definition, using 4.6.8, it is clear that D(K) = D(W*(K)).
6.5.4 Proposition. Let K be a compact type Kac algebra. Then, the discrete type Kac algebra D(K) associated by 6.5.2 is the dual Kac algebra K. Proof. By 6.2.6 (ii), it is clear that the representation p defined in 6.4.4 is then nothing but the Fourier representation. The definitions of rd and nd given in 6.5.2 are then the same as the definition of F (3.2.2 (iv)) and k (3.3.1). The result comes then from 2.7.7.
6.5.5 Theorem. Let G be a locally compact group. We shall denote by Gd the discrete topological group having the same underlying group. Then, we have: D(Hs(G)) = D(W*(G)) = Ka(Gd)
-
Proof. By construction, D(H3(G)) = (D(W*(G)) is built up with finitedimensional irreducible representations of £(G)* = A(G). By 4.3.3, they are nothing but the points of the group G. The representation p, as defined in 6.5.1, is therefore, for f in A(G) or in B(G), given by:
Xf) = sEG 0 AS)
228
6. Special Cases
It is then clear that the von Neumann algebra generated by p is £°O (G). Also, we have seen in 2.6.6 (ii) that the Kronecker product of the characters
s and t of A(G) is equal to the product is (s, t E G). Therefore, for f in A(G) or in B(G), we have c(p x p)(f) = I'a(f ). By 5.5.7, we then have D(1H[s (G)) = Ka(Gd), which completes the proof. 6.5.6 Theorem. Let G be a locally compact group. There exists a compact group
bG, unique up to an isormorphism, and a continuous morphism p : G -+ bG such that for every compact group K, and every morphism a : G --+ K, there is a morphism 0 : bG --+ K such that ,a o p = a. In that situation, we have: D(Ha(G)) = Ks(bG)
and p(G) is dense in bG. Proof. Let us compute D(]H[a (G)). By construction, it is the von Neumann algebra generated by the representation p = EBVE Jv, where J is the set of (equivalence classes) of finite-dimensional irreducible representations of G. Let vl, v2 be in J. For any fin L1(G), 1 in 02 in (A,,2), we have: ((111 X v2)(f), e1 (9 02) = (V1*(01)112*(02), f)
by 1.4.3
= (V2*(02)V1*(01)J)
because L°O(G) is abelian ((V2 X v1)(f), 02 0 01) (';(v2 X VOY ), 010 02)
Therefore, by linearity and continuity, we have vl x v2 = c(v2 X vl ). It implies that c(p x p) = p x p and, therefore, that the Kac algebra D(1H[a (G))
is symmetric. By 4.2.5, there exists a locally compact group bG such that IID(HQ(G)) is equal to K8(bG); moreover, as it is of discrete type, bG is actually compact by 6.3.3 and 6.2.2. Finally, bG being the intrinsic group of IlD(Ha (G)), it contains the subgroup {pG(s), s E G}, where PG is the unitary representation of G deduced from the representation p associated to Ha (G) by 6.5.1. Thus, PG is a continuous morphism from G to bG, from what we get by 5.1.4 (i) the existence of an H-morphism Ks (p) from W * (G) to W*(bG) such that: Ks(#'7rG(S)) = -7rbG(P(s))
Now, let a : G -+ K and Ids (a) the ]H[-morphism from W * (G) to W * (K) defined by: K.q(a)(7rG(s)) = '7rK(a(S))
6.5 Characterisation of Compact Type Kac Algebras
229
Because of K being compact, by 6.2.7 (ii), saK is an ]HI-isomorphism from W * (K) to H8 (K), and AK is the sum of finite-dimensional irreducible representations vi, by 6.2.7 (iii). For every i, vi o a is a finite-dimensional representation of G, then there exists 7ri : AP = £(bG)) -+ £(H,,) such that: ri(AbG(P(S))) = Vi 0 a(s)-
There exists therefore 7r : D(lHIa(G)) --+ £(K) such that: ,7r(AbG(P(s))) = AK(a(s)) = SAK o WK(a(s)) = saK o K8(a)(lrG(s))
As bG is compact, SAbG is a IHI-isomorphism from W*(bG) to K8(bG) which, by definition, is equal to D(1HIa (G)). As 7r is an IHI-morphism, we get, using 5.1.4 (iii), that there exists a continuous morphism /3 : bG K such that and we have, for all s in G: K8 (0) = 7r o SAbG , saK o SAK o K8($) o Ks(P)7rG(5) = 7r o SAbG o 7bG(P(s)) = 7r o AbG(P(s)) = S,\K o Ks(a)7rG(s)
or, as SAK is injective:
Ks(#)Ks(p) = K,(a) that is, by 5.1.4 (iii):
