ALGEBRAIC VERSIONS OF A FINITE-DIMENSIONAL QUANTUM

GROUPOID

Dmitri Nikshych^{∗}  Leonid Vainerman^{†}

August 11, 1998

Abstract

We establish the equivalence of three versions of a finite dimensional quantum groupoid: a generalized Kac algebra introduced by T. Yamanouchi, a weak C^{∗}-Hopf algebra introduced by G. Böhm– F. Nill–K. Szlachányi (with an involutive antipode), and a Kac bimodule – an algebraic version of a Hopf bimodule, the notion introduced by J.-M. Vallin. We also study the structure and construct examples of finite dimensional quantum groupoids.

INTRODUCTION

0.1. Our initial point is the following theorem conjectured by A. Ocneanu (see [19], the postface in [7] and remark 4.7a in [9]) and proved by W. Szymanski [22], R. Longo [12] and M.-C. David [4]:

if M_{0} ⊂ M_{1} ⊂ M_{2} ⊂ M_{3} is a Jones’s tower of type II_{1} von Neumann factors with finite index [9], which is irreducible (i.e., M′_{0} ∩ M_{1} = C) and of depth 2 (i.e., M′_{0} ∩ M_{3} is a factor), then M′_{1} ∩ M_{3} has a natural finitedimensional (f-d) Kac algebra [11], [7] structure which acts outerly on M_{2} in such a way that the resulting cross-product is isomorphic to M_{3}. This gives an intrinsic characterization of inclusions of the mentioned type.

∗UCLA, Dept. of Math., Los Angeles, CA 90095-1555;E-mail: nikshych@math.ucla.edu

†URA CNRS (Case 191), Université Pierre et Marie Curie, 4, place Jussieu, F-75252, Paris Cedex 05 France; E-mail: vainerma@math.jussieu.fr

A similar result for infinite index inclusions was conjectured in [10] and proved for arbitrary type factors in [6], [5] (see also [17]).

It is natural to extend the above theorem to reducible (i.e., ℂ ⊂ M′_{0}∩M_{1}) inclusions of depth 2 – for example, an inclusion given by a cross-product M_{0} ⊂ M_{0} ×_{α} G (G is a finite group) is always of depth 2 and irreducible iff an action α is outer [6], [8], [17]. But for this one should replace a structure of a Kac algebra on M′_{1} ∩ M_{3} by some more general structure.

0.2. One of the suitable structures is a f-d generalized (gen). Kac algebra [27] – a C^{∗}-bialgebra with a (generally) non-unital coproduct, an involutive antipode and a Haar trace. Exactly as for Kac algebras [11], [7], the corresponding duality theory is based on the usage of the fundamental operator W. W^{∗} satisfies the pentagonal relation [1], it is a partial isometry and not necessarily a unitary. However, one can construct a gen. Kac algebra from a given f-d multiplicative partial isometry [26]. Since a gen. Kac algebra is commutative iff it is isomorphic to the algebra of complex-valued functions on a finite groupoid (see [27], Theorem 7.32), one can consider two above notions as “measurable versions” of a f-d quantum groupoid – “measurable”, because their definitions contain explicitly (a gen. Kac algebra) or implicitly (a multiplicative partial isometry) a Haar measure.

The corresponding infinite-dimensional structures are Hopf bimodules [24] and pseudo-multiplicative unitaries [25] introduced using a fibered product of von Neumann algebras [21]. In their terms a structure of von Neumann algebra inclusions of depth 2 and any index with an operator-valued weight verifying some regularity conditions was studied [8]. On the other hand, a f-d weak C^{∗}-Hopf algebra [2], [3] generalizing a Kac algebra and formulated in purely algebraic terms – a C^{∗}-bialgebra with an antipode and a counit instead of a Haar measure – was used in [18] in order to show that any inclusion given by a cross-product with such an algebra is of depth 2.

Let us mention also some other points of view on quantum groupoids - [13], [15], [23].

In what follows all C^{∗}-algebras and linear spaces over ℂ are f-d.

0.3. In this paper we establish the equivalence of three versions of a f-d quantum groupoid: gen. and weak Kac algebra (a weak C^{∗}-Hopf algebra with an involutive antipode), and a Kac bimodule – an “algebraization” of a Hopf bimodule. We also study the structure and construct examples of f-d quantum groupoids (some other examples are given in [8], [3], [2]).

The paper is organized as follows. In Section 1 (Preliminaries) we discuss, following [21], [8], [26], a relative tensor product of Hilbert modules and a fibered product of f-d C^{∗}-algebras. Next, important notions of Cartan subalgebras (this term is borrowed from [20], [26]) and counital maps in C^{∗}-bialgebras are introduced. In their terms we define two auxiliary structures – counital C^{∗}-bialgebras and Hopf bimodules, study their elementary properties and prove their equivalence.

Section 2 is central. First we discuss weak Kac algebras. The theory of general weak C^{∗}-Hopf algebras ([2], [3]) is much more complicated than the one of weak Kac algebras, that is why we give the proofs of all the statements. The most important results are existence and uniqueness of a Haar projection, normalized Haar trace and a Haar conditional expectation on any weak Kac algebra. This material can be read independently, one can look at the section of preliminaries in the case of necessity.

Then the equivalence of the categories of gen. and weak Kac algebras is established. For having a possibility to extend this theory to infinitedimensional case, we formulate it also using the equivalent language of Kac bimodules – Hopf bimodules [8], [24], [26] equipped with a counit.

It is natural to consider a f-d quantum groupoid as non-trivial if it is noncommutative and non-cocommutative (i.e., does not come from groupoids) and is not a usual Kac algebra. Such examples are presented in Section 3. In particular, a finite group acting on a weak Kac algebra gives, via crossproduct construction, a new weak Kac algebra. In such a way, we construct a series of concrete non-trivial quantum groupoids of dimensions n^{3} (n ≥ 2).

Then we show that for any f-d C^{∗}-algebra A there exists a unique weak Kac algebra structure (non-trivial when A is non-abelian) on the full matrix algebra M_{n}(ℂ) (where n = dim A) with Cartan subalgebras isomorphic to A. This gives a classification of weak Kac algebras simple as C^{∗}-algebras.

Acknowledgement. The first author would like to thank E. Effros for stimulating discussions. The second author is deeply grateful to M. Enock and J.-M. Vallin for many important discussions and also to Max-Planck-Institut für Mathematik in Bonn for excellent conditions for his work.

1 Preliminaries

1.1 Relative tensor product of Hilbert modules

1.1.1. Let θ be a faithful trace on a C^{∗}-algebra N, L_{2}(N) a Hilbert space given by the GNS-construction for (N, θ), λ (resp., ρ) a faithful unital ∗- representation (resp., anti-∗-representation) of N on L_{2}(N) by operators of left (resp., right) multiplication. Clearly, the commutant λ(N)′ = ρ(N).

Let α be a faithful unital ∗-representation of N in some other Hilbert space H. Then H can be viewed as a left N-module_{α}H : n·_{α}ζ = α(n)ζ (∀n ∈ N, ζ ∈ H). An operator R(ζ) : L_{2}(N) → H (n → α(n)ζ) is such that R(ζ)λ(n′) = α(n′)R(ζ) (∀n′ ∈ N), so, for any ζ, η ∈ H, operator R(ζ)R(η)^{∗} belongs to the commutant α(N)′ and < ζ, η >_{N}= R(η)^{∗}R(ζ) belongs to the commutant λ(N)′ which is isomorphic to the opposite C^{∗}-algebra N^{o}.

In a similar way, if β be a faithful unital anti-∗-representation of N in H, then H can be viewed simultaneously as a right N-module H_{β} or as a left N^{o}-module. An operator R^{o}(ζ) : L_{2}(N) → H (n → β(n^{∗})ζ) is such that R^{o}(ζ)ρ(n′) = β(n^{′∗})R^{o}(ζ) (∀n′ ∈ N), so, for any ζ, η ∈ H, operator R^{o}(ζ)R^{o}(η)^{∗} ∈ β(N)′ and < ζ, η >_{N^{o}}= R^{o}(η)^{∗}R^{o}(ζ) ∈ λ(N).

1.1.2. If K is another Hilbert space carrying a faithful unital ∗-repre-sentation γ of N, we obtain a relative tensor product H _{β}⊗_{γ}_{N} K from H ⊗ K

introducing the following inner product (and factorizing, if necessary):

(ζ_{1} ⊗ η_{1}|ζ_{2} ⊗ η_{2}) = (γ(< ζ_{1}, ζ_{2} >_{N^{o}})η_{1}|η_{2}) (∀ζ_{1}, ζ_{2} ∈ H, η_{1}, η_{2} ∈ K).

Then there exists a linear operator A_{β,γ} : H ⊗ K → H _{β}⊗_{γ}_{N} K, the image

of ζ ⊗ η being denoted by ζ _{β}⊗_{γ}_{N} η (∀ζ ∈ H, η ∈ K). One can also define

H _{β}⊗_{γ}_{N} K by introducing the following inner product:

(ζ_{1} ⊗ η_{1}|ζ_{2} ⊗ η_{2}) = (β(< η_{1}, η_{2} >_{N})ζ_{1}|ζ_{2}) (∀ζ_{1}, ζ_{2} ∈ H, η_{1}, η_{2} ∈ K),

this leads to a definition of a flip σ_{β,γ} : H _{β}⊗_{γ}_{N} K → K _{γ}⊗_{β}_{N^{o}} H by σ_{β,γ} :

ζ _{β}⊗_{γ}_{N} η → η _{γ}⊗_{β}_{N^{o}} ζ (∀ζ ∈ H, η ∈ K). This allows to define a flip Σ_{β,γ} :

L(H _{β}⊗_{γ}_{N} K) → L(K _{γ}⊗_{β}_{N^{o}} H) by Σ_{β,γ}(X) = σ_{β,γ}^{∗}Xσ_{β,γ} (∀X ∈ L(H _{β}⊗_{γ}_{N} K)).

Recall also that β(n)ζ _{β}⊗_{γ}_{N} η = ζ _{β}⊗_{γ}_{N} γ(n)η (∀ζ ∈ H, η ∈ K, n ∈ N).

1.1.3. Finally, if_{α}H_{β} (resp., _{γ}K_{δ}) is an (N, N)-bimodule with respect to a ∗-representation α (resp., γ) and an anti-∗-representation β (resp., δ), then

H _{β}⊗_{γ}_{N} K is also (N, N)-bimodule: n ·_{α} [ζ _{β}⊗_{γ}_{N} η] ·_{δ} n′ = [α(n)ζ] _{β}⊗_{γ}_{N}[δ(n′)η]

(∀ζ ∈ H, η ∈ K, n, n′ ∈ N).

The described construction does not depend on the choice of a faithful trace on N (up to an isomorphism of Hilbert spaces).

1.2 Fibered product of C^{∗}-algebras

Given two C^{∗}-algebras M_{1}, M_{2} such that β(N) ⊂ M_{1}, γ(N) ⊂ M_{2}, we construct their fibered product - a canonical algebra (M_{1}) _{β}∗_{γ}_{N}(M_{2}) isomorphic

to the reduction of M_{1} ⊗ M_{2} by some projection e_{β,γ} depending on β, γ.

1.2.1. First, we define (M′_{1}) _{β}⊗_{γ}_{N}(M′_{2}) as a C^{∗}-algebra generated by

x _{β}⊗_{γ}_{N} y (∀x ∈ M′_{1}, y ∈ M′_{2}) and then (M_{1}) _{β}∗_{γ}_{N}(M_{2}) := [(M′_{1}) _{β}⊗_{γ}_{N}(M′_{2})]′.

One can verify that:

[(M_{1}) _{β}∗_{γ}_{N}(M_{2})]^{o} = (M^{o}_{1}) _{β}∗_{γ}_{N^{o}}(M^{o}_{2}),
Σ_{β,γ}((M_{1}) _{β}∗_{γ}_{N}(M_{2})) = (M_{2}) _{γ}∗_{β}_{N^{o}}(M_{1}),
(M_{1} ∩ β(N)′) _{β}⊗_{γ}_{N}(M_{2} ∩ γ(N)′) ⊂ (M_{1}) _{β}∗_{γ}_{N}(M_{2}),
(M_{1}) _{β}∗_{γ}_{N}(N) = (M_{1} ∩ β(N)′) _{β}⊗_{γ}_{N} 1_{K} = (1_{H}) _{β}⊗_{γ}_{N}(M_{2} ∩ γ(N)′).

If β′ (resp., γ′) is an injective unital anti-∗-homomorphism (resp., ∗- homomorphism) of N into another C^{∗}-algebra L_{1} (resp., L_{2}), and Φ : M_{1} → L_{1} (resp., Ψ : M_{2} → L_{2}) is a unital ∗-homomorphism such that Φ ∘ β = β′ (resp., Ψ ∘ γ = γ′), then one can define a unital ∗-homomorphism Φ _{β}∗_{γ}_{N} Ψ : (M_{1}) _{β}∗_{γ}_{N}(M_{2}) → (L_{1}) _{β′}⊗_{γ′}_{N}(L_{2}). This means, in particular, that if

β (resp.,γ) is an injective unital anti-∗-homomorphism (resp., ∗-homomor-phism) of N into M_{1} (resp., M_{2}), then one can define (M_{1}) _{β}∗_{γ}_{N}(M_{2}) (to be

a C^{∗}-algebra) without any references to a specific Hilbert space.

Obviously, if τ : P → N is an isomorphism of C^{∗}-algebras, then one can

identify H _{β}⊗_{γ}_{N} K with H _{β∘τ} ⊗_{γ∘τ}_{P} K, (M′_{1}) _{β}⊗_{γ}_{N}(M′_{2}) with (M′_{1}) _{β∘τ} ⊗_{γ∘τ}_{P} (M′_{2}),

and (M_{1}) _{β}∗_{γ}_{N}(M_{2}) with (M_{1}) _{β∘τ} ∗_{γ∘τ}_{P} (M_{2}).

