John Quigg

Full and Reduced C*-Coactions

Full and reduced (7*-coactions are shown to be essentially equivalent as far as the representations and cocrossed products are concerned, at least in the presence of nondegeneracy. This is shown to be particularly true for a special class of full coactions which are given the name normal.

C*-actions of r-discrete groupoids and inverse semigroups

Groupoid actions on C*-bundles and inverse semigroup actions on C *-algebras are closely related when the groupoid is r -discrete.

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 × δ G × δˆ r N . 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.

DUALITY OF RESTRICTION AND INDUCTION FOR C*-COACTIONS

Consider a coaction δ of a locally compact group G on a C-algebra A, and a closed normal subgroup N of G. We prove, following results of Echterhoff for abelian G, that Mansfield’s imprimitivity between A ×δ| G/N and A ×δ G ×δ,r N implements equivalences between Mansfield induction of representations from A×G/N to A×G and restriction of representations fromA×G×rN to A × G, and between restriction of representations from A × G to A × G/N and Green induction of representations from A × G to A × G ×r N . This allows us to deduce properties of Mansfield induction from the known theory of ordinary crossed products.

Hecke C*-algebras, Schlichting completions and Morita equivalence

The Hecke algebra of a Hecke pair ( G , H ) is studied using the Schlichting completion (Ḡ, ), which is a Hecke pair whose Hecke algebra is isomorphic to and which is topologized so that is a compact open subgroup of Ḡ. In particular, the representation theory and C *-completions of are addressed in terms of the projection using both Fells and Rieffels imprimitivity theorems and the identity . An extended analysis of the case where H is contained in a normal subgroup of G (and in particular the case where G is a semi-direct product) is carried out, and several specific examples are analysed using this approach.

Induced Coactions of Discrete Groups on

Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a C*-coaction of a quotient group G/N of a discrete group G to a C*-coaction of G itself on an induced C*-algebra. 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 of the original and the induced coactions 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.

C^*

We define the action of a locally compact group G on a topological graph E . This action induces a natural action of G on the C * -correspondence ℋ( E ) and on the graph C * -algebra C * ( E ). If the action is free and proper, we prove that C * ( E )⋊ r G is strongly Morita equivalent to C * ( E / G ) . We define the skew product of a locally compact group G by a topological graph E via a cocycle c : E 1 → G . The group acts freely and properly on this new topological graph E × c G . If G is abelian, there is a dual action on C * ( E ) such that . We also define the fundamental group and the universal covering of a topological graph.

-Algebras

For any maximal coaction (A, G, S) and any closed normal subgroup N of G, there exists an imprimitivity bimodule Y G G/N (A) between the full crossed product A × δ G × δ| N and A × δ| G/N, together with Inf δ| - δ dec compatible coaction Sy of G. The assignment (A,δ) → (Y G G/N (A), δ Y ) implements a natural equivalence between the crossed-product functors x G x N and x G/N, in the category whose objects are maximal coactions of G and whose morphisms are isomorphism classes of right-Hilbert bimodule coactions of G.

Group actions on topological graphs

Cuntz and Li have defined a C*-algebra associated to any integral domain, using generators and relations, and proved that it is simple and purely infinite and that it is stably isomorphic to a crossed product of a commutative C*-algebra. We give an approach to a class of C*-algebras containing those studied by Cuntz and Li, using the general theory of C*-dynamical systems associated to certain semidirect product groups. Even for the special case of the Cuntz-Li algebras, our development is new.

Mansfield's imprimitivity theorem for full crossed products

For a countable discrete space V, every nondegenerate separable C*-correspondence over c_0(V) is isomorphic to one coming from a directed graph with vertex set V. In this paper we demonstrate why the analogous characterizations fail to hold for higher-rank graphs (where one considers product systems of C*-correspondences) and for topological graphs (where V is locally compact Hausdorff), and we discuss the obstructions that arise.