/3op=a
We can also deduce p(G) = bG which completes the proof. 6.5.7 Proposition. Let (M, 1', n) be a co-involutive Hopf-von Neumann alge-
bra. Let p be the representation defined in 6.5.1, D be the von Neumann algebra generated by p, H be a Hilbert space on which D has a standard representation, (D, 1'd, Kd, pd) be the discrete type Kac algebra associated as in 6.4.5 (iii). Let us recall that, by 6.3.4, we have 5Pd = trH I D. Let us assume that the set of elements w in M* such that p(w) be a Hilbert-Schmidt operator is dense in M*. Then, there exists a morphism 4$ from D to M such that 4$(x(9)) = ip*(9) for all 9 in D* when A is the Fourier representation of the Kac algebra D. Moreover, 4$ is an H-morphism from (1), 1'd, Kd) to (M, T, ). Proof. For all w in M*, 91, 92 in M*, we have: (P* (01 *02))
(P(W), 01 *02)
_ (rdp(w), e1 (9 e2) _ ((P x P)(w), B2 0 B1) _ (p*(92)p*(O1),W)
by 6.4.5 (iii) by 1.4.3
230
6. Special Cases
then: p*(e1*e2) = p*(92)P*(O1)
The mapping r p* from D* to M is then multiplicative. We have also, with 9 in M* : (P(W), 0')
(P*
_ ( KdP(w)*, B) 0
(P(Lij
0 re), 0)
(P(CA;) 1
0) -
by 6.4.2 (i) by 1.2.5
(P* (0) CD) (P* (0) *' Lij)
then:
by 1.2.5
p*(eo) = p*(e)*
And so rcp* is a representation of D. Moreover, we have: ((ICP*)*(W),e) = (icp*(9),w) = (p*(9),w 0 10 = (p(w 0 10, 0) = (ICO(W),e)
Therefore we have for all w in M* : (icp*)*(w) = KO(w)
As, by 6.3.3 and 6.1.4, 'Pd is a lcd-invariant trace, we see that we have:
{weM; P(w)E
91.p}
(icp*)*(w)E'Yt,}
and, then, by 3.1.3, x p* is quasi-equivalent to a subrepresentation of A. So, there exists : D -+ M such that, for all 9 in D*, we have: (A(9)) = rcp*(9)
and we have, too, iP(1) = 1. Then, we have, for w1, w2 in M* :
(r(A(9)
W1
w2) = (ficp*(9),wi (9 w2) = (p*(9),w2 0 rc*w1 0 rc) = (p(w2 0 K*W1 0 rc), 9)
= (p(w2 0 rc)p(wl 0 rc), 9)
_ ((l'cp*)*(w2)(icp*)*(w1),9) _ ((icp* x rcp*)(9),w2 (9 WO
by 1.4.3
_ ((!P oA xiP oA)(9),w2®w1) _ ((iP ®iP)(A x A)(9), w2 ®wl) _ ((p (D !V)rda(e), wi (9 W2)
by 1.4.5 (ii) by 6.4.5 (i)
6.5 Characterisation of Compact Type Kac Algebras
231
therefore, by continuity, we have: T4$ = (4$ 0O)Td
We have, for any0inD*, by 3.3.1 and 6.3.2: 4$(kcdA(e)) = 4$(x(9 o lcd)) = rip*(e o rid) = p*(9) = mP(A(e)) Therefore 4$ icd = rc4$, which completes the proof.
6.5.8 Theorem. Let (M, F, rc) be a co-involutive Hopf-von Neumann algebra. Let p be the representation defined in 6.5.1, H be a Hilbert space on which
p(M*)" has a standard representation. Then, the following assertions are equivalent:
(i) The representation p is faithful and the set of elements w in M*, such that p(w) is a Hilbert-Schmidt operator in H, is dense in M*. (ii) There exists a state So on M such that (M, F, rc, Sp) is a compact type Kac algebra. Then, p is the Fourier representation of (M, I', K, cc), and H is isomorphic to Hip.
Proof. Let us assume (i). We have the morphism 4$ as defined in 6.5.7. As p is faithful, the representation rcp* generates M and 4$ is surjective. Let P be the support of !. We have (M, I', rc) = (D,Fd, Kd)p. Now, (D, I'd) lcd, Wd) is a compact type Kac algebra and so is (D, I'd , Kd , SPd )P which is (ii). Let us assume (ii). Let K be the compact type Kac algebra (M, I', n, ip). By 6.2.6 (ii), the representation p is equal to the Fourier representation of K and, by 4.1.3 (ii) and 6.2.5 (iii), we have (i), which completes the proof. 6.5.9 Corollary. Let (M, T, rc) be an abelian co-involutive Hopf-von Neumann algebra. Let p be the representation defined in 6.5.1, H be a Hilbert space on which p(M*)" has a standard representation. Then, the following assertions are equivalent:
(i) The representation p is faithful and the set of elements w in M* such that p(w) is a Hilbert-Schmidt operator on H is dense in M*. (ii) There exists a compact group G such that (M, T, K) ^j Ha(G). Proof. It is a consequence of 6.5.8, 4.2.5 (ii) and 6.2.2.