1.2.2. Let now H = K and e_{β,γ} be the support of the above operator A_{β,γ}. Then one can show that e_{β,γ} ∈ β(N) ⊗ γ(N) and e_{β,γ}(β(n) ⊗ 1_{H}) = e_{β,γ}(1_{H} ⊗ γ(n)) (∀n ∈ N). Vice versa, let M_{1} and M_{2} be two C^{∗}-algebras acting in H such that β(N) ⊂ M_{1} and γ(N) ⊂ M_{2}, and let e_{β,γ} ∈ β(N) ⊗ γ(N) be a projection satisfying the above relation. Then automatically e = (id ⊗ θ)(e_{β,γ}) ∈ Z(β(N)) and

||(e_{β,γ})^{\frac{1}{2}} (ζ ⊗ η)||^{2} = (β(< η, η >_{N})e^{\frac{1}{2}} ζ, e^{\frac{1}{2}} ζ) (∀ζ, η ∈ H),

which equals to ||e^{\frac{1}{2}} ζ _{β}⊗_{γ}_{N} η||^{2}. So e^{\frac{1}{2}} is invertible and there exists an

isometry V_{β,γ} : H _{β}⊗_{γ}_{N} H → H ⊗ H with final support e_{β,γ}, defined by:

V_{β,γ}(ζ _{β}⊗_{γ}_{N} η) = e_{β,γ}(e^{− \frac{1}{2}} ζ ⊗ η) (∀ζ, η ∈ H).

This isometry satisfies the following relations:

V_{β,γ}(x _{β}⊗_{γ}_{N} y) = (x ⊗ y)V_{β,γ} (∀x ∈ β(N)′, y ∈ γ(N)′);
(β(n) ⊗ 1_{H})V_{β,γ} = (1_{H} ⊗ γ(n))V_{β,γ} (∀n ∈ N);
(M′_{1}) _{β}⊗_{γ}_{N}(M′_{2}) = V_{β,γ}^{∗}(M′_{1} ⊗ M′_{2})V_{β,γ};
(M_{1}) _{β}∗_{γ}_{N}(M_{2}) = V_{β,γ}^{∗}(M_{1} ⊗ M_{2})V_{β,γ},

and (M_{1}) _{β}∗_{γ}_{N}(M_{2}) is isomorphic to the reduced C^{∗}-algebra (M_{1} ⊗ M_{2})_{e_{β,γ}}.

Let i_{β,γ} be a non-unital injection of (M_{1}) _{β}∗_{γ}_{N}(M_{2}) into M_{1} ⊗ M_{2}.

1.3 Counital C^{∗}-bialgebras

Definition 1.3.1 a) A pair (M, Δ) is said to be a gen. C^{∗}-bialgebra if M is a f-d C^{∗}-algebra and a coproduct Δ : M → M ⊗ M is an injective ∗- homomorphism (not necessarily unital) such that (Δ⊗id)∘Δ = (id⊗Δ)∘Δ.

b) A triplet A = (M, Δ, S) is said to be a gen. coinvolutive C^{∗}-bialgebra if (M, Δ) is a gen. C^{∗}-bialgebra and a co-involution S : M → M is a unital anti-∗-automorphism such that S^{2} = id and (S ⊗ S) ∘ Δ = ς ∘ Δ ∘ S, where ς : M ⊗ M → M ⊗ M is a usual flip.

c) A homomorphism of gen. coinvolutive C^{∗}-bialgebras (M_{1}, Δ_{1}, S_{1}) and (M_{2}, Δ_{2}, S_{2}) is a unital ∗-homomorphism of C^{∗}-algebras π : M_{1} → M_{2} such that (π ⊗ π) ∘ Δ_{1} = Δ_{2} ∘ π, π ∘ S_{1} = S_{2} ∘ π. Clearly, gen. coinvolutive C^{∗}-bialgebras form a category.

d) [27] A quadruplet GK = (M, Δ, S, ϕ) is said to be a gen. Kac algebra if A = (M, Δ, S) is a gen. coinvolutive C^{∗}-bialgebra and a Haar trace ϕ is a faithful trace on M such that ϕ ∘ S = ϕ and

(id ⊗ ϕ)[(1 ⊗ y)Δ(x)] = S[(id ⊗ ϕ)(Δ(y)(1 ⊗ x))] (∀x, y ∈ M).

Gen. Kac algebra is abelian, i.e., M is abelian (resp., symmetric, i.e., ς ∘Δ = Δ), iff it is isomorphic to the algebra of functions on a finite groupoid (resp., to the groupoid algebra of a finite groupoid) [27].

Given a gen. C^{∗}-bialgebra, e = Δ(1), let us define two sets: N_{s} = {x ∈ M|Δ(x) = e(1⊗x) = (1⊗x)e} and N_{t} = {x ∈ M|Δ(x) = e(x⊗1) = (x⊗1)e}.

Proposition 1.3.2 [26] For any gen. C^{∗}-bialgebra we have:

a) N_{s} and N_{t} are C^{∗}-subalgebras of M.

b) N_{s} and N_{t} commute: nn′ = n′n (∀n ∈ N_{s}, n′ ∈ N_{t}).

Proof. a) It is clear that N_{s} and N_{t} are linear ∗-invariant subspaces of M containing 1. For any x, y ∈ N_{t} we have:

Δ(xy) = Δ(x)Δ(y) = Δ(x)e(y ⊗ 1)
= Δ(x)(y ⊗ 1) = e(xy ⊗ 1),

so N_{t} is a C^{∗}-subalgebra of M; similarly for N_{s}.

b) For all n ∈ N_{s}, n′ ∈ N_{t} we have:

Δ(nn′) = Δ(n)e(n′ ⊗ 1) = Δ(n)(n′ ⊗ 1)
= e(1 ⊗ n)(n′ ⊗ 1) = e(n′ ⊗ 1)(1 ⊗ n)
= Δ(n′)(1 ⊗ n) = Δ(n′n),

and the injectivity of Δ gives the result.

N_{s} and N_{t} are said to be source and target Cartan subalgebras respectively. In any generalized coinvolutive C^{∗}-bialgebra S(N_{s}) = N_{t}.

Definition 1.3.3 A gen. coinvolutive C^{∗}-bialgebra is said to be counital if the target counital map ε_{t} := μ(id ⊗ S)Δ : M → M (where μ : M ⊗ M → M is the multiplication in M) satisfies the following conditions:

1. ε_{t}(1) = 1.

2. Imε_{t} ⊂ N_{t}.

3. (id ⊗ ε_{t})Δ(x) = e(x ⊗ 1) (∀x ∈ N_{s}).

A definition of a morphism of generalized counital C^{∗}-bialgebras is obvious.

Remark 1.3.4 a) Clearly, source counital map ε_{s} := μ(S ⊗ id)Δ : M → M has similar properties:

1’. ε_{s}(1) = 1.

2’. Imε_{s} ⊂ N_{s}.

3’. (ε_{s} ⊗ id)Δ(x) = (1 ⊗ x)e (∀x ∈ N_{t}).

4. S ∘ ε_{t} = ε_{s} ∘ S.

b) From 3 and 3’ we have (id ⊗ ε_{t})e = e = (ε_{s} ⊗ id)e, together with 2 and 2’ this gives e ∈ N_{s} ⊗ N_{t}.

c) (Δ ⊗ id)(e) = (id ⊗ Δ)(e) = (e ⊗ 1)(1 ⊗ e) = (1 ⊗ e)(e ⊗ 1).

Indeed, let e = ∑_{i=1}^{n} x_{i} ⊗ y_{i} (where x_{i} ∈ N_{s}, y_{i} ∈ N_{t} - such a representation is possible by b)). Then, using the definition of N_{s}, we have:

(Δ ⊗ id)(e) = ∑_{i=1}^{n} Δ(x_{i}) ⊗ y_{i} = ∑_{i=1}^{n} e(1 ⊗ x_{i}) ⊗ y_{i} = (e ⊗ 1)(1 ⊗ e).

The other equalities are now obvious.

d) One can easily verify that:

ε_{t}(x)^{∗} = ε_{t}(S(x)^{∗}), ε_{t}(xy) = ε_{t}(xε_{t}(y)), ε_{t}(xn) = ε_{t}(xS(n))

for all x, y ∈ M, n ∈ N_{t}, and similar relations for ε_{s}.

Proposition 1.3.5 ε_{t}(nS(n′)x) = n · ε_{t}(x) · n′ (∀x ∈ M, n, n′ ∈ N_{t}).

Proof. The definitions of N_{t}, N_{s} and the properties of Δ, S give:

ε_{t}(nS(n′)x) = μ(id ⊗ S)Δ(nS(n′)x)
= μ(id ⊗ S)[(n ⊗ 1)e(1 ⊗ S(n′))Δ(x)]
= n · μ[(id ⊗ S)Δ(x)(1 ⊗ n′)] = n · ε_{t}(x) · n′.

Remark 1.3.6 a) Let us introduce the following structure of an (N_{t}, N_{t})- bimodule _{t}M_{s} on M : n ·_{t} x ·_{s} n′ = nS(n′)x (∀x ∈ M, n, n′ ∈ N_{t}). Then Proposition 1.3.5 means that ε_{t} : _{t}M_{s} → N_{t} is the (N_{t}, N_{t})-bimodule map.

b) One can prove the similar property for ε_{s}:

ε_{s}(xnS(n′)) = S(n) · ε_{s}(x) · S(n′) (∀x ∈ M, n, n′ ∈ N_{t}).

c) Proposition 1.3.5 and axiom 1 imply that ε_{t} equals to identity on N_{t} and to S on N_{s}; similarly ε_{s} equals to identity on N_{s} and to S on N_{t}.

Proposition 1.3.7 In any gen. counital C^{∗}-bialgebra

e(S(n) ⊗ 1) = e(1 ⊗ n), (S(n) ⊗ 1)e = (1 ⊗ n)e (∀n ∈ N_{s}).

Proof. Clearly, the second relation can be deduced from the first. Then, using successively the axiom 3, the definition of N_{t}, Proposition 1.3.5 and axiom 1, we have for any n ∈ N_{t}:

e(S(n) ⊗ 1) = (id ⊗ ε_{t})Δ(S(n)) = (id ⊗ ε_{t})e(1 ⊗ S(n)) = e(1 ⊗ n).

Definition 1.3.8 Let (M, Δ) be a gen. counital C^{∗}-bialgebra.

a) A projection Λ ∈ M is said to be a Haar projection if

xΛ = ε_{t}(x)Λ (∀x ∈ M), S(Λ) = Λ, ε_{t}(Λ) = 1.

b) A functional (resp., a faithful trace) ϕ on M is said to be a Haar functional (resp., a Haar trace) if

(id ⊗ ϕ)Δ = (ε_{t} ⊗ ϕ)Δ, ϕ ∘ S = ϕ,

ϕ is said to be normalized if (id ⊗ ϕ)e = 1.

c) A conditional expectation E_{t} : M → N_{t} (resp., E_{s} : M → N_{s}) is said to be target (resp., source) Haar conditional expectation on (M, Δ) if (id ⊗ E_{t}) ∘ Δ = Δ ∘ E_{t} (resp., (E_{s} ⊗ id) ∘ Δ = Δ ∘ E_{s}).

Obviously, the identities symmetric to a) and b) hold true for ε_{s}.

1.4 Counital Hopf bimodules

Definition 1.4.1 [24], [8]. A collection B = (N, M, t, s, Δ) is said to be a Hopf bimodule, if N, M are C^{∗}-algebras, s (resp., t, Δ) is an injective unital anti-∗-homomorphism (resp., ∗-homomorphism) of N → M (resp.,

N → M, M → M _{s}∗_{t}_{N} M) such that

s(n)t(n′) = t(n′)s(n) ∀n, n′ ∈ N,

Δ(t(n)) = t(n) _{s}⊗_{t}_{N}(1), Δ(s(n)) = (1) _{s}⊗_{t}_{N} s(n) (∀n ∈ N);

(Δ _{s}∗_{t}_{N} id) ∘ Δ = ((id) _{s}∗_{t}_{N} Δ) ∘ Δ.

Let us clarify this definition. The structures of left, right N-module and (N, N)-bimodule on M (_{t}M, M_{s} and _{t}M_{s} respectively), can be defined by:

n·_{t}m = t(n)m, m·_{s}n′ = ms(n′), n·_{t}m·_{s}n′ = t(n)ms(n′) ∀m ∈ M, n, n′ ∈ N.

Then the fibered product M _{s}∗_{t}_{N} M is the (N, N)-bimodule

tM _{s}∗_{t}_{N} M_{s} : n ·_{t} (m _{s}∗_{t}_{N} m′) ·_{s} n′ = (n ·_{t} m) _{s}∗_{t}_{N}(m′ ·_{s} n′),

for all m, m′ ∈ M, n, n′ ∈ N. So one can construct two fibered products: (M _{s}∗_{t}_{N} M) _{s}∗_{t}_{N} M and M _{s}∗_{t}_{N}(M _{s}∗_{t}_{N} M); both of them are isomorphic

to p(M ⊗ M ⊗ M)p, where p = (e_{s,t} ⊗ 1)(1 ⊗ e_{s,t}) = (1 ⊗ e_{s,t})(e_{s,t} ⊗ 1). Two first conditions mean that Δ :_{t} M_{s} →_{t} M _{s}∗_{t}_{N} M_{s} is an (N, N)-bimodule

morphism; after that the last one is clear.

