Coactions Of Groups Research Articles

A new approach to induction and imprimitivity results

Given a closed quantum subgroup of a locally compact quantum group, we study induction of unitary corepresentations of the quantum subgroup to the ambient quantum group. More generally, we study induction given a coaction of the quantum subgroup on a C * -algebra. We prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups. This means that we define quantum homogeneous spaces as C * -algebras and that we prove Morita equivalence of crossed products and homogeneous spaces. We essentially use von Neumann algebraic techniques to prove these Morita equivalences between C * -algebras.

Group Entwining Structures and Group Coalgebra Galois Extensions#

The notions of group coalgebra Galois extension and group entwining structure are defined. It is proved that any group coalgebra Galois extension induces a unique group-entwining map ψ = {ψα, β}α, β∈π compatible with the right group coaction, generalizing the recent work of Brzeziński and Hajac [Brzeziński, T., Hajac, P. M. (1999). Coalgebra extensions and algebra coextensions of Galois type. Comm. Algebra 27:1347–1368].

Full duality for coactions of discrete groups

Using the strong relation between coactions of a discrete group $G$ on $C^*$-algebras and Fell bundles over $G$ we prove a new version of Mansfield's imprimitivity theorem for coactions of discrete groups. Our imprimitivity theorem works for the universally defined full crossed products and arbitrary subgroups of $G$ as opposed to the usual theory of [16], [11] which uses the spatially defined reduced crossed products and normal subgroups of $G$. Moreover, our theorem factors through the usual one by passing to appropriate quotients. As applications we show that a Fell bundle over a discrete group is amenable in the sense of Exel [7] if and only if the double dual action is amenable in the sense that the maximal and reduced crossed products coincide. We also give a new characterization of induced coactions in terms of their dual actions.

Equalizers and coactions of groups

If $f:G\to H$ is a group homomorphism and $p,q$ are the projections from the free product $G*H$ onto its factors $G$ and $H$ respectively, let the group ${\cal E}_f\subseteq G*H$ be the equalizer of $fp$ and $q:G*H\to H$. Then $p$ restricts to an epimorph

Co-actions of groups

Let f : G → H be a fixed homomorphism and p′ : G * H → G and p″ : G * H → H the two projections of the free product. Then a co-action relative to f is a homomorphism s : G → G * H such that p′s = id and p″s = f. We study this notion and investigate the following questions. What restrictions does s place on the structure of the group G? What form does s take in special cases? When does s induce a co-multiplication on H? What is the relation between associativity of s and associativity of the induced co-multiplication m on H? What are the properties of the operation of Hom(H, B) on Hom(G, B) induced by s : G → G * H? In addition, we give several diverse examples of co-actions in the last section.

Induced Coactions of Discrete Groups on C*-Algebras

AbstractUsing the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a C*-coaction δ: D → D ⊗C*(G/N) of a quotient group G/N of a discrete group G to a C*-coaction Ind δ: Ind D → D ⊗C*(G) of G. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products Ind D ×IndδG and D ×δG/N are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions.

Imprimitivity for C*-coactions of non-amenable groups

We give a condition on a full coaction (A, G, δ) of a (possibly) non-amenable group G and a closed normal subgroup N of G which ensures that Mansfield imprimitivity works; i.e. that A×δ[mid ]G/N is Morita equivalent to A×δG×δˆrN. This condition obtains if N is amenable or δ is normal. It is preserved under Morita equivalence, inflation of coactions, the stabilization trick of Echterhoff and Raeburn, and on passing to twisted coactions.

Discrete coactions on C*-algebras

AbstractWe will consider coactions of discrete groups on C*-algebras and imitate some of the results about compact group actions on C*-algebras. In particular, the crossed product of a reduced coaction ∈ of a discrete amenable group G on A is liminal (respectively, postliminal) if and only if the fixed point algebra of ∈ is. Moreover, we will also consider ergodic coactions on C*-algebras.

Models for coactions of finite groups on the AFD factor of type II1

Models for coactions of finite groups on the AFD factor of type II1

Constant solutions of reflection equations and quantum groups

To the Yang–Baxter equation an additional relation can be added. This is the reflection equation that appears in various places, with or without spectral parameter, e.g., in factorizable scattering on a half-line, integrable lattice models with nonperiodic boundary conditions, noncommutative differential geometry on quantum groups, etc. Two forms of spectral-parameter-independent reflection equations are studied, chosen by the requirement that their solutions be comodules with respect to the quantum group coaction leaving invariant the reflection equations. For a variety of known solutions of the Yang–Baxter equation the constant solutions of the reflection equations are given. Various quadratic algebras defined by the reflection equations are also given explicitly.

Equivariant completely bounded operators

An equivariant completely bounded linear operator between two C*-algebras acted on by an amenable group is shown to lift to a completely bounded operator between the crossed products that is equivariant with respect to the dual coactions. A similar result is proved for coactions and dual actions. It is shown that the only equivariant linear operators that lift twice through the action and dual coaction of an infinite group are the completely bounded ones. 1. Introduction. Let Φ: A -» B be a bounded linear map between C* -algebras, and let a: G —• A\x\A and β: G —> Auti? be actions of an amenable locally compact group G. If Φ is a homomorphism and is equivariant—i .e., Φ(as(a)) = βs(Φ(a)) for s e G, a e A —then it extends to a homomorphism Φ x i from the crossed product A xaG to B XJ^ G. On the other hand, if C is another C*-algebra, then for any completely bounded map Φ: A —• B there is a bounded operator Φ x i: A® C -^ B ® C; indeed, by taking C to be the algebra 3£ of compact operators, we can see that the complete boundedness of Φ is necessary for this to be true. We shall combine these results to prove that any equivariant completely bounded operator extends to a bounded map on the crossed product, formulate and prove the analogous results for crossed products by coactions of nonabelian groups, and investigate the extent to which complete boundedness is a necessary hypothesis. We shall take similar approaches to the problems of lifting through actions and coactions. First, we prove equivariant versions of Stinespring's theorem [23] on completely positive maps into 3§{%f), and we then show how to modify Wittstock's theorem [25] to write an equivariant symmetric completely bounded map into 3S{^) as a difference of equivariant completely positive ones. From these we can use standard symmetrisation techniques to deduce that an equivariant completely bounded operator Φ: A —•ϊ%W) can be realized in the form Φ(a) = Tπ(a)V, where π is part of a covariant representation (π, U) of (A9G,ά) in some larger Hubert space. We can then define