6.5.10 Corollary. Let (M, F, x) be a symmetric co-involutive Hopf-von Neumann algebra. Let p be the representation defined in 6.5.1, H be a Hilbert space on which p(M*)rr has a standard representation. Then the following assertions are equivalent: (i) The representation p is faithful and the set of elements w in M* such that p(w) is a Hilbert-Schmidt operator on H is dense in M*.
232
6. Special Cases
(ii) There exists a discrete group G such that (M, I', r) ^ H, (G). Proof. It is a consequence of 6.5.8, 4.2.5 (i), 6.3.3 and 6.3.2.
6.6 Finite Dimensional Kac Algebras 6.6.1 Theorem. Let K = (M,1', K, W) be a Kac algebra. The following assertions are equivalent: (i) The Kac algebra K is both of compact and discrete type. (ii) The von Neumann algebra M is finite-dimensional. In this case, if H is a Hilbert space on which M has a standard representation, we have cp = trH I M. Proof. Let us assume (i). By 6.3.4 (iv), we have co = trH I M. As, by definition co(1) is finite, we get dim H < oo and (ii) follows. Let us assume (ii). Then, we have dim A(M*) = dim M. Therefore, A(M* )
is equal to its closure M, which implies that A(M*) contains the unity, and then that M* is unital. Thus, we get that dim M = dim M and that K is of discrete type. By iterating the argument we find that K is also of discrete type and using 6.3.3 we get (i). 6.6.2 Lemma. Let (M, T, i) be a co-involutive Hopf-von Neumann algebra. Let us assume that M is finite-dimensional, therefore equal to ®i 1,C(Hi) with di = dim Hi < oo. Let pi be the projection on Hi. The multiplication on M, induces a linear mapping m : MOM -+ M such that m(a ® b) = ab for all a, b in M. Moreover ic 0 i is a linear mapping from M 0 M to M ® M. For , ri in H = ®Z 1 Hi, x in M, we have: n
(m(ic 0 i)r(x)e I q) =
n
(m(i 0 )r(x)e I77) =
di
E:11 ki *Q i=1 k=1
k
x
di
E L llui *11 ki k i=1 k=1
x
where the {}1
Proof. The vectors {}1
((mX)Iii) = (apbpe I rl P
6.6 Finite Dimensional Kac Algebras
233
(bpeta)
= P
= 1: J>p6I6k)Cap?7I P
ilk
P
ilk
=
C(a® ®bP)( k ®6) 171 ®6k)
E(X(6ki 0 0177 0 60 ilk
It implies that: sk 77
om=
E ilk
k
k
and we have: SA,,7
omo(c®i)of =J:(i),7®Q, )o1' ilk
ilk
771k
k
E flo ilk
'1'k
k
The same way, we get:
,7omo(i®,)or=1:Q ilk
k
*,fl° i 774k
which completes the proof.
6.6.3 Lemma. To the hypothesis and notations of 6.6.2, we add up the assumption that it exists a homomorphism e from M to C which, considered as an element of M*, is a unit of the algebra M* and verifies also, for all x in M: m(k 0 i)f(x) = C(x)1 (So) e is a co-unit, and # an antipode, in the sense of [1] or [154].) Then, we have, for all x in M: Ek(=) = E(=)
M(i 0 1C)r(x) =,F(X)l
Proof. Since a is the unit of M*, we have E° = e. As e is a homomorphism, it is positive, therefore: EK(z) = e(s*)- = E(=) and we have (i).
6. Special Cases
234
It is clear that me = icm(x 0 ic), therefore we have for all x in M: M(K 0 K)(K 0 i).r(x)
M(i 0 1C).r(X)
K,nc(w 0 )r(=)
r.)4;.r(x)
#%M(i
= Km(i
by 1.2.5
K)(K 0 w)P(rc(x))
t)r(K(=))
Km(K
= k(6(x)1) =e(x)1
by assumption
which completes the proof of (ii).
6.6.4 Proposition. Let (M, I', i) satisfying the hypothesis and notations of 6.6.3. Then M* is a Krein algebra. Proof. Let pE be the support of e. As pE belongs to the centre of M, there exists io, 1 < io < n, such that Hio = C and p,.- = pio; so M* satisfies 6.4.2 (i). By 1.2.11(i) and (ii), we get that M* satisfies 6.4.2 (ii) and (iii). If in Hj , for j i, we have fl i 4 = 0 for all 1 (1 < 1 < dii ); therefore we get :
,fly= i omo(i (9 i)or= q' I
il k
_
lk
Di
fl°= I) P
q
by 6.6.2
i1
2C
i P
by the preceding
P
and then: fl°i P
i
l) P
4i p
(x)
q
l
O M O (IC 0 0 O r(x) =
I Qh = 6q,le(x)
which means that M* satisfies 6.4.2 (v). Let be a unit vector of Hi ® Hj, decomposed into e _ Eq,8 aq,sej 0 and such that P(x)C = e(x)e. We have:
r(x) 1: aq,86q ® es
6p ®6r
=,6(x)ap,r
q,a
or:
aq,8 (r(x), q,8
n q
p
r
= c(x)ap,r
6.6 Finite Dimensional Kac Algebras
then: q,s
aq,sR q' P= * 12 > ; = apse
and also: aq,B ° q,8
: * ` `q sP *
tJs r = ap,r
Oi
Sj
p
Then we have:
q,8
aq,8 E J°i1 p= * flgi
; = Eap,rno.