If B = (N, M, t, s, Δ) is a Hopf bimodule, then B^{o} = (N^{o}, M, s, t, Σ_{s,t}∘Δ) is also a Hopf bimodule – symmetrized to the first one; obviously, Σ_{s,t} ∘ Δ : M → M _{t}∗_{s}_{N^{o}} M. If N is abelian and t = s, then the two above Hopf bimodules

coincide; we call such a Hopf bimodule symmetric.

Definition 1.4.2 Let B = (N, M, t, s, Δ) be a Hopf bimodule. A coinvolution is such a unital anti-∗-automorphism S : M → M that S^{2} = id,

S ∘ t = s, and (S _{s}∗_{t}_{N} S) ∘ Δ = Σ_{s,t} ∘ Δ ∘ S, where Σ_{s,t} : M _{s}∗_{t}_{N} M = M _{t}∗_{s}_{N^{o}} M

is a flip introduced in 1.1, 1.2. Then the collection B = (N, M, t, Δ, S) is said to be a coinvolutive Hopf bimodule.

Let us clarify this definition. The right-hand side of the latest equality is the unital injective anti-∗-homomorphism of C^{∗}-algebras M and M _{t}∗_{s}_{N^{o}} M.

If one considers S as an isomorphism between M and M^{o}, then the map (S _{s}∗_{t}_{N} S) in the left-hand side of the above equality is, according to 1.2,

the isomorphism of C^{∗}-algebras M _{s}∗_{t}_{N} M and M^{o} _{t}∗_{s}_{N} M^{o}, the last algebra

coincides with [M _{t}∗_{s}_{N^{o}} M]^{o}. Thus, the left-hand side of the discussed equality

is the unital injective anti-∗-homomorphism of C^{∗}-algebras M and M _{t}∗_{s}_{N^{o}} M.

Definition 1.4.3 A coinvolutive Hopf bimodule B = (N, M, t, Δ, S) is said to be counital if the target counital map ϵ_{t} := μ[(id) _{s}∗_{t}_{N} S] ∘ Δ : M → M (where (id) _{s}∗_{t}_{N} S : M _{s}∗_{t}_{N} M → M _{s}∗_{s}_{N} M^{o}, the last one is a subalgebra in

M ⊗ M^{o} (see 1.2), and μ : M ⊗ M^{o} → M is the linear map given by the multiplication in M) satisfies the following conditions:

1. ϵ_{t}(1) = 1.

2. Imϵ_{t} ⊂ t(N).

3. ϵ_{t} : _{t}M_{s} → t(N) is the (N, N)-bimodule map.

A definition of a morphism of counital Hopf bimodules is clear.

Lemma 1.4.4 The categories of counital Hopf bimodules and gen. counital C^{∗}-bialgebras are equivalent.

Proof. a) Given a counital Hopf bimodule B = (N, M, t, Δ, S), let us consider the projection e_{s,t} ∈ s(N) ⊗ t(N) such that e_{s,t}(S(n) ⊗ 1) = e_{s,t}(1 ⊗ t(n)) (∀n ∈ N) and M _{s}∗_{t}_{N} M = (M ⊗ M)_{e_{s,t}}, and the canoni-

cal injection of C^{∗}-algebras i_{s,t} : M _{s}∗_{t}_{N} M → M ⊗ M (see 1.2). Then

Δ̃ = i_{s,t} ∘ Δ : M → M ⊗ M is an injective ∗-homomorphism (not necessarily unital) such that:

(Δ̃ ⊗ id) ∘ Δ̃ = (id ⊗ Δ̃) ∘ Δ̃,
Δ̃(t(n)) = (t(n) ⊗ 1) · e_{s,t}, Δ̃(s(n)) = (1 ⊗ s(n)) · e_{s,t}, (∀n ∈ N),

from where Δ̃(1) = e_{s,t} ∈ M ⊗ M. Now it is clear that (M, Δ̃, S) is a gen. coinvolutive C^{∗}-bialgebra such that ϵ_{t} = ε_{t} and t(N) ⊂ N_{t}, so the conditions 1 and 2 of Definition 1.3.3 are satisfied. But the definition of N_{t}, the structure of ε_{t} and the property ε_{t}(1) = 1 imply that ε_{t} is identical on N_{t}; from Imε_{t} ⊂ t(N) we have t(N) = N_{t}, so s(N) = N_{s}.

Finally, using the above properties of e_{s,t}, Δ̃ and ϵ_{t}, we have:

(id ⊗ ε_{t})Δ̃(s(n)) = (id ⊗ ϵ_{t})e_{s,t}(1 ⊗ s(n)) = e_{s,t}(1 ⊗ t(n))
= e_{s,t}(s(n) ⊗ 1) (∀n ∈ N).

b) Given a gen. counital C^{∗}-bialgebra (M, Δ̃, S) with Cartan subalgebras N_{t} and N_{s}, put N = N_{t}, t = id_{N}, s = S. Let H = L_{2}(M) be the Hilbert space given by the GNS-construction for (M, ϕ), where ϕ is a fixed faithful trace on M. Then, using the properties of e (Remark 1.3.4 b), Proposition 1.3.7), one can establish the following isomorphisms (see 1.1,1.2):

e(H ⊗ H) ≅ H _{s}⊗_{t}_{N} H, (e ⊗ 1)(1 ⊗ e)(H ⊗ H ⊗ H) ≅ H _{s}⊗_{t}_{N} H _{s}⊗_{t}_{N} H, (M′ ⊗

M′)e ≅ M′ _{s}⊗_{t}_{N} M′, (M′ ⊗ M′ ⊗ M′)(e ⊗ 1)(1 ⊗ e) ≅ M′ _{s}⊗_{t}_{N} M′ _{s}⊗_{t}_{N} M′ (spa-

tially). Then M _{s}∗_{t}_{N} M is isomorphic to the reduced C^{∗}-algebra (M ⊗ M)_{e}

and M _{s}∗_{t}_{N} M _{s}∗_{t}_{N} M is isomorphic to the reduced C^{∗}-algebra (M ⊗ M ⊗

M)_{(e⊗1)(1⊗e)}.

Taking in mind the above isomorphisms, Definition 1.3.3 and Proposition 1.3.5, one can see that the collection (N, M, t, Δ, S) is a counital Hopf bimodule. The statements concerning morphisms are obvious.

Obviously, e_{s,t} = 1 ⊗ 1 iff N = ℂ.

Definition 1.4.5 Let B = (N, M, t, s, Δ) be a Hopf bimodule. A conditional expectation E_{t} : M → t(N) (resp., E_{s} : M → s(N)) is said to be a target (resp., source) Haar conditional expectation on B if:

[(id) _{s}∗_{t}_{N} E_{t}] ∘ Δ = Δ ∘ E_{t}, (resp.,[(E_{s}) _{s}∗_{t}_{N}(id)] ∘ Δ = Δ ∘ E_{s}).

Here (id) _{s}∗_{t}_{N}(E_{t}) is the restriction of the map id ⊗ E_{t} : M ⊗ M → M →

t(N) to M _{s}∗_{t}_{N} M; since E_{t} is a conditional expectation, this restriction is

a well defined map from M _{s}∗_{t}_{N} M = (M ⊗ M)_{e_{s,t}} to M _{s}∗_{t}_{N} t(N) = (M ⊗

t(N))_{e_{s,t}}. Similarly for (E_{s}) _{s}∗_{t}_{N}(id).

Clearly E is a source (resp., target) Haar conditional expectation on a counital Hopf bimodule B iff it is a source (resp., target) Haar conditional expectation on the corresponding gen. counital C^{∗}-bialgebra.

Remark 1.4.6 There exists a duality theory for gen. Kac algebras [27]: for any K = (M, Δ, S, ϕ) one can construct a dual gen. Kac algebra K̂ = (M̂, Δ̂, Ŝ, ϕ̂) in such a way that the algebra dual for K̂ is isomorphic to K. It is shown in [26] that for any gen. Kac algebra: a) N_{t} = M ∩ M̂ is a target Cartan subalgebra and the conditions of Proposition 1.2 are satisfied;

b) there exists the unique right Haar conditional expectation E : M → N_{t}, which can be defined as the orthogonal projection on N_{t} in the Hilbert space given by the GNS-construction for (M, ϕ) such that ϕ ∘ E = ϕ.

2 Weak and generalized Kac algebras, Kac bimodules

2.1 Weak Kac algebras

Definition 2.1.1 A counit for a gen. coinvolutive C^{∗}-bialgebra (M, Δ, S) is such a linear map ε : M → ℂ that (ε ⊗ id)Δ = (id ⊗ ε)Δ = id and:

1). ε(S(x)) = ε(x), ε(x^{∗}) = \overline{ε(x)}.

2). (ε ⊗ ε)((x ⊗ 1)e(1 ⊗ y)) = ε(xy).

3). (ε_{s} ⊗ id)Δ(x) = (1 ⊗ x)e (recall that e := Δ(1)).

Here x, y ∈ M, ε_{s} = μ(S ⊗ id)Δ, μ : M ⊗ M → M is a multiplication in M.

A collection W K = (M, Δ, S, ε) satisfying 1)-3) is said to be a weak Kac algebra. A homomorphism π : W K_{1} → W K_{2} of weak Kac algebras is such a homomorphism of their gen. coinvolutive C^{∗}-bialgebras (Definition 1.3.1 c)), that for the counits ε_{2} ∘ π = ε_{1}. Clearly weak Kac algebras form a category.

The above notion is the special case (S^{2} = id) of the notion of a weak C^{∗}- Hopf algebra introduced in [3]. It becomes a usual Kac algebra iff either e = 1 ⊗ 1, or ε(xy) = ε(x)ε(y), or μ(S ⊗ id)Δ(x) = ε(x)1 (∀x ∈ M).

Proposition 2.1.2 The set of axioms 2) and 3) is equivalent to the following set of axioms:

A2. (ε ⊗ id)((x ⊗ 1)e(1 ⊗ y)) = (ε ⊗ id)((x ⊗ 1)Δ(y)),

A3. (id ⊗ ε ⊗ id)[(e ⊗ 1)(1 ⊗ Δ(x))] = e(1 ⊗ x),

A4. ε_{s}(x) = (id ⊗ ε)((1 ⊗ x)e) (∀x, y ∈ M).

Proof. Indeed, applying (id ⊗ ε) to 3), we get A4. This, in its turn, gives

A3’. (id ⊗ ε ⊗ id)[(1 ⊗ Δ(x))(e ⊗ 1)] = (1 ⊗ x)e,

which is clearly equivalent to A3.

Finally, using the above identities and the standard notation Δ(x) = x_{(1)} ⊗ x_{(2)} (∀x ∈ M), we have an equality A2’:

(ε ⊗ id)((1 ⊗ x)e(y ⊗ 1)) = (ε ⊗ ε ⊗ id)((1 ⊗ Δ(x))(e ⊗ 1)(y ⊗ 1 ⊗ 1))
= (ε ⊗ ε)((1 ⊗ x_{(1)})e(y ⊗ 1))x_{(2)}
= (ε ⊗ id)(Δ(x)(y ⊗ 1)).

which is equivalent to A2. Vice versa, applying ε to A2, we get 2), and we get 3) combining A3’ and A4.

The following relations are straightforward corollaries of A2-A4:

A3”. (id ⊗ ε_{t})Δ(x) = e(x ⊗ 1).

A4’. ε_{t}(x) = (ε ⊗ id)(e(x ⊗ 1)) ( here ε_{t} = μ(id ⊗ S)Δ).

Remark 2.1.3 a) Let us equip the dual linear space M̂ with a product and a coproduct obtained by transposing the product and the coproduct of M by means of the canonical pairing <, >: M̂ × M → ℂ. The unit of M̂ is ˆ1 = ε, the coinvolution Ŝ and the involution ∗ of M̂ are defined by

< Ŝ(α), x >=< α, S(x) >, < α^{∗}, x >= \overline{< α, S(x)^{∗} >} (∀x ∈ M, α ∈ M̂).

One can see that (M̂, Δ̂, Ŝ, ε̂) is also a weak Kac algebra. Indeed, using linear maps η : λ → λe from ℂ to M ⊗ M and ε ∘ μ : M ⊗ M → ℂ, let us rewrite the above axioms as follows:

Ã2. (ε ∘ μ ⊗ μ)(id ⊗ η ⊗ id) = (ε ∘ μ ⊗ id)(id ⊗ Δ).

Ã3. (id ⊗ ε ∘ μ ⊗ id)(η ⊗ Δ) = (id ⊗ μ)(η ⊗ id).

Ã4. μ(S ⊗ id)Δ = (id ⊗ ε ∘ μ)(id ⊗ ς ⊗ id)(id ⊗ η).

Now it is clear that A2 and A3 are dual to one another, A4 is selfdual.

b) The particular case of A3, in which x = 1:

A3^{∗}. (id ⊗ ε ⊗ id)[(e ⊗ 1)(1 ⊗ e)] = e

is sufficient for getting A3. Indeed, using axioms A2, A3*, one has ∀x ∈ M:

(id ⊗ ε ⊗ id)[(e ⊗ 1)(1 ⊗ Δ(x))] = 1_{(1)} ⊗ (ε ⊗ id)[(1_{(2)} ⊗ 1)Δ(x)]
= 1_{(1)} ⊗ (ε ⊗ id)[(1_{(2)} ⊗ 1)e(1 ⊗ x)]
= 1_{(1)} ⊗ (ε ⊗ id)[(1_{(2)} ⊗ 1)e] · x
= e(1 ⊗ x).

Example 2.1.4 As groups and their duals are trivial examples of usual Kac algebras, groupoids and their duals give trivial examples of weak Kac algebras. Let G be a finite groupoid (see [20] for definitions and terminology).