*fl 1
P
q
p
.
P
Thanks to 6.4.2 (v), we have: no
ap,r w k
p
1dP
=
a1,8 s168 8
26
Let V be the linear mapping Hi - Hi defined by (V6' I
For all x=$xi in M, r,s such that 1
(xV* rI d) = xi
I
1
ap,r (xi 6p I j ) p
E ap,r(xid I 4) P
X, E ap,r f?l
t
P
(x,
anoal) 8(x,
8
P
no
;) 6s , r
a1,8(x(x*)Cs I d)8
_
a1,8(x(xxr I s)
8
= I VQ) = (V*x(x)d I ch
d, we have:
235
236
6. Special Cases
and so we get, by linearity: xiV* = V*rc(xi)pj
from which we infer that !C(pi) = pj and that the operator V is the operator Vi defined in 1.2.11 (ii); therefore, the vector is equal to qq ® V q and, so, there is a one dimensional projection p in Hi ®Hj such that T(x)p = e(x)p
for all x in M if and only if j = i', and this projection is then unique. So, M* satisfies 6.4.2 (iv), and the result is proved. 6.6.5 Kac-Paljutkin's Theorem ([75]). Let (M, I', rc) be a co-involutive Hopfvon Neumann algebra such that M is finite dimensional. Then the following assertions are equivalent:
(i) There exists a semi-finite, faithful, normal weight on M+ such that (M, T, r,,, W) is a Kac algebra.
(ii) There exists a homomorphism e from M to C which, considered as an element of M*, is a unit of M*, and which satisfies, for x in M: M(Ic 0 i)F(X) = E(X)l
The quadruple (M,f,e,i'c) is then a Hopf-algebra in the sense of [1] or [154].
In that situation, the Haar weight is then equal to trH I M, where H is a Hilbert space on which M has a standard representation; it is therefore a finite trace. Proof. Let us assume (i). Let A be the Fourier representation of K, let us put x i, j = A* (w ; i ), where space Hip. By 1.4.2, we have:
is a basis of the finite-dimensional Hilbert
T(xi,j) = E,, xi,k 0 xk,j k
* K(xi,j) = xj'i
and then: m(rt' ®Z)r'(xi,j) _
xlc,ixk,j k
As A is a finite-dimensional representation with a unitary generator, by 1.5.7, we have: m(ic 0 i).P(xi,j) = Si,j1 And, by 6.6.1, we obtain that K is of discrete type. Therefore, by 6.3.5 there exists a homomorphism e in M* which is a unit of M*. Then, we have:
C(xi,j)=(e,A
(xj,i))=(A(e))w;,i)=6i,j
6.6 Finite Dimensional Kac Algebras
237
Then, for all i, j, we have: m(x ® i)r(xi,j) = E(xi, j )1 and, every element of A * (M*) being a linear combination of x i j, by linearity we get (ii).
The implication (ii) = (i) is given by 6.4.3 and 6.6.5. 6.6.6 Definitions and Notations ([1], [154] ). Let A be a complex algebra with
unit 1A. The product and the unit allow us to define linear mappings, mA from the tensor product A ® A to A and 77A from C to A, by, for any a, b in A and a in C: mA(a(9 b)=ab 77A(a) = a1A
A coproduct over A is a multiplicative linear mapping d from A to A ® A, such that: The dual space A*, equipped with the transposed d* : A* 0 A* ---> A*, is then a complex algebra. Moreover, let us suppose that there exists on A a co-unit e, i.e. an algebra morphism A -> C, such that:
(e0 i)d=(ice)d=i which implies that d is injective. Then, by transposing, we get an application e* from C to A*, which gives an unit &* (1) to the algebra A*. Moreover, the transposed mappings mA and qA are, respectively, a coproduct over A*, and a co-unit on A*.
Let us suppose now that there is an antipode j, i.e. a linear application j : A -+ A such that: mA(7 0 i)d = MA(z 0 .7)d = 77AE
Then, by transposing, we get that j * is an antipode on A*.
The quadruple (A, d, e, j) is called a complex Hopf algebra; then the quadruple (A*, mA, 77A, j*) will be called the dual complex Hopf algebra. If A is finite dimensional, it is clear that the bidual complex Hopf algebra is equal
to the initial one. If A is an involutive algebra, and d, e, j preserve the involution, we shall say that (A, d, e, j) is a *-Hopf algebra.
6. Special Cases
238
6.6.7 Lemma ([1], [154]). Let (A, d, e, j) be a finite-dimensional complex Hopf algebra. Then, we have:
cd(j®j)d=dj (i) (ii) The application j is antimultiplicative. (iii) ej =6 j(1) = 1
(iv)
.
Proof. As mA®A(i®i®s) = (mA®i)(i®c)(i®mA®i), for any x in A, we have:
0 j)d)(d 0 i)d(x) = (mA 0 i)(i 0 s)(i 0 mq 0 i)(i 0 i 0 J 0 j)(' 0 i 0 d)(i 0 d)d(x)
'mA(&A(%(9 i (2) s(J
= (+^a 0 i)(i 0 t)(i 0 MA 0 i)(i 0 i 0 7 0 j)(+ 0 d 0 i)(i 0 d)d(x) by 6.6.6
(mA 0 O(z 0 s)(z 0 mA(z 0 j)d 0 j)(i 0 d)d(x) = (MA 0 z)(z (9 t)(i (9 CIA (9 j)d(x)
by 6.6.6 by 6.6.6
= MA(a 0 j)d(x) 0 1A = E(x)(lA 0 1A)
by 6.6.6
= (MA 0 z)(z 0 s)(z 0 77aE 0 .i)d(x)
Moreover, as mA(DA(d 0 d) = d o mA, we have, too:
MaOA(d.7 0 d)d(x) = d o m,qU 0 i)d(x) = d(e(a)lA) = E(x)(lA 0 1A) And then, as: (mA(&A 0 i 0 z)(d9 0 d 0 4;(7 0 j)d)(d 0 i)d(x) = 1AOA 0 (e 0 4;(j 0 j)d)d(x) = lAOA 0 c(.9 0 .7)d(x)
and: (z 0 i 0 mAOA)(di 0 d 0 4;(.7 0 j)d)(d 0 i)d(x) = (dj (9 e)d(x) (9 lA(&A = d1(x) (9 lA(&A
we get:
s(.9 0 j)d(x) = MAOA(mAOA 0 i 0 z)(dj 0 d 0 4;(.7 0 j)d)(d 0 i)d(x) = MAOA(z 0 i 0 ma(&A)(di 0 d 0 c(j 0 j)d)(d 0 i)d(x)
= dj(x) which gives (i).
6.6 Finite Dimensional Kac Algebras
239
Let us now apply (i) to the dual Hopf algebra; we get, for all f in A*:
s(j* ®j*)m* (f) = M*
j*(f)
which, for all x, y in A, implies:
f(i(v)i (=)) = ((j' i')mn(f),y (9 s) = J(i(=v)) which implies (ii). We have, too: E(s) = C(X)C(1A) = E(E(s)lA) - E(mAUOi)d(x)) = (eOe)(j®i)d(x) = ej(x)
which is (iii). Applying (iii) to the dual Hopf algebra, we get, for all f in A*:
fi(l) = Ai) which implies (iv).
6.6.8 Lemma. Let (A, d, e, j) be a finite-dimensional *-Hopf algebra; then:
j2=i. Proof. Let x be in A; we have: n+A((J 0 i)rd(x)) = ma(s(i 0 j)d(x)) = (mA((i 0 j)d(x*)))* = (6(X*)1A)* = C(X)1A So, we have:
MA((j2 0 j)d(x)) = rnA((j 0 i)(j 0 j)d(x)) =+na(i 0 !)saj(z)) = e9(=)la = e(x)lq
by 6.6.7 (i) by 6.6.7 (iii)
Moreover, we have:
(mA 0 i)(j2 0 j 0 i)(d 0 i)d(x) = (m,, 0 i)((i 2 0 j )d 0 =)d(=) = MA(77AC 0 i)d(x)
= la 0 (E (9 i)d(s) =1A(9 2
6. Special Cases
240
and:
mA)(72 0 j 0 i)(i 0 d)d(x) = (j2 0 mAU 0 i)d)d(x) = (j2 0 7IAE)d(x)
= j2(x) 0 1A and so:
i)(j2 (9 j 0 i)(d 0 i)d(x)
x=
= mA(z 0 MA)(j2 0.7
i)(i 0 d)d(x)
j2 (X)
6.6.9 Theorem. Let (A,d,E,j) be a finite-dimensional *-Hopf algebra, such that A is a semi-simple algebra. Then, A is a finite sum of finite-dimensional matrix algebras (see, for instance [49], appendix IIa), and so, it is a finitedimensional von Neumann algebra, and its predual A* is equal to the dual A*; let us put: n
A = L(Hi)
n
i-1
with di = dim Hi < oo, and where Tri is the canonical trace on C(Hi). Let us denote by I the linear isomorphism from A to A* defined, for all x in A, by I(x) = wx (with the notations of 6.1.1). We have then: (i) The trace p is equal to 17A o I (ii) The quadruple (A, d, j, gyp) is a finite-dimensional Kac algebra.