(a) ℂG, the groupoid C^{∗}-algebra of G, has a structure of a cocommutative weak Kac algebra given by

Δ(g) = g ⊗ g, S(g) = g^{−1}, ε(g) = 1 for all g ∈ G.

In this case ε_{s}(g) = g^{−1}g and ε_{t}(g) = gg^{−1} are familiar source and target maps. The source and target Cartan subalgebras coincide with ℂG^{0}, where G^{0} is the unit space of G.

(b) ℂ(G), the C^{∗}-algebra of complex-valued functions on G, has a structure of a commutative weak Kac algebra given by

Δ(δ_{g}) = ∑_{xy=g} δ_{x}⊗δ_{y}, S(δ_{g}) = δ_{g^{−1}}, ε(δ_{g}) = δ_{g,t(g)} = δ_{g,s(g)} for all g ∈ G.

In this case ε_{s}(δ_{g}) = ∑_{s(x)=g} δ_{x} and ε_{t}(δ_{g}) = ∑_{t(x)=g} δ_{x}. The source and target Cartan subalgebras are N_{s} = {f ∈ ℂ(G) | f(g) = f(s(g)) ∀g ∈ G} and N_{t} = {f ∈ ℂ(G) | f(g) = f(t(g)) ∀g ∈ G}.

The above weak Kac algebras are clearly dual to each other.

Remark 2.1.5 It follows from [27] and Theorem 2.3.5 below that every cocommutative (resp. commutative) weak Kac algebra is isomorphic to ℂG (resp. ℂ(G)) for some groupoid G.

Given two weak Kac algebras M_{1} and M_{2}, one can construct their tensor product M_{1} ⊗ M_{2} and direct sum M_{1} ⊕ M_{2} in an obvious way.

Let us write e = ∑_{i=1}^{n} x_{i} ⊗ y_{i} with minimal possible n (i.e., with both {x_{i}} and {y_{i}} linearly independent). Obviously, one can choose x_{i} and y_{i} in such a way that x_{i} = x_{i}^{∗}, y_{i} = y_{i}^{∗}.

Proposition 2.1.6 ε(y_{i}x_{k}) = ε(x_{k}y_{i}) = δ_{ik}.

Proof. From A3^{∗} one gets:

∑_{i,j=1}^{n} x_{i} ⊗ ε(y_{i}x_{j})y_{j} = ∑_{i=1}^{n} x_{i} ⊗ y_{i}, ∑_{i,j=1}^{n} x_{i} ⊗ ε(x_{j}y_{i})y_{j} = ∑_{i=1}^{n} x_{i} ⊗ y_{i},

so ∑_{j=1}^{n} ε(y_{k}x_{j})y_{j} = y_{k} = ∑_{j=1}^{n} ε(x_{j}y_{k})y_{j}, from where the result follows.

Let us denote N_{s} = span{x_{1}, … x_{n}} and N_{t} = span{y_{1}, … y_{n}}. Then it is clear that e ∈ N_{s} ⊗ N_{t}, that both N_{s}, N_{t} are ∗-invariant and contain 1.

Proposition 2.1.7 a) ε_{s} (resp. ε_{t}) is a linear map from M to N_{s} (resp. N_{t}); b) ε_{s}|_{N_{s}} = id, ε_{t}|_{N_{t}} = id.

Proof. a) is clear from A4 and A4’. b): ε_{s}(x_{k}) = ∑_{i=1}^{n} x_{i}ε(x_{k}y_{i}) = x_{k}, similarly ε_{t}(y_{l}) = y_{l} (∀k, l = 1, ..., n).

Proposition 2.1.8 Δ(ε_{s}(x)) = (1 ⊗ ε_{s}(x))e, Δ(ε_{t}(x)) = (ε_{t}(x) ⊗ 1)e.

Proof. The equality (Δ ⊗ id)(e) = (id ⊗ Δ)(e) gives:

∑_{i=1}^{n} Δ(x_{i}) ⊗ y_{i} = ∑_{i=1}^{n} x_{i} ⊗ Δ(y_{i}),

from where, using Proposition 2.1.6

Δ(x_{k}) = ∑_{i=1}^{n} Δ(x_{i})ε(x_{k}y_{i}) = ∑_{i=1}^{n} x_{i} ⊗ (id ⊗ ε)[(1 ⊗ x_{k})Δ(y_{i})] ∈ N_{s} ⊗ M.

Similarly Δ(y_{k}) ∈ M ⊗ N_{t}. Now Axioms 3), A3” and Proposition 2.1.7 give the result.

Corollary 2.1.9 Any weak Kac algebra is a gen. counital C^{∗}-bialgebra.

Proof. Relations 1 and 3 of Definition 1.3.3 follow from A4 and A4’. Since N_{s}, N_{t} are ∗-invariant, Proposition 2.1.8 shows that they belong to the corresponding Cartan subalgebras, then Proposition 2.1.7 a) gives the relation 2 of Definition 1.3.3.

Moreover, N_{s} and N_{t} coincide with the corresponding Cartan subalgebras, so they do not depend on the choice of a representation of e by means of {x_{i}, y_{i}}_{i=1}^{n}. Indeed, applying μ(S ⊗ id) to both sides of the equality in the definition of N_{s}, one can see from A4 that ε_{s} is identical on this algebra. Similarly ε_{t} is identical on N_{t}.

Proposition 2.1.10 ε(x) = T r L_{x} (∀x ∈ N_{t}), where the linear operator L_{x} acting in N_{t} is defined by L_{x}(n) = xn (∀n ∈ N_{t}), T r L_{x} is its usual trace.

Proof. A4’ shows that ε ∘ ε_{t} = ε, then from Corollary 2.1.9 and Remark 1.3.4 d) we have for all x ∈ M, y ∈ N_{t}: ε(xy) = ε(yS(x)).

Then in the representation (S ⊗ id)e = ∑_{i=1}^{n} y′_{i} ⊗ y_{i} = ∑_{i=1}^{n} y_{i} ⊗ y′_{i}, where y′_{i} = S(x_{i}), y_{i} ∈ N_{t}, we have from Proposition 2.1.6 : ε(y′_{k}y_{i}) = ε(y_{i}y′_{k}) = δ_{i,k} (∀i, k = 1, ..., n). This means that ε is central on N_{t} and that ∑_{i=1}^{n} y′_{i} · y_{i} = ∑_{i=1}^{n} y_{i} · y′_{i} = 1. Now one can identify N_{t} with its dual linear space by means of the non-degenerated duality given by < y, z >= ε(yz) = ε(zy) (∀y, z ∈ N_{t}). Then, by the definition of a trace, for every x ∈ N_{t} we have (using the centrality of ε): T r(L_{x}) = ∑_{i=1}^{n} < xy_{i}, y′_{i} >= ε(x(∑_{i=1}^{n} y_{i} · y′_{i})) = ε(x). In particular, ε(1) = dim N_{t}.

So the restriction of ε to N_{t} is a faithful trace. On the other hand, using Axioms 1),2) of Definition 2.1.1, one can easily show that ε(ε_{t}(x)^{∗}ε_{t}(y)) = ε(x^{∗}y) (∀x, y ∈ M), so ε itself is a positive functional on M.

Proposition 2.1.11 Let N_{s} ≅ ⊕_{α=1}^{K} M_{n_{α}}ℂ and { f_{pq}^{α} }_{p,q=1...n_{α}}^{α=1...K} be a system of matrix units in N_{s}. Then

e = ∑_{α} \frac{1}{n_{α}} ∑_{pq} f_{pq}^{α} ⊗ S(f_{qp}^{α}).

Proof. Let us write e = ∑_{αpq} f_{pq}^{α} ⊗ g_{pq}^{α}, where g_{pq}^{α} ∈ N_{t}. Using Proposition 1.3.7 we can write

e = ∑_{αpq} f_{pq}^{α} ⊗ S(f_{pp}^{α})g_{pq}^{α}S(f_{qq}^{α}) = ∑_{αpq} c_{pq}^{α} f_{pq}^{α} ⊗ S(f_{qp}^{α}),

for some scalars c_{pq}^{α}. Proposition 2.1.6 and Remark 1.3.4(d) imply that

1 = c_{pq}^{α}ε(f_{pq}^{α}S(f_{qp}^{α})) = c_{pq}^{α}ε(f_{pq}^{α}f_{qp}^{α}) = c_{pq}^{α}ε(f_{pp}^{α}),

but ε(f_{pp}^{α}) = n_{α} by Proposition 2.1.10, therefore c_{pq}^{α} = \frac{1}{n_{α}}, from where the result follows.

Remark 2.1.12 Morphisms of weak Kac algebras preserve Cartan subalgebras.

Indeed, let ρ : M → M̃ be a morphism and Ñ_{s}, Ñ_{t} be Cartan subalgebras of M̃. Let us write Δ(1) = ∑_{i=1}^{n} x_{i} ⊗ y_{i} with {x_{i}}_{i=1}^{n} linearly independent. Then N_{s} = span{x_{i}} and since Δ(1M̃) = ∑_{i=1}^{n} ρ(x_{i}) ⊗ ρ(y_{i}) we have Ñ_{s} = span{ρ(x_{i})}, i.e., ρ|_{N_{s}} is surjective. On the other hand, by Proposition 2.1.10 n = dim N_{s} = ε_{M}(1) = εM̃(1) = dim Ñ_{s}, so ρ|_{N_{s}} is injective and N_{s} ≅ Ñ_{s}. Similarly, N_{t} ≅ Ñ_{t}.

Corollary 2.1.13 Weak Kac algebras with Cartan subalgebras isomorphic to the given C^{∗}-algebra N form a subcategory of the category of weak Kac algebras. This subcategory is closed under duality.

We will show in Section 3 that any C^{∗}-algebra N can appear as a Cartan subalgebra of some weak Kac algebra.

2.2 The counital representation

Let M = ⊕_{i∈I} M_{d_{i}}(ℂ) be the C^{∗}-algebra of a weak Kac algebra, P_{i} (i ∈ I, I is a finite set) be the minimal central projections (i.e., selfadjoint idempotents) of M; π_{i} be the class of the irreducible representation x ↦ P_{i}x ∈ M_{d_{i}}(ℂ), x ∈ M. Clearly, {π_{i}, i ∈ I} is the set of all classes of irreducible representations of M. Let χ_{i} be the character of π_{i}.

Observe that the classes of non-degenerate representations of M form a ring K_{0}(M) : if ρ_{i} : M → B(H_{i}), i = 1, 2 are two representations, then

(ρ_{1} ⊕ ρ_{2})(x) := ρ_{1}(x) ⊕ ρ_{2}(x) and (ρ_{1} × ρ_{2})(x) := (ρ_{1} ⊗ ρ_{2})Δ(x)

are representations of M in Hilbert spaces H_{1} ⊕ H_{2} and (ρ_{1} ⊗ ρ_{2})e(H_{1} ⊗ H_{1}) respectively. The set of all irreducible classes {π_{i}, i ∈ I} forms a linear basis of K_{0}(M). For the properties of K_{0}-rings of usual Kac algebras and semisimple Hopf algebras see [16].

The class of a representation ρ is completely determined by its (normalized) character χ_{ρ}, the product and the direct sum of representations correspond to the product and the direct sum of characters respectively. Let ρ^{∗} be the representation corresponding to the character χ_{ρ} ∘ S, then the map ρ ↦ ρ^{∗} defines an antimultiplicative involution in K_{0}(M).

Consider the counital representation π_{ε} of M associated by the GNS-con-struction with the positive functional ε. Proposition 2.1.10 shows that ε is faithful on N_{t} and we also have ε((ε_{t}(x) − x)^{∗}(ε_{t}(x) − x)) = 0 (∀x ∈ M), therefore, π_{ε} acts in the Hilbert space N_{t} equipped with a scalar product (x, y) := ε(y^{∗}x) in the following way:

π_{ε}(x)y = ε_{t}(xy) for all x ∈ M, y ∈ N_{t}.

Proposition 2.2.1 (cf. [3]) The class of π_{ε} is a unit for K_{0}(M), i.e., π_{ε}×ρ and ρ × π_{ε} are equivalent to ρ for any representation ρ.

Proof. Let us compute the character χ_{ε} = Tr π_{ε}. Let e = ∑_{k} x_{k} ⊗ y_{k}, then the set {y_{k}} forms a basis of N_{t}. Therefore, for all x ∈ M we have

π_{ε}(x)y_{k} = ε_{t}(xy_{k}) = ∑_{i} ε(x_{i}xy_{k})y_{i},

and

χ_{ε}(x) = ∑_{k} ε(x_{k}xy_{k}) = ε(μ(e(x ⊗ 1)))
= ε ⊗ ε((1 ⊗ x_{(1)})e(x_{(2)} ⊗ 1)) = ε(μ ∘ Δ(x)).

Using this formula we compute

(χ_{ε} ⊗ χ)Δ(x) = (ε ⊗ χ)((μ ⊗ id)(Δ ⊗ id)Δ(x))
= (ε ⊗ χ)((x_{(1)} ⊗ 1)e(1 ⊗ x_{(2)})) = χ(x),

for all characters χ. Similarly, one can show that (χ ⊗ χ_{ε})Δ(x) = χ(x). Hence, π_{ε} × ρ, ρ × π_{ε}, and ρ are equivalent for all representations ρ.

As a representation of a f-d C^{∗}-algebra, π_{ε} is equivalent to the sum of irreducible representations with some multiplicities:

π_{ε} ≃ ⊕_{i∈S} ν_{i}π_{i},

where S ⊂ I and ν_{i} ≥ 1 for all i ∈ S.

Proposition 2.2.2 (cf. [3], 2.4.) The representation π_{ε} is multiplicity free, i.e. ν_{i} = 1 for all i ∈ S. We have π_{i}^{∗} = π_{i} for any i ∈ S and π_{ε}^{∗} = π_{ε}.

Proof. Since π_{ε} is the unit of K_{0}(M), we have π_{ε}^{∗} = π_{ε} and π_{j} = π_{j} ×π_{ε} = ∑_{i∈S} π_{j} × ν_{i}π_{i} for all j ∈ I. In particular, the right-hand side of the last equality must be irreducible. This is the case iff there exists a unique index u(j) ∈ S such that ν_{u(j)} = 1 and π_{j} × π_{i} = δ_{i u(j)} π_{j} for all i ∈ S.

On the other hand, for any irreducible π_{j} one has π_{j} × π_{j}^{∗} ≠ 0. Indeed, if (P_{j} ⊗ S(P_{j}))e = 0, then applying μ(id ⊗ S) we get P_{j} = 0, a contradiction. This implies π_{u(j)} = π_{j}^{∗} and, hence, π_{j} × π_{j}^{∗} = π_{j} for all j ∈ S. Applying the involution to the last equality we get π_{j} = π_{j}^{∗} and ν_{u(j)} = ν_{j} = 1 (∀j ∈ S).

Remark 2.2.3 Let p_{ε} be the support of ε, i.e., a non-zero projection p minimal with respect to the property ε(xp) = ε(px) = ε(x) (∀x ∈ M) (which is equivalent to ε_{s}(px) = ε_{s}(x) and ε_{t}(xp) = ε_{t}(x)). Proposition 2.2.2 shows that p_{ε} = ∑_{i∈S} p_{i}, where p_{i} are minimal in M projections whose central supports P_{i} are mutually orthogonal. Obviously ε∘S = ε implies S(p_{ε}) = p_{ε}.

Since π_{ε} is multiplicity free, we have

dim N_{t} = dim H_{ε} = ∑_{i∈S} (dim P_{i}M)^{\frac{1}{2}} = ∑_{i∈S} dim Mp_{i} = dim Mp_{ε}.

Therefore, the map ε_{t} : Mp_{ε} → N_{t} is a linear isomorphism of vector spaces. In other words, {x ∈ M|ε_{t}(x) = 0} = {x ∈ M|xp_{ε} = 0}, this implies (x − ε_{t}(x))p_{ε} = 0 and xp_{ε} = ε_{t}(x)p_{ε} (∀x ∈ M). Replacing x by S(x) and applying S to the last equality, we also get p_{ε}x = p_{ε}ε_{s}(x).

Proposition 2.2.4 Let I_{s} = { y ∈ M | yx = yε_{s}(x), ∀x ∈ M } and I_{t} = { y ∈ M | xy = ε_{t}(x)y, ∀x ∈ M }. Then I_{s} ∩ I_{t} = { y ∈ M | y = p_{ε}yp_{ε} }.

Proof. Clearly, I_{s} is a left ideal in M, so there exists a projection p ∈ M such that I_{s} = Mp. Since p_{ε} ∈ I_{s}, we have p_{ε} = p_{ε}p = p_{ε}pp_{ε} and p′ = p − p_{ε} ∈ I_{s} is a projection orthogonal to p_{ε}. Hence,

p′ = p′^{2} = p′ε_{s}(p′) = p′ε_{s}(p_{ε}p′) = 0.

So p = p_{ε} and I_{s} = Mp_{ε}, similarly I_{t} = p_{ε}M; these equalities give the result.

Theorem 2.2.5 Given a weak Kac algebra, there exist (see Definition 1.3.8) a) a unique Haar projection Λ = p_{ε} ∈ M;

b) a unique normalized Haar functional ϕ_{ε} = p̂_{ε} ∈ M̂.

Proof. It suffices to prove part a) since b) is just its dual version. Proposition 2.2.4 implies that any Λ satisfying 1.3.8 a) has the form Λ = p_{ε}Λp_{ε} = ∑_{i∈S} λ_{i}p_{i} for some scalars λ_{i}; then, due to the above relations, ε(p_{i}) = ε(Λp_{i}) = λ_{i}ε(p_{i}) from where λ_{i} = 1 and Λ = p_{ε}. Since p_{ε} satisfies 1.3.8 a), the proof is completed.

Note that ϕ_{ε} is the Haar projection in M̂.

Now let us introduce and study a counital quotient of a weak Kac algebra arising from its counital representation π_{ε}.

Proposition 2.2.6 (a) Let {π_{i}}_{i∈S} be the set of irreducible representations of M contained in the decomposition of π_{ε}. Then for any i ∈ S M_{i} = P_{i}M is a weak Kac algebra with a comultiplication Δ_{i}(x) = (P_{i} ⊗ P_{i})Δ(x), an antipode S_{i}(x) = S(x), and a counit ε_{i}(x) = ε(x).

(b) Let P_{ε} = ∑_{i∈S} P_{i} be the central support of ε. Then x ↦ P_{ε}x is a surjective morphism of weak Kac algebras M and M_{ε} = ⊕_{i∈S} M_{i}, i.e., M_{ε} is a quotient of M.

Proof. Proposition 2.2.2 implies that π_{i} = π_{i}^{∗} and π_{i} × π_{j} = δ_{ij}π_{i} for all i, j ∈ S. This means that (P_{i} ⊗ P_{i})Δ = (P_{i} ⊗ P_{i})Δ ∘ P_{i} and S ∘ P_{i} = P_{i} ∘ S. In particular, Δ_{i}(x) ∈ P_{i}M ⊗ P_{i}M and S_{i}(x) ∈ P_{i}M (∀x ∈ P_{i}M). We also have that (P_{ε} ⊗ P_{ε})Δ = Δ_{i}. x ∈ P_{i}M.

Clearly, (P_{i}M, Δ_{i}, S_{i}) is a gen. coinvolutive C^{∗}-bialgebra. Due to the definitions of Δ_{i}, S_{i}, ε_{i} and the centrality of P_{i}, ε_{i} is a counit for P_{i}M satisfying the axioms of Definition 2.1.1, which proves (a). Obviously, M_{ε} = ⊕_{i} M_{i} is a weak Kac algebra. To see that x ↦ P_{ε}x is a (clearly surjective) morphism of weak Kac algebras, it suffices to note that (P_{ε} ⊗ P_{ε})Δ(x) = ∑_{i} Δ_{i}(P_{ε}x), P_{ε}S(x) = ∑_{i} S_{i}(P_{ε}x), and ε(x) = ∑_{i} ε_{i}(P_{ε}x) ∀x ∈ M.

Definition 2.2.7 The weak Kac algebra M_{ε} defined in Proposition 2.2.6 is said to be a counital quotient of M.

By Proposition 2.2.6(a), M_{ε} is a direct sum of weak Kac algebras which are simple as algebras. We will classify all such algebras in Section 3.

Remark 2.2.8 M_{ε} is a minimal quotient weak Kac algebra of M, i.e. if ρ : M → M̃ is a surjective morphism of weak Kac algebras, then ρ(M_{ε}) ≅ M̃_{ε}.

Indeed, ε(x) = ε̃(ρ(x)) (∀x ∈ M) implies ρ(P_{ε}) = P̃_{ε}, so ρ|_{M_{ε}} : M_{ε} → M̃_{ε} is surjective. If it is not injective then ρ(P_{i}) = 0 for some i ∈ S, which contradicts to ε(P_{i}) ≠ 0. Therefore, ρ|_{M_{ε}} is an isomorphism.

2.3 Haar traces

In fact, ϕ_{ε} = pε̂ is a normalized Haar trace, in order to show that we need some preliminary statements. We use the notations L_{x} : y → xy, R_{α}^{∗} : y → (id ⊗ α)Δ(y) (∀x, y ∈ M, α ∈ M̂) [14]. These two maps are clearly ∗- representations of the ∗-algebras M and M̂ respectively in End(M).

Proposition 2.3.1 a) R_{α}^{∗}L_{x} = α_{(1)}(x_{(2)})L_{x_{(1)}}R_{α_{(2)}}^{∗}, b) R_{ε̂_{t}(α)}^{∗} = L_{(id⊗α)e}, where ε̂_{t}(α) = (ε̂⊗ idM̂)(Δ̂(ε)(α ⊗ ε)).

Proof. a) We have

R_{α}^{∗}L_{x}y = (id ⊗ α)Δ(xy) = x_{(1)}y_{(1)}α(x_{(2)}y_{(2)})
= x_{(1)}y_{(1)}Δ̂(α)(x_{(2)} ⊗ y_{(2)}) = x_{(1)}y_{(1)}α_{(1)}(x_{(2)})α_{(2)}(y_{(2)})
= α_{(1)}(x_{(2)})L_{x_{(1)}}R_{α_{(2)}}^{∗}y.

b) Clearly the map ε̂_{t} is dual to the map ε_{t}, and we have, using A3”:

R_{ε̂_{t}(α)}^{∗}(x) = (id ⊗ ε̂_{t}(α))Δ(x) = (id ⊗ α ∘ ε_{t})(Δ(x))
= (id ⊗ α)(e(x ⊗ 1)) = L_{(id⊗α)e}(x).

Corollary 2.3.2 R_{α_{(1)}}^{∗}L_{x}R_{Ŝ(α_{(2)})}^{∗} = L_{x_{(1)}}α(x_{(2)}).

Proof. Indeed, using the notation (Δ̂ ⊗ idM̂)Δ̂(α) = (idM̂ ⊗ Δ̂)Δ̂(α) = α_{(1)} ⊗ α_{(2)} ⊗ α_{(3)} and Proposition 2.3.1, we have:

R_{α_{(1)}}^{∗}L_{x}R_{Ŝ(α_{(2)})}^{∗} = L_{x_{(1)}}α_{(1)}(x_{(2)})R_{α_{(2)}Ŝ(α_{(3)})}^{∗} = L_{x_{(1)}}α_{(1)}(x_{(2)})R_{ε̂_{t}(α_{(2)})}^{∗}
= L_{x_{(1)}}α_{(1)}(x_{(2)})L_{(id⊗α_{(2)})e} = L_{x_{(1)}}α_{(1)}(x_{(2)})L_{1_{(1)}}α_{(2)}(1_{(2)})
= L_{x_{(1)}1_{(1)}}α(x_{(2)}1_{(2)}) = L_{x_{(1)}}α(x_{(2)}).

Proposition 2.3.3 Let us consider the following faithful trace on M:

θ(x) = T r(L_{x}),

where T r is the usual trace of a linear operator. Then we have:

(θ ⊗ id)(Δ(x)) = (θ ⊗ S)(e(x ⊗ 1)) = (θ ⊗ ε_{s} ∘ S)(Δ(x)) (∀x ∈ M).

Proof. Using Proposition 2.3.1, Corollary 2.3.2 and the properties of T r, one has for any x ∈ M, α ∈ M̂:

(θ ⊗ α)(Δ(x)) = T r[L_{x_{(1)}}α(x_{(2)})] = T r[R_{α_{(1)}}^{∗}L_{x}R_{Ŝ(α_{(2)})}^{∗}]
= T r[R_{Ŝ(α_{(2)})α_{(1)}}^{∗}L_{x}] = T r[R_{ε̂_{t}(Ŝ(α))}^{∗}L_{x}]
= T r[L_{(id⊗Ŝ(α))e}L_{x}] = T r[L_{(id⊗α∘S)(e(x⊗1))}]
= (θ ⊗ α ∘ S)(e(x ⊗ 1)).

Remark 2.3.4 Since (θ ⊗ id)(Δ(x)) ∈ N_{s}, θ is a Haar trace (θ ∘ S = θ from S^{2} = id - see [11], Lemma 3.1), but generally it is not normalized.

Theorem 2.3.5 Let (M, Δ, S, ε) be a weak Kac algebra and ϕ_{ε} a Haar trace on it. Then the collection (M, Δ, S, ϕ_{ε}) is a gen. Kac algebra.

Proof. It is enough to show that

(ϕ_{ε} ⊗ id)[(y ⊗ 1)Δ(x)] = S[(ϕ_{ε} ⊗ id)(Δ(y)(x ⊗ 1))] (∀x, y ∈ M).

Indeed, using successively A3”, A4’ and Definition of ε_{t}, one has:

(ϕ_{ε} ⊗ id)[(y ⊗ 1)Δ(x)] = (ϕ_{ε} ⊗ id)[Δ(x)e(y ⊗ 1)]
= (ϕ_{ε} ⊗ id)[Δ(x)(id ⊗ ε_{t})Δ(y)]
= (ϕ_{ε} ⊗ id)[Δ(x)(id ⊗ μ(id ⊗ S)Δ)Δ(y)]
= (ϕ_{ε} ⊗ id)[Δ(xy_{(1)})]S(y_{(2)})
= (ϕ_{ε} ⊗ id)[e(xy_{(1)} ⊗ 1)]S(y_{(2)})
= S[(ϕ_{ε} ⊗ id)(Δ(y)(x ⊗ 1))].

Now let us compute Δ(p_{ε}).

Lemma 2.3.6 Let A = M_{d}(ℂ), B = M_{d}(ℂ), and S : A → B be an antiautomorphism of algebras. Let Q ∈ A ⊗ B be a projection of rank 1 in A ⊗ B such that μ(S ⊗ id_{B})Q = 1_{B} and μ(id_{A} ⊗ S^{−1})Q = 1_{A}. Then Q = \frac{1}{d} ∑_{km} e_{km} ⊗ S(e_{mk}) for any system \{ e_{km} \}_{k,m=1}^{d} of matrix units in A.