(iii) The dual A* is a von Neumann algebra, moreover, the quadruple (A*,m,j*, E o I-1) is a finite-dimensional Kac algebra, and the Fourier representation A is a H-isomorphism between this Kac algebra and the dual Kac algebra Proof. For all x in A, we have: 17A 01(x) = rjA(wx) = wx(1) = Sp(x)
which gives (i).
By 6.6.7 (i), (ii) and 6.6.8, (A, d, j) is a finite-dimensional co-involutive Hopf-von Neumann algebra, and so, by 6.6.4, we get (ii). As A* is finite-dimensional, we get, by 4.1.3, that A is an isomorphism from the algebra A* to A. So A* is a finite-dimensional von Neumann algebra, and we can apply (i) and (ii) to the quadruple (A*, mA, r)A, j*), and so we get that (A*, mA, c oI-1) is a finite-dimensional Kac algebra. Let now 01, 92 be in A* , w in A*; we have: (ccIA(w), e1 (9 e2) = ((a x A)(w), 01 0 02)
by 3.2.2 (iv) and (ii)
6.6 Finite Dimensional Kac Algebras
= (A*(0j)A*(02)7W) = MA(A* (01) 0 A* (02))
241
by 1.4.3 1
W)
= (((A 0 A) m*A(W)7 010 02)
from which we get that:
=cdA which gives the result, thanks to 5.5.6.
6.6.10 Corollaries. (i) Let I be a finite set, and d, e, j such that (C', d, e, j) is a *-Hopf algebra. Then I has a group structure, with identity element e, such that, for all f in CI: df (sit) = f(st) e(f) = f (e)
(f)(s) = AS-
1)
(ii) Let (A, d, .6, j) be a finite dimensional symmetric *-Hopf algebra, such that A is a semi-simple algebra. Then the set {x E A; x 54 0, dx = x 0 x} is a finite group G, the algebra A is the vector space CG, and d, e, j verify, for all s in G:
Proof. By 6.6.8 and 4.2.5 (ii), there is a group I' such that (C', d, j) is IHI-isomorphic with lHIa(G). That allows us to put on I a group structure satisfying (i).
By 6.6.8 and 3.6.10, the set Ix E A; x 54 0, dx = x 0 x} is the intrinsic group G of the co-involutive Hopf-von Neumann algebra (A, d, j), and, by 6.6.8 and 4.2.5 (i), we get that (A, d, j) is H-isomorphic to H5(G), which gives (ii).
The Cat's head began fading away ... (Alice's Adventures in Wonderland) O teachers, are my lessons done I cannot do another one They laughed and laughed and said Well, child, are your lessons done (Leonard Cohen, Teachers)
Postface
The Kac algebras, to the study of which the authors have had a major contribution since the beginning of the subject, have been introduced as a natural framework for duality of type Pontrjagin-van Kampen. A different approach, linking Kac algebras to the classical groups via quantization, has been described in the preface of this book. I would like to mention here the way in which Kac algebras appear as symmetry structures. Like groups, Kac algebras can act automorphically on algebras. Let K be a Kac algebra, which, for convenience, will be assumed to be finite-dimensional,
acting on a factor A; one can then construct the crossed product B of A by K, which contains A. We shall consider an outer action, which means that
the relative commutant A' n B is scalar. The relative position of A in B can then be used to recover the structure of K. In fact, more generally, we shall consider an inclusion of factors A C B, with A' n B = C, where again for convenience, we assume B to be finitely generated over A (or, in other words, we assume that the Jones index [B : A] is finite). Then the bimodule endomorphisms M = End(ABA) have a bialgebra structure. If we write: M = End(ABA) = A, fl (B OA B)
so that any x in M has the form x = >L1 bi ®A ci, with bi and c2 in B, and satisfies ax = xa for all a in A, then we have, for x = Ei bi ®A ci and y = Ej dj ®A ej as above, the multiplication of endomorphisms, given by: xy = 1: biEA(cid j) ®A ei i, j
where EA is the conditional expectation on A, and a convolution operation defined by:
X * y = 1: bi dj ®A ejci = > bi yci i (the convolution is well defined since y commutes with A).
244
Postface
When B is the crossed product of A by K, then this recovers the bialgebra structure of K (up to completions) in an invariant way. In the general case, one can show that, under our hypothesis, B is the crossed product of A by a Kac algebra K with a cocycle twisted action, if and only if End(AB (&A BB) is
a factor. This type of result generalizes beyond our finite index assumption, and shows that the Kac algebras arise naturally by means of their actions, as symmetry structures associated to inclusions of von Neumann algebras. I can only hope thus that the authors will continue their beautiful monograph with a study of the Kac algebras in action. Paris, October 1991
Adrian Ocneanu
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JIH
MexaHHKa. 3an. HayxiH. ceMHH. JIeHHHrpa.i. OTIeJI. MaTeM. I4HCT. HM. CTexnoaa (JIOMI4) 95(1980), 129-160; translated in: Solutions of the Yang-Baxter equation. J. Soviet Maths 19(1982), 1596-1620. [86] 11.11. KyJIHm, H.10. PemeTHxHH (P.P. Kulish, N.Yu. Reshetikhin): KBaHTOBaR H
JIHHetIHaR 3a,Iiaua ,IJIR ypaBHeHHA cHHyc-FopttoH H BLICmHe npe.l cTaBJIeHHR.
B c6.: BonpocbI KB. TeopHH HOJIR H CTaT. (4H3HKH. 3an. HayMH. ceMHH. JIeHHHrpa,q. OT)jea. MaTeM. I4HCT. HM. CTeKJIOBa (JIOMI4) 101(1981), 101110; translated in: Quantum linear problem for the sine-Gordon equation and higher representations. J. Soviet Maths 23(1983), 2435-2441. [87] P.P. Kulish, N.Yu. Reshetikhin and E.K. Sklyanin: Yang-Baxter equation and representation theory I. Lett. Math. Physics 5(1981), 393-403. [88] M.B. Landstad: Duality theory for covariant systems. Trans. A.M.S. 248(1979), 223267.
[89] M.B. Landstad: Duality for dual covariance algebras. Comm. Math. Phys. 52(1977), 191-202.
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[91] J.-H. Lu and A. Weinstein: Poisson-Lie groups, dressing transformations and Bruhat decompositions. J. Diff. Geom. 31(1990), 501-526. [92] G. Lusztig: Quantum deformations of certain simple modules over enveloping algebras. Adv. Math. 70(1988), 237-249. [93] G. Lusztig: Modular representations and quantum group. Contemp. Math. 82(1989), 59-77. [94] G.W. Mackey: Borel structures in groups and their duals. Trans. A.M.S. 85(1957), 134-165. [95] S. Mac Lane: Categories for the Working Mathematician. Springer, New York, 1971. [96] V.B. de Magalhaes Iorio: Hopf-(C*-algebras and locally compacts groups. Pacific J. of Maths 87 (1980), 75-96. [97] S. Majid: Hopf-von Neumann algebra bicrossproducts, Kac algebra bicrossproducts and the classical Yang-Baxter equation. J. Funct. Anal. 95(1991), 291-319. [98] S. Majid and Ya.S. Soibelman: Rank of quantized universal enveloping algebras and modular functions. Com. Math. Phys. 137(1991), 249-262.
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34(1979), 13-63; translated in: The quantum method of the inverse problem and the Heisenberg XYZ model. Russian Math. Surveys 34(1979), 11-68. [166] T. Tannaka: Uber den Dualitatssatz der nichtkommutativen topologischen Gruppen. Tohoku Math. J. 45(1938), 1-12. [167] N. Tatsuuma: A duality theorem for locally compact groups I,II. Proc. Japan Acad. 41(1965), 878-882; 42(1966), 46-47. [168] N. Tatsuuma: A duality theorem for locally compact groups. J. Math. Kyoto Univ. 6(1967), 187-293. [169] V.G. Turaev: The Yang-Baxter equation and invariants of links. Inv. Math. 92(1988), 527-553. [170] J1. 14. BattHepMaH (L.I. Vainerman): Xapa1KTepH3anxHJI ,nBO#CTBeHHbIX K JIOKaJIbHO KOMIIaKTHbIM rpynnaM. I yHKIn. aHaJIH3 H ero IIpaJi. 8-1(1974), 75-
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rHIIepsomnAeKCHbIX cHcTeM. I43B. By3oB.MaT. 29-2(1985), 12-21; translated in:
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in Sel. Math. Sov. [182] J1.14. BaItHepMaH, r.J1. JIHTBHHOB (L.I. VaYnerman, G.L. Litvinov): IInaHmepensi H 4opMyJia o6pangeHHA AJIA onepaTopoB o6o6ceHHoro uiABHra.