Proof. If {e_{km} }_{k,m=1}^{d} is a system of matrix units in A, then { S(e_{nl}) }_{l,n=1}^{d} is a system of matrix units in B. Since Q has rank 1, we can write it as

Q = ∑_{klmn} γ_{kl}β_{mn} e_{km} ⊗ S(e_{nl}),

for some β_{mn}, γ_{kl} ∈ ℂ (k, l, m, n = 1 … d). By the hypothesis,

1_{B} = μ(S ⊗ id_{B})Q = ∑_{klmn} γ_{kl}β_{mn} S(e_{nl}e_{km}) = (∑_{k} γ_{kk}) ∑_{mn} β_{mn} S(e_{nm}),

so β_{mn} = δ_{mn} β for some β ∈ ℂ. Similarly, γ_{kl} = δ_{kl} γ for some γ ∈ ℂ. Thus, Q = γβ ∑_{km} e_{km} ⊗ S(e_{mk}) and condition Q^{2} = Q gives γβ = \frac{1}{d}.

Lemma 2.3.7 Let P_{ε} = ∑_{i∈S} P_{i} be the central support of p_{ε}. Then

(P_{ε} ⊗ P_{ε})Δ(p_{ε}) = ∑_{i∈S} \frac{1}{d_{i}} ∑_{kl} e_{kl}^{(i)} ⊗ S(e_{lk}^{(i)}),

where { e_{kl}^{(i)} }_{k,l=1}^{d_{i}} is any system of matrix units in P_{i}M.

Proof. The proof of Proposition 2.2.2 shows that P_{i} = S(P_{i}), i ∈ S and (P_{ε} ⊗ P_{ε})Δ(p_{ε}) = ∑_{i∈S} Q_{i}, where Q_{i} = (P_{i} ⊗ P_{i})Δ(p_{ε}) is a projection of rank 1 in P_{i}M ⊗ P_{i}M such that μ(S ⊗ id)Q_{i} = P_{i}ε_{s}(p_{ε}) = P_{i}. Similarly μ(id ⊗ S)Q_{i} = P_{i}. Applying Lemma 2.3.6 we get Q_{i} = \frac{1}{d_{i}} ∑_{kl} e_{kl}^{(i)} ⊗ S(e_{lk}^{(i)}), from where the result follows.

Lemma 2.3.8 (θ ⊗ id)Δ(p_{ε}) = (id ⊗ θ)Δ(p_{ε}) = 1.

Proof. Let us choose a system of matrix units { e_{kl}^{(i)} }_{k,l=1}^{d_{i}} in P_{ε}M in

such a way that p_{i} = e_{11}^{(i)} and S(e_{lk}^{(i)}) = e_{kl}^{(i)} for all i, k, l. We have, using Remark 2.3.4, relation Δ(p_{ε}) = (P_{ε} ⊗ P_{ε})Δ(p_{ε}) and Lemma 2.3.7:

(θ ⊗ id)Δ(p_{ε}) = (θ ⊗ ε_{s})Δ(p_{ε}) = (θ ⊗ ε_{s})((1 ⊗ p_{ε})Δ(p_{ε}))
= ∑_{i∈S} \frac{1}{d_{i}} ∑_{kl} θ(e_{lk}^{(i)}) ε_{s}(e_{11}^{(i)}e_{kl}^{(i)}) = ∑_{i∈S} \frac{1}{d_{i}} ∑_{kl} d_{i} δ_{kl} ε_{s}(δ_{1k}e_{1l}^{(i)})
= ∑_{i∈S} ε_{s}(e_{11}^{(i)}) = ε_{s}(p_{ε}) = ε_{s}(1) = 1.

The second part is similar.

Let us generalize now the formula known for Kac algebras ([7], 6.3.7).

Proposition 2.3.9 For any system { e_{kl}^{(i)}; k, l = … d_{i}, i ∈ I } of matrix units in M

Δ(p_{ε}) = ∑_{i∈I} \frac{1}{d_{i}} ∑_{kl} e_{kl}^{(i)} ⊗ S(e_{lk}^{(i)}).

In particular, ςΔ(p_{ε}) = Δ(p_{ε}).

Proof. Let us write Δ(p_{ε}) = ∑_{ij∈I} R_{ij}, where R_{ij} is a projection in P_{i}M ⊗ P_{j}M. ¿From Lemma 2.3.8 we have

1 = (id ⊗ θ)Δ(p_{ε}) = Σ_{i} (Σ_{j} d_{j}(id ⊗ χ_{j})R_{ij})) P_{i}
1 = (θ ⊗ id)Δ(p_{ε}) = Σ_{j} (Σ_{i} d_{i}(χ_{i} ⊗ id)R_{ij})) P_{j}.

Applying χ_{i} to the first equality and χ_{j} to the second one, we get :

d_{i} = Σ_{j} d_{j} rank(R_{ij}), d_{j} = Σ_{i} d_{i} rank(R_{ij}).

Note that ε_{s}(p_{ε}) = 1 implies that R_{ii^{∗}} ≠ 0 (we can assume that the involution acts on the set I). Using this fact and the above relations, we conclude that rank(R_{ij}) = δ_{ij^{∗}}. Thus, μ(S ⊗ id)R_{ii^{∗}} = μ(id ⊗ S)R_{ii^{∗}} = P_{i} and application of Lemma 2.3.6 completes the proof.

Corollary 2.3.10 ϕ_{ε} is the normalized Haar trace.

Proof. Since ϕ_{ε} = p̂_{ε} is a Haar projection of M̂, the centrality follows from Proposition 2.3.9. In order to see that ϕ_{ε} is positive and faithful let us take a system { f_{kl}^{(i)} } of matrix units in M̂. Then for any x ∈ M:

ϕ_{ε}(xx^{∗}) = < x ⊗ x^{∗}, Δ̂(p̂_{ε}) >= ∑_{i∈I} \frac{1}{d_{i}} ∑_{kl} f_{kl}^{(i)}(x) Ŝf_{lk}^{(i)}(x^{∗})
= ∑_{i∈I} \frac{1}{d_{i}} ∑_{kl} | f_{kl}^{(i)}(x) |^{2} ≥ 0,

which equals to 0 iff x = 0.

Remark 2.3.11 Let us describe all Haar traces on a weak Kac algebra. Proposition 2.3.9 shows that ςΔ(p_{i}) = Δ(p_{i}) (∀i ∈ S), therefore, any linear combination ∑_{i∈S} λ_{i}p_{i} with positive coefficients λ_{i} defines a trace on M̂. But the proof of Theorem 2.2.5 shows that any Haar trace is of this form. Such a trace is faithful iff all λ_{i} > 0.

2.4 Haar conditional expectations

Proposition 2.4.1 Given a weak Kac algebra (M, Δ, S, ε) and a normalized Haar trace ϕ_{ε} on it, there exist unique faithful target and source Haar conditional expectations E_{t} : M → N_{t} and E_{s} : M → N_{s} such that ϕ_{ε} = ϕ_{ε} ∘ E_{t} and ϕ_{ε} = ϕ_{ε} ∘ E_{s}. They could be defined as follows:

E_{t}(x) = (id⊗ϕ_{ε})Δ(x) = (S⊗ϕ_{ε})((1⊗x)e), E_{s}(x) = (ϕ_{ε}⊗id)Δ(x) (∀x ∈ M).

We also have E_{t} ∘ S = S ∘ E_{s} and

(E_{t} ⊗ E_{t})((Δ(x)(y ⊗ z)) = ς(E_{t} ⊗ E_{t})((S(y) ⊗ x)Δ(z)) (∀x, y, z ∈ M).

Proof. Clearly both E_{t} and E_{s} are linear unital ∗-maps from M to N_{t} and N_{s} respectively such that E_{t}|N_{t} = id_{N_{t}}, E_{s}|N_{s} = id_{N_{s}}, and that E_{t} ∘ S = S ∘ E_{s}. We also have ∀x ∈ M, n ∈ N_{t}:

E_{t}(x) · n = (S ⊗ ϕ)((1 ⊗ x)e) · n = (S ⊗ ϕ)((1 ⊗ x)(S(n) ⊗ 1)e)
= (S ⊗ ϕ)((1 ⊗ xn)e) = E_{t}(xn)

and

ϕ_{ε} ∘ E_{t}(x) = (ϕ_{ε} ∘ S ⊗ ϕ_{ε})((1 ⊗ x)e) = ϕ_{ε}(x).

Now it follows from ([9], 2.6.2) that E_{t} is indeed a faithful conditional expectation. The relations (id⊗E_{t})∘Δ = Δ∘E_{t} and (E_{s} ⊗id)∘Δ = Δ∘E_{s} are obvious from the coassociativity of Δ. In order to prove the last relation, it suffices to show that (α ⊗ β)[(E_{t} ⊗ E_{t})(Δ(x)(y ⊗ z))] = (α ⊗ β)[ς(E_{t} ⊗ E_{t})(S(y)⊗x)Δ(z))] for any linear functionals α, β on N_{t}. But these functionals can be represented as α(·) = ϕ_{ε}(n·), β(·) = ϕ_{ε}(n′·) for some n, n′ ∈ N_{t}. Then, using the properties of traces, conditional expectations, the invariancy of ϕ_{ε} and Propositions 2.1.8, 1.3.7, we have ∀x, y, z ∈ M:

(α ⊗ β)[(E_{t} ⊗ E_{t})(Δ(x)(y ⊗ z))] =
= (ϕ_{ε} ⊗ ϕ_{ε})[(n ⊗ n′)(E_{t} ⊗ E_{t})(Δ(x)(y ⊗ z))]
= (ϕ_{ε} ⊗ ϕ_{ε})[(E_{t} ⊗ E_{t})((n ⊗ n′)Δ(x)(y ⊗ z))]
= (ϕ_{ε} ⊗ ϕ_{ε})[(n ⊗ n′)Δ(x)(y ⊗ z)]
= (ϕ_{ε} ⊗ ϕ_{ε})[(n ⊗ 1)eΔ(x)(y ⊗ zn′)]
= (ϕ_{ε} ⊗ ϕ_{ε})[Δ(nx)(y ⊗ zn′)]
= (ϕ_{ε} ⊗ ϕ_{ε})[(S(y) ⊗ nx)Δ(zn′)]
= (ϕ_{ε} ⊗ ϕ_{ε})[(S(y) ⊗ nx)Δ(z)(n′ ⊗ 1)]

= (ϕ_{ε} ⊗ ϕ_{ε})[(n′ ⊗ n)(S(y) ⊗ x)Δ(z)]
= (ϕ_{ε} ⊗ ϕ_{ε})[(E_{t} ⊗ E_{t})((n ⊗ n′)ς((S(y) ⊗ x)Δ(z))]
= (ϕ_{ε} ⊗ ϕ_{ε})[(n ⊗ n′)ς((E_{t} ⊗ E_{t})((S(y) ⊗ x)Δ(z))]
= (α ⊗ β)[ς(E_{t} ⊗ E_{t})((S(y) ⊗ x)Δ(z))]

Remark 2.4.2 For a general Haar trace ϕ the conditional expectations E_{t} and E_{s} satisfying all the above properties are the orthogonal projectors on N_{t} and N_{s} respectively in the Hilbert space given by the GNS-construction for (M, ϕ), but they cannot be given by the above formulae (see [26]).

Proposition 2.4.3 a) In a weak Kac algebra any of the relations e(1⊗x) = 0, e(x ⊗ 1) = 0 implies x = 0.

b) If y ∈ M and one of the elements e(1⊗y), e(y ⊗1), (1⊗y)e, (y ⊗1)e is positive, then y is positive.

Proof. a) Since μ(S ⊗ id)(e(1 ⊗ x)) = x (∀x ∈ M), the first of the statements a) is clear. The second one can be proved similarly.

b) If e(y ⊗ 1) is positive and ϕ_{ε} is the normalized Haar trace, then:

(ϕ_{ε} ⊗ ϕ_{ε})((x^{∗} ⊗ 1)e(y ⊗ 1)(x ⊗ 1)) = ϕ_{ε}(x^{∗}yx) ≥ 0 (∀x ∈ M).

This means that y is positive. Now all the other statements are clear. The following statement for gen. Kac algebras was obtained in [26].

Proposition 2.4.4 a) the map y → E_{t}^{o}(y) = μ(S⊗id)((1⊗y)e) is a faithful conditional expectation from M to M ∩ N′_{t} satisfying the relations

e(1 ⊗ y)e = e(1 ⊗ E_{t}^{o}(y)) = (1 ⊗ E_{t}^{o}(y))e.

b) For any fixed Haar trace ϕ, E_{t}^{o}(y) is the unique faithful conditional expectation from M to M ∩ N′_{t} such that ϕ = ϕ ∘ E_{t}^{o}.

Proof. a) For any representation e = ∑_{i=1}^{n} x_{i} ⊗ y_{i} with minimal possible n we have, using Proposition 1.3.7:

(1 ⊗ E_{t}^{o}(y))e = (1 ⊗ ∑_{i=1}^{n} S(x_{i})yy_{i})e = ∑_{i=1}^{n}(1 ⊗ S(x_{i})y)(1 ⊗ y_{i})e

= ∑_{i=1}^{n}(S(y_{i}) ⊗ S(x_{i})y)e = ς(S ⊗ S)(e)(1 ⊗ y)e
= e(1 ⊗ y)e (∀y ∈ M).

Similarly one can prove the other relation. These relations and Proposition 1.3.7 imply for any y ∈ M, n ∈ N_{t}:

(1 ⊗ E_{t}^{o}(y)n)e = (1 ⊗ E_{t}^{o}(y))(1 ⊗ n)e = (S(n) ⊗ 1)(1 ⊗ E_{t}^{o}(y))e
= (S(n) ⊗ 1)e(1 ⊗ y)e = (1 ⊗ n)e(1 ⊗ y)e
= (1 ⊗ nE_{t}^{o}(y))e.