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306(1989), 269-272; translated in: q-analogues of Clebsch-Gordan coefficients and the algebra of functions on the quantum group SU(2). Soviet Math. Dokl. 39(1989), 467-470. [184] JI.J1. BaxcMaH H JI.M. KoporoAcsHtt (L.L. Vaksman and L.I. Korogodskii): Anre6pa orpaHHqeHHbIX (PyHKIXHit Ha KBaHTOBO1 rpynne ABHxceHHt IIJIOCKOCTH H
q-aHaJiorH 4yHKIrH1l BeCCeJA. floss. AxaA. Hays CCCP 304(1989); translated in: An algebra of bounded functions on the quantum group of the motion of the plane, and q-analogues of Bessel functions. Soviet Math. Dokl. 39(1989), 173-177. [185] JI.JI. BascMaH H $I.C. Cott6enbMaH (L.L. Vaksman and Ya.S. Soibel'man): Anre6pa (pyHxn(HIl Ha KBaHTOBOtt rpynne SU(2). 'yHxu. aHaJH3 H ero IIpHJI. 22-
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Index
Abelian co-involutive Hopf-von Neumann algebra Achieved left Hilbert algebra . . . . . . . Affiliated . . . . . . . . . . . . . . . Analytic element . . . . . . . . . . . . Antipode . . . . . . . . . . . . . . .
Bochner's theorem C * -algebra
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Coextension of a H-morphism . . . . Co-involutive Hopf-von Neumann algebra Commutant Kac algebra . . . . . . Compact type Kac algebra . . . . . Complex Hopf algebra . . . . . . . Conditional expectation . . . . . . Connection relations . . . . . . .
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2.1.8 (ii) .
Discrete type Kac algebra . . . . . . . Dual co-involutive Hopf-von Neumann algebra Dual Kac algebra . . . . . . . . . Dual morphism . . . . . . . . . . . Dual weight . . . . . . . . . . . . .
Ernest algebra . . . . . . Ernest's theorem . . . . . Extended positive part of M Extension of a H-morphism . Extension of a representation Eymard algebra . . . . . Eymard's duality theorem . Eymard's theorem . . . .
1.1.1 (i) . 5.2.3
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1.6.8 4.7.3 2.1.8 (i) . 5.2.3 4.6.8 . . 3.4.4 . 4.3.3 . 3.4.6 . .
256
Index
. Fourier algebra . . . . Fourier-Plancherel mapping . Fourier-Plancherel transform Fourier representation . . . Fourier-Stieltjes algebra Fourier-Stieltjes representation Fourier transform . . . . Fundamental operator . . .
3.3.4 .
3.5.3
4.3.7 2.5.3 1.6.9 1.6.9 .
4.3.7
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2.4.2
Generator
1.5.2
H-morphism . . . . . . . Haar weight . . . . . . . . Heisenberg bicharacter . . . Heisenberg's commutation relation Heisenberg pairing operator . . Heisenberg's theorem . . . . Hopf algebra . . . . . . . *-Hopf algebra . . . . . . . Hopf-von Neumann algebra . .
1.2.6 2.2.1 4.6.6 (ii)
4.6.7 4.6.2 4.6.1 6.6.6 6.6.6 1.2.1
Induced von Neumann algebra Intrinsic group . . . . . .
1.1.1 (ii)
1.2.2
Johnson's theorem
5.6.9
Kac algebra . . . . . . . . Kac-Paljutkin's theorem . . . K-morphism . . . . . . . Krein algebra . . . . . . . Krein's theorem . . . . . . Kronecker product . . . . Kubo-Martin-Schwinger condition
2.2.5 6.6.5 5.1.1
6.4.2 6.4.6 1.4.1
2.1.1 (iv)
Left Hilbert algebra Left-invariant weight Left regular representation
2.1.1 (iii) 2.2.1 1.1.5
Modular automorphism group
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Operator-valued weight Opposite Kac algebra
Pentagonal relation
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2.4.4
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Peter-Weyl's theorem . . . Plancherel weight . . Pontrjagin's duality theorem Positive definite elements
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Radon-Nykodim derivative of weights Reduced Kac algebra . . . . . . Reduced von Neumann algebra . . Represent able . . . . . . . . . Square-integrable element of M* Standard von Neumann algebra Stone's theorem . . . . . . Strict ]HI-morphism
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6.2.7 (iv) . 3.6.11 4.3.8 . .
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257
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1.3.1
2.1.1 (v) . 2.2.6 1.1.1 (ii) .
1.3.6
Sub-Kac algebra . . . . . . . . . . . . . Symmetric co-involutive Hopf-von Neumann algebra
Tannaka's theorem . . . . . . . . . Takesaki's theorem . . . . . . . . . . . . . . . . . Tatsuuma's theorem Tensor product of von Neumann algebras . . . Tensor product of weights . . . Tensor product of operator-valued weights Unimodular Kac algebra Von Neumann algebra
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6.2.8 4.2.4 . . 4.7.4 1.1.1 (ii) 2.1.1 (iv) 2.1.8 (iii)
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1. 1. 1 (i)
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W*-algebra . . . . . . . . . . . . . Walter's theorem on Fourier algebras . . . Walter's theorem on Fourier-Stieltjes algebras . . . . . Weight . . . . . . . . . Weil's theorem . . . . . . . . . . . Wendel's duality theorem . . . . . . . Wendel's theorem . . . . . . 0
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4.5.9 .
5.5.11