Using Proposition 2.4.3 a), we see that E_{t}^{o}(y)n = nE_{t}^{o}(y), so E_{t}^{o}(y) ∈ M ∩N′_{t}. If x, x′ ∈ M ∩ N′_{t}, y ∈ M, then

E_{t}^{o}(xyx′) = ∑_{i=1}^{n} S(x_{i})(xyx′)y_{i} = x(∑_{i=1}^{n} S(x_{i})yy_{i})x′ = xE_{t}^{o}(y)x′.

If now y = zz^{∗} ∈ M is positive, then

e(1 ⊗ E_{t}^{o}(zz^{∗})) = e(1 ⊗ zz^{∗})e = e(1 ⊗ z)[e(1 ⊗ z)]^{∗} ≥ 0.

So, according to Proposition 2.4.3 b), E_{t}^{o}(zz^{∗})) is positive, and if E_{t}^{o}(zz^{∗})) = 0, then Proposition 2.4.3 a) implies z = 0. Since E_{t}^{o}(1) = ε_{t}(e) = 1, then E_{t}^{o} is indeed the faithful conditional expectation from M to M ∩ N′_{t}.

b) For any x ∈ M one has: ϕ ∘ E_{t}^{o}(x) = ϕ(∑_{i=1}^{n} y_{i}S(x_{i})x) = ϕ(x), so the uniqueness of E_{t}^{o} is a consequence of ([9], 2.6.2.)

2.5 Generalized Kac algebras

Let K = (M, Δ, S, ϕ) be a gen. Kac algebra and K̂ = (M̂, Δ̂, Ŝ, ϕ̂) the corresponding dual gen. Kac algebra [27]. As a linear space, M̂ can be identified with M because any linear functional α on M can be represented as α(x) = ϕ(xy) = ϕ(yx) for some y ∈ M. Let us consider the following linear functional ε : M → C : ε(x) = ϕ(ˆ1x) (∀x ∈ M), where ˆ1 ∈ M corresponds to the unit in M̂. From [27], §5 we have S(ˆ1) = (ˆ1)^{∗} = ˆ1,

(ϕ ⊗ ϕ)[Δ(ˆ1)(x ⊗ y)] = ϕ(xS(y)) (∀x, y ∈ M),

which implies respectively ε∘S(x) = ε(x), ε(x^{∗}) = \overline{ε(x)} and (id⊗ε)[Δ(x)] = x (∀x ∈ M). The last one gives: (ε ⊗ id)[Δ(x)] = x (∀x ∈ M).

Now let us show that (M, Δ, S, ε) is a weak Kac algebra. According to Proposition 2.1.2 , it suffices to verify axioms A2-A4.

One can calculate, using the invariancy of ϕ and the latest relations:

(ϕ ⊗ id)[Δ(ˆ1)(x ⊗ y)] = (ϕ ⊗ id)[Δ(ˆ1)(x ⊗ 1)]y
= (ϕ ⊗ S)[(ˆ1 ⊗ 1)Δ(x)]y
= [S ∘ (ε ⊗ id)Δ(x)]y = S(x)y (∀x, y ∈ M),

which gives (ϕ ⊗ id)[Δ(ˆ1)A] = μ(S ⊗ id)(A) (∀A ∈ M ⊗ M). Similarly we have: (id ⊗ ϕ)[AΔ(ˆ1)] = μ(id ⊗ S)(A) (∀A ∈ M ⊗ M).

Let us first verify A4, using the invariancy of ϕ and the latest relations:

(id ⊗ ε)[(1 ⊗ x)Δ(1)] = (id ⊗ ϕ)[(1 ⊗ ˆ1x)Δ(1)] = (S ⊗ ϕ)[Δ(ˆ1x)]
= μ(S ⊗ id)Δ(x) (∀x ∈ M).

Then let us verify A2, using the same reasoning and also A4:

(ε ⊗ id)[(x ⊗ 1)Δ(y)] = (ϕ ⊗ id)[(ˆ1x ⊗ 1)Δ(y)]
= (ϕ ⊗ S)[Δ(ˆ1x)(y ⊗ 1)]
= (ϕ ⊗ S)[Δ(ˆ1)Δ(x)(y ⊗ 1)]
= S[μ(S ⊗ id)(Δ(x)(y ⊗ 1))]
= S[μ(S ⊗ id)Δ(x)]y
= (ϕ ⊗ S)[Δ(ˆ1x)] · y
= (ϕ ⊗ id)[(ˆ1x ⊗ 1)Δ(1)] · y
= (ε ⊗ id)[(x ⊗ 1)e(1 ⊗ y)].

Finally, it suffices to verify A3* (see Remark 2.1.3 b)). But this is obvious if one uses the following results obtained in [26]: a) N_{t} = M ∩ M̂ is a target Cartan subalgebra; b) e ∈ N_{s} ⊗ N_{t}. Indeed:

(id ⊗ ε ⊗ id)[(e ⊗ 1)(1 ⊗ e)] = (id ⊗ ε)(e(1 ⊗ 1_{(1)})) ⊗ 1_{(2)}
= (id ⊗ ε)(Δ(1_{(1)})) ⊗ 1_{(2)} = e.

Thus, we get the following

Theorem 2.5.1 Let (M, Δ, S, ϕ) be a gen. Kac algebra, ε(x) = ϕ(ˆ1x) (∀x ∈ M), where ˆ1 ∈ M is the unit in M̂. Then (M, Δ, S, ε) is a weak Kac algebra.

Corollary 2.5.2 On any gen. weak Kac algebra there is a unique Haar normalized trace.

Indeed, to every gen. Kac algebra, there corresponds (Theorem 2.5.1) a weak Kac algebra with (clearly unique) counit ε and then the unique normalized Haar trace on it (Theorem 2.2.5, Corollary 2.3.10).

Definition 2.5.3 A homomorphism π of gen. Kac algebras with normalized Haar traces ψ_{1} and ψ_{2} is such a homomorphism of their gen. coinvolutive C^{∗}-bialgebras (Definition 1.3.1 c)), that ψ_{2} ∘ π = ψ_{1}.

Clearly gen. Kac algebras form a category; Corollary 2.5.2 and Theorem 2.5.1 describe its equivalence to the category of weak Kac algebras.

2.6 Kac bimodules

The structure of a counital C^{∗}-bialgebra is not selfdual: there is a unit for its algebra but not necessarily a counit for its coalgebra. Now let us show that the existence of a counit of certain structure (motivated by Lemma 2.1.10), gives a weak Kac algebra structure.

Let θ_{t} and θ_{s} be the traces of left regular representations of Cartan subalgebras N_{t} and N_{s} respectively. Since S : N_{s} → N_{t} is an anti-∗-isomorphism, we have θ_{t} ∘ S = θ_{s}. Also θ_{s} ∘ ε_{s}(x) = θ_{s}(Sx_{(1)}x_{(2)}) = θ_{t} ∘ S(Sx_{(1)}x_{(2)}) = θ_{t}(Sx_{(2)}x_{(1)}) = θ_{t}(x_{(1)}Sx_{(2)}) = θ_{t} ∘ ε_{t}(x) (∀x ∈ M).

Denoting ε := θ_{s} ∘ ε_{s} = θ_{t} ∘ ε_{t} : M → ℂ, we have immediately ε ∘ S = ε and, using Remark 1.3.4 d) and self-adjointness of θ_{t}:

ε(x) = θ_{t} ∘ (ε_{t}(x^{∗})^{∗}) = \overline{θ_{t} ∘ ε_{t}(x^{∗})} = \overline{ε(x^{∗})}.

Theorem 2.6.1 A counital C^{∗}-bialgebra (M, Δ, S) is a weak Kac algebra if and only if (id ⊗ ε) ∘ Δ = id.

Proof. Any weak Kac algebra is a counital C^{∗}-bialgebra satisfying the above condition (Corollary 2.1.9, Proposition 2.1.10). Conversely:

1) Using Remark 1.3.4 c) and the condition of the theorem, we have the axiom A3* of a weak Kac algebra:

(id ⊗ ε ⊗ id)[(e ⊗ 1)(1 ⊗ e)] = (id ⊗ ε ⊗ id)(Δ ⊗ id)(e) = e.

2) From Remark 1.3.4 d) and Proposition 1.3.7 we have:

(ε ⊗ id)[Δ(y)(x ⊗ 1)] = (θ_{t} ∘ ε_{t} ⊗ id)[Δ(y)(S ∘ ε_{t}(x) ⊗ 1)] = yε_{t}(x) (x, y ∈ M);

for y = 1 this gives the axiom A4’ of a weak Kac algebra equivalent to A4.

3) Using the above relation 2) and the axiom A4’, we have the axiom A2’ of a weak Kac algebra (which is equivalent to A2):

(ε ⊗ id)[Δ(y)(x ⊗ 1)] = yε_{t}(x) = (ε ⊗ id)((1 ⊗ y)e(x ⊗ 1)),

The final result follows from Proposition 2.1.2 and Remark 2.1.3 b).

Let us remark that the condition of Theorem 2.6.1 makes sense also for counital Hopf bimodules (see step a) of the proof of Lemma 1.4.4).

Definition 2.6.2 A counital Hopf bimodule B = (N, M, t, Δ, S) is said to be a Kac bimodule if it satisfies the condition of Theorem 2.6.1.

Now, taking into account Lemma 1.4.4 and Theorem 2.6.1, we get

Theorem 2.6.3 The categories of Kac bimodules and weak Kac algebras are equivalent.

3 Examples

3.1 Actions of finite groups on weak Kac algebras

Definition 3.1.1 A right action of a finite group G on a weak Kac algebra (Definition 2.1.1) W K = (M, Δ, S, ε) is a right action M ◁ G of G on a C^{∗}-algebra M [14] such that, for all m ∈ M, g ∈ G:

Δ(m ◁ g) = Δ(m) ◁ (g ⊗ g), S(m ◁ g) = S(m) ◁ g, ε(m ◁ g) = ε(m).

In other words, a right action of G is exactly a right action of a cocommutative Kac algebra K_{s}(G) = (CG, Δ_{s}, S_{s}, ε_{s}) with Δ_{s}(g) = g ⊗ g, S_{s}(g) = g^{−1}, ε(g) = 1 [7] on a C^{∗}-algebra M such that Δ(m◁g̃) = Δ(m)◁Δ_{s}(g̃), S(m◁ g̃) = S(m) ◁ g̃, ε(m ◁ g̃) = ε(m) (∀m ∈ M, g̃ ∈ CG).

Given a finite group G acting from the right on a weak Kac algebra WK, let us consider a collection W K_{G} = (M_{G}, Δ_{G}, S_{G}, ε_{G}). Here M_{G} = M#G is a crossproduct C^{∗}-algebra [14] generated by elements of the form m ⊗ g (∀m ∈ M, g ∈ G) with operations

(m ⊗ g)(n ⊗ h) = (m ◁ h)n ⊗ gh, 1_{M_{G}} = 1 ⊗ 1_{G}, (m ⊗ g)^{∗} = (m ◁ g^{−1})^{∗} ⊗ g^{−1},

where m ∈ M, g ∈ G, 1_{G} is a unit of G. A coproduct Δ_{G}, an antipode S_{G} and a counit ε_{G} are defined respectively by: Δ_{G}(m ⊗ g) = (id ⊗ ς ⊗ id_{G})(Δ(m) ⊗ g ⊗ g), S_{G}(m ⊗ g) = S(m ◁ g^{−1}) ⊗ g^{−1}, where ς : M ⊗ CG → CG ⊗ M is the usual flip, and ε_{G}(m ⊗ g) = ε(m).

Lemma 3.1.2 W K_{G} = (M_{G}, Δ_{G}, S_{G}, ε_{G}) is a weak Kac algebra.

Proof. a) Δ_{G} is coassociative as tensor product of two coassociative coproducts Δ and Δ_{s} [14] and obviously multiplicative. Let m, n ∈ M, g, h ∈ G.

Relations S_{G}(1_{M_{G}}) = 1_{M_{G}} and S_{G}^{2} = id_{M_{G}} are obvious; we also have: S_{G}[(m ⊗ g)(n ⊗ h)] = S_{G}((m ◁ h)n ⊗ gh) = S(((m ◁ h)n) ◁ (gh)^{−1} ⊗ (gh)^{−1} = S((m ◁ g^{−1})(n ◁ (gh)^{−1})) ⊗ (gh)^{−1}.

On the other hand : S_{G}(n ⊗ h)S_{G}(m ⊗ g) = (S(n ◁ h^{−1}) ⊗ h^{−1})(S(m ◁ g^{−1}) ⊗ g^{−1}) = ([(S(n ◁ h^{−1})] ◁ g^{−1})(S(m ◁ g^{−1}) ⊗ h^{−1}g^{−1}) = S((m ◁ g^{−1})(n ◁ (gh)^{−1})) ⊗ (gh)^{−1}. Thus, S_{G} is antimultiplicative.

(Δ_{G} ∘ S_{G})(m ⊗ g) = Δ_{G}(S(m ◁ g^{−1}) ⊗ g^{−1}) = (id ⊗ ς ⊗ id_{G})(ς_{M}(S ⊗ S)Δ(m ◁ g^{−1}) ⊗ g^{−1} ⊗ g^{−1}) = ς_{M_{G}}(S_{G} ⊗ S_{G})Δ_{G}(m ⊗ g), where the usual flips act: ς_{M} in M ⊗ M and ς_{M_{G}} in M_{G} ⊗ M_{G}.

Thus, (M_{G}, Δ_{G}, S_{G}) is a gen. coinvolutive C^{∗}-bialgebra.

b) Clearly, ε_{G} is a counit with respect to Δ_{G}. Let us verify the axiom 2):

ε_{G}((m ⊗ g)(n ⊗ h)) = ε((m ◁ h)n) = (ε ⊗ ε)[((m ◁ h) ⊗ 1)e(1 ⊗ n)]
= (ε ⊗ ε)([((m ⊗ 1)e) ◁ (h ⊗ h)](1 ⊗ n))
= (ε_{G} ⊗ ε_{G})[(m ⊗ g ⊗ 1_{M_{G}})Δ_{G}(1_{M_{G}})(1_{M_{G}} ⊗ n ⊗ h)].

c) Finally, let us verify the axiom 3) (μ_{G} is the multiplication in M_{G}):

(μ_{G} ⊗ id_{M_{G}}))(S_{G} ⊗ id_{M_{G}} ⊗ id_{M_{G}})(Δ_{G} ⊗ id_{M_{G}})Δ_{G}(m ⊗ g) =

= (μ_{G} ⊗ id_{M_{G}})(S_{G} ⊗ id_{M_{G}} ⊗ id_{M_{G}})(m_{(1)} ⊗ g ⊗ m_{(2)} ⊗ g ⊗ m_{(3)} ⊗ g)
= (id ⊗ ς ⊗ id_{G})[(μ ⊗ id)(S ⊗ id ⊗ id)(Δ ⊗ id)Δ(m) ⊗ 1_{G} ⊗ g]
= (id ⊗ ς ⊗ id_{G})[(1 ⊗ m)e ⊗ 1_{G} ⊗ g] = (1 ⊗ 1_{G} ⊗ m ⊗ g)Δ_{G}(1 ⊗ 1_{G}).

Remark 3.1.3 One can show that the Haar trace on W K_{G} is just a dual trace for the Haar trace on WK with respect to the action of G:

ϕ_{G}(m ⊗ g) = δ_{1_{G},g}ϕ(m) (∀m ∈ M, g ∈ G).

Now we describe a series of concrete examples of nontrivial (non-commu-tative and non-symmetric) weak Kac algebras of dimensions n^{3} (n ≥ 3).

Let us start with a weak Kac algebra ℂ(K_{n}) of functions on a transitive principal groupoid K_{n} on a set of n elements (for various examples of groupoids see [20]). Then M = span{e_{ij}}_{i,j=1}^{n} is a commutative C^{∗}- algebra of functions on the set {e_{ij}} equipped with a coproduct Δ : e_{ij} → ∑_{k=1}^{n} e_{ik} ⊗ e_{kj}, an antipode S : e_{ij} → e_{ji} and a counit ε : e_{ij} → δ_{ij} (i, j, k ∈ {1, ..., n}, δ_{ij} is the Kronecker symbol).

Let us consider the action of the cyclic group G = Z/nZ on WK given by e_{ij} ◁ α := e_{i+1,j+1} (α is a generator of G, i, j ∈ {1, ..., n} and the summation is modulo n). Applying Lemma 3.1.2 (all its conditions are satisfied), we get a description of the resulting weak Kac algebra in terms of matrix units f_{ij}^{k} := e_{j,j+k} ⊗ α^{j−i} of C^{∗}-algebra M_{G} = M#G ≅ ⊕_{k=1}^{n} M_{n}(ℂ):

Δ_{G}(f_{ij}^{k}) = ∑_{r=1}^{n} f_{ij}^{r} ⊗ f_{i+r,j+r}^{k−r}, S_{G}(f_{ij}^{k}) = f_{i+k,j+k}^{n−k}, ε_{G}(f_{ij}^{k}) = δ_{n,k},

(i, j, k ∈ {1, ..., n} and the summation is modulo n. Then one can show that: a) the above weak Kac algebra is non-trivial;

b) its Haar trace is the canonical trace on M_{G}; c) ε_{t}(f_{ij}^{k}) = δ_{n,k} ∑_{r=1}^{n} f_{j+r,j+r}^{n−r}, ε_{s}(f_{ij}^{k}) = δ_{n,k} ∑_{r=1}^{n} f_{i,i}^{r};

d) N_{t} = span{∑_{r=1}^{n} f_{j+r,j+r}^{n−r}}, N_{s} = span{∑_{r=1}^{n} f_{i,i}^{r}}.

For n = 2 the above non-trivial weak Kac algebra is of dimension 8. A non-trivial Kac algebra of dimension 8 is constructed in [11]. Let us show that this dimension is minimal possible.

Proposition 3.1.4 All weak Kac algebras of dimension < 8 are either commutative or cocommutative (therefore, come from groupoids or groups).

Proof. This statement is well-known for usual Kac algebras, so we may assume that dimension of Cartan subalgebras ≥ 2. Let M be a non-commutative weak Kac algebra of dimension < 8, then M ≅ ℂ^{k} ⊕ M_{2}(ℂ), 0 ≤ k ≤ 3 as a C^{∗}-algebra. Let us show that M is cocommutative.

For k = 0 this follows from the classification of elementary weak Kac algebras in section 3.2. For k > 0 it suffices to prove that M is a direct sum of two weak Kac algebras, since any weak Kac algebra of dimension ≤ 3 is cocommutative. Clearly, the last property is equivalent to existence of a non-trivial projection q in N_{s} ∩ N_{t} ∩ Center(M), the hyper-center of M [18].

Due to Proposition 2.3.9 and the multiplicativity of Δ, for the counital quotient M_{ε} ⊂ M (Proposition 2.2.6) we have M_{ε} ≠ M_{2}(ℂ), and M_{ε} ≠ ℂ since it must contain a copy of a Cartan subalgebra by Remark 2.1.12. Using Proposition 2.3.9 again, we see that all other possibilities lead to the existence of a proper minimal subprojection p_{i} (Remark 2.2.3) of the Haar projection p_{ε} ∈ M, such that ςΔ(p_{i}) = Δ(p_{i}) (i.e. p_{i} is cocommutative). Since S(p_{i}) = p_{i}, Proposition 2.2.2 implies (S ⊗id_{M})Δ(p_{i}) = (id_{M} ⊗S)Δ(p_{i}) and q = ε_{s}(p_{i}) = ε_{t}(p_{i}) is a non-trivial projection in the hyper-center of M.

3.2 Elementary weak Kac algebras

In Subsection 2.2 we have seen that every counital quotient of a weak Kac algebra is a direct sum of weak Kac algebras which are simple as algebras.

Definition 3.2.1 A weak Kac algebra is said to be elementary if it is simple as an algebra, i.e., is isomorphic to the full matrix algebra M_{n}(ℂ).

The next theorem classifies all elementary weak Kac algebras and shows that any f-d C^{∗}-algebra can appear as a Cartan subalgebra of an elementary weak Kac algebra in a unique way.

Theorem 3.2.2 Given a f-d C^{∗}-algebra A, there exists a unique elementary weak Kac algebra M(A) = (M_{n}(ℂ), Δ, S, ε) (n = dim A) with Cartan subalgebras isomorphic to A.

Proof. 1) For A = ⊕_{α} M_{n_{α}}, n = ∑_{α} n_{α}^{2} let us write down M(A) explicitly, if matrix units {E_{ijα}^{klβ}}_{i,j=1...n_{α}, k,l=1...n_{β}}^{α,β=1...N} in M_{n}(ℂ) are defined by

E_{ijα}^{klβ} E\begin{matrix}ĩ & j̃ & α̃ \\ k̃ & l̃ & β̃ \end{matrix} = δ_{ββ̃} δ_{kk̃} δ_{ll̃}E_{ijα}^{ĩj̃α̃}, (E_{ijα}^{klβ})^{∗} = E_{klβ}^{ijα}.

Equip M_{n}(ℂ) with the following structure of a weak Kac algebra :

Δ(E_{ijα}^{klβ}) = \frac{1}{\sqrt{n_{α}n_{β}}} ∑_{uv} E_{iuα}^{kvβ} ⊗ E_{ujα}^{vlβ},
S(E_{ijα}^{klβ}) = E_{lkβ}^{jiα},
ε(E_{ijα}^{klβ}) = \sqrt{n_{α}n_{β}} δ_{ij} δ_{kl}.

The verification of axioms is a straightforward computation. We have

e = Δ(1) = ∑_{α} \frac{1}{n_{α}} ∑_{pq} (∑_{i} E_{ipα}^{iqα}) ⊗ (∑_{j} E_{pjα}^{qjα}),

therefore, F_{pqα} = ∑_{i} E_{ipα}^{iqα} and G_{pqα} = ∑_{j} E_{qjα}^{pjα} are matrix units for the source and target Cartan subalgebras respectively, isomorphic to A.

2) The uniqueness. If M̃(A) = (M_{n}(ℂ), Δ̃, S̃, ε̃) is another weak Kac algebra with Ñ_{s} ≅ Ñ_{t} ≅ ⊕_{α} M_{n_{α}}(ℂ), by the observation from Remark 2.2.3, ∑_{i∈S} n_{α}^{2} = dim A = deg π_{ε} = n.

The inclusion N_{s} ⊂ M is determined up to a conjugation by the vector (d_{1}, … d_{n}) where d_{α} is the multiplicity of M_{n_{α}}(ℂ) in M. By Proposition 1.3.2, N_{t} ⊂ N′_{s} ∩ M = ⊕_{α} M_{d_{α}}(ℂ), therefore we may assume that n_{α} ≤ d_{α}. Since the inclusion N_{s} ⊂ M is unital, ∑ d_{α}n_{α} = n, so 0 = ∑ d_{α}n_{α} − ∑ n_{α}^{2} = ∑(d_{α} − n_{α})n_{α}, and d_{α} = n_{α} (∀α). Thus, N_{t} = N′_{s} ∩ M, and N_{s} = N′_{t} ∩ M.

Hence, we can write N_{s} = ⊕_{α}N_{sα} and N_{t} = ⊕_{α}N_{tα}, where N_{sα} ≅ N_{tα} ≅ M_{n_{α}}(ℂ), N_{sα}N_{tβ} = 0 if α ≠ β, and N_{sα}N_{tα} ≅ M_{n_{α}^{2}}(ℂ). Let {F_{pqα}}_{p,q=1...n_{α}} be a system of matrix units in N_{sα}, α = 1 … K. Proposition 2.1.11 gives

ẽ = ∑_{α} \frac{1}{n_{α}} ∑_{pq} F_{pqα} ⊗ S(F_{qpα}).

Note that E_{ijα}^{klα} = S̃(F_{kiα})F_{jlα} are matrix units in N_{sα}N_{tα} such that

F_{pqα} = ∑_{i} E_{ipα}^{iqα},  S̃(F_{qpα}) = ∑_{j} E_{pjα}^{qjα}

Let {E_{ijα}^{klβ}} be the system of matrix units in M_{n}(ℂ) defined as in 1) which extends the union of the above systems {E_{ijα}^{klα}}, α = 1 … K. Using the definition of a Cartan subalgebra we compute

Δ̃(F_{pqα}) = \frac{1}{n_{α}} ∑_{p̃q̃} F_{p̃q̃α} ⊗ E_{p̃pα}^{q̃qα},
Δ̃(E_{lpα}^{mqα}) = Δ̃(S̃(F_{mlα}))Δ̃(F_{pqα}) = \frac{1}{n_{α}} ∑_{uv} E_{luα}^{mvα} ⊗ E_{upα}^{vqα} = Δ(E_{lpα}^{mqα}).

Since Δ̃ is a coassociative ∗-homomorphism, we get

Δ̃(E_{ijα}^{klβ}) = \frac{λ_{α}^{β}}{\sqrt{n_{α}n_{β}}} ∑_{uv} E_{iuα}^{kvβ} ⊗ E_{ujα}^{vlβ}, = λ_{α}^{β}Δ(E_{ijα}^{klβ})

for some \{λ_{α}^{β}\}_{α,β=1...N} with |λ_{α}^{β}| = 1, λ_{β}^{α} = \overline{λ_{α}^{β}}, and λ_{α}^{β} λ_{β}^{γ} = λ_{α}^{γ} (∀α, β, γ). The properties of antipode and counit imply: S̃(E_{ijα}^{klβ}) = (λ_{α}^{β})^{−2} S(E_{ijα}^{klβ}) and ε̃(E_{ijα}^{klβ}) = (λ_{α}^{β})^{−1}ε(E_{ijα}^{klβ}), so the map E_{ijα}^{klβ} ↦ λ_{α}^{β}E_{ijα}^{klβ} defines an isomorphism of weak Kac algebras M(A) = (M_{n}(ℂ), Δ, S, ε) and M̃(A) = (M_{n}(ℂ), Δ̃, S̃, ε̃).

Corollary 3.2.3 The dual weak Kac algebra M̂(A) has an algebra structure ⊕_{αβ} M_{n_{α}n_{β}}(ℂ); there are comatrix units {C_{ijα}^{klβ}}_{i,j=1...n_{α}, k,l=1...n_{β}}^{α,β=1...N} of M̂(A),

Δ(C_{ijα}^{klβ}) = ∑_{pqγ} C_{ijα}^{pqγ} ⊗ C_{pqγ}^{klβ}, ε(C_{ijα}^{klβ}) = δ_{αβ}δ_{ik}δ_{jl},

such that

C_{iuα}^{kvβ} · C_{ũjα̃}^{ṽlβ̃} = \frac{1}{\sqrt{n_{α}n_{β}}} δ_{αα̃}δ_{ββ̃}δ_{uũ}δ_{vṽ} C_{ijα}^{klβ},
1 = ∑_{αβ ij} \sqrt{n_{α}n_{β}} C_{iiα}^{kkβ},

(C_{ijα}^{klβ})^{∗} = C_{jiα}^{lkβ}, S(C_{ijα}^{klβ}) = C_{lkβ}^{jiα}.

Proof. A straightforward computation using the definition of dual weak Kac algebra.

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