451 research outputs found
Representations of Hecke algebras and dilations of semigroup crossed products
We consider a family of Hecke C*-algebras which can be realised as crossed
products by semigroups of endomorphisms. We show by dilating representations of
the semigroup crossed product that the category of representations of the Hecke
algebra is equivalent to the category of continuous unitary representations of
a totally disconnected locally compact group.Comment: 16 page
Proper actions, fixed-point algebras and naturality in nonabelian duality
Suppose a locally compact group G acts freely and properly on a locally
compact Hausdorff space X, and let gamma be the induced action on C_0(X). We
consider a category in which the objects are C*-dynamical systems (A, G, alpha)
for which there is an equivariant homomorphism of (C_0(X), gamma) into the
multiplier algebra M(A). Rieffel has shown that such systems are proper and
saturated, and hence have a generalized fixed-point algebra A^alpha which is
Morita equivalent to A times_{alpha,r} G. We show that the assignment (A,
alpha) maps to A^alpha is functorial, and that Rieffel's Morita equivalence is
natural in a suitable sense. We then use our results to prove a categorical
version of Landstad duality which characterizes crossed products by coactions,
and to prove that Mansfield imprimitivity for crossed products by homogeneous
spaces is natural.Comment: 19 pages; minor revisio
Crossed Products by Dual Coactions of Groups and Homogeneous Spaces
Mansfield showed how to induce representations of crossed products of
C*-algebras by coactions from crossed products by quotient groups and proved an
imprimitivity theorem characterising these induced representations. We give an
alternative construction of his bimodule in the case of dual coactions, based
on the symmetric imprimitivity theorem of the third author; this provides a
more workable way of inducing representations of crossed products of
C*-algebras by dual coactions. The construction works for homogeneous spaces as
well as quotient groups, and we prove an imprimitivity theorem for these
induced representations.Comment: LaTeX-2e, 19 pages, requires pb-diagram.sty ((E) University of
Paderborn, Germany (K,R) University of Newcastle, Australia
Obstructions to a general characterization of graph correspondences
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.Comment: major revision; stated some results in greater generalit
Extension problems and non-abelian duality for -algebras
Suppose that is a closed subgroup of a locally compact group . We show
that a unitary representation of is the restriction of a unitary
representation of if and only if a dual representation of a
crossed product is regular in an appropriate sense. We
then discuss the problem of deciding whether a given representation is regular;
we believe that this problem will prove to be an interesting test question in
non-abelian duality for crossed products of -algebras.Comment: Substantial changes from the previous versio
Naturality and Induced Representations
We show that induction of covariant representations for C*-dynamical systems
is natural in the sense that it gives a natural transformation between certain
crossed-product functors. This involves setting up suitable categories of
C*-algebras and dynamical systems, and extending the usual constructions of
crossed products to define the appropriate functors. From this point of view,
Green's Imprimitivity Theorem identifies the functors for which induction is a
natural equivalence. Various spcecial cases of these results have previously
been obtained on an ad hoc basis.Comment: LaTeX-2e, 24 pages, uses package pb-diagra
A Categorical Approach to Imprimitivity Theorems for C*-Dynamical Systems
Imprimitivity theorems provide a fundamental tool for studying the
representation theory and structure of crossed-product C*-algebras. In this
work, we show that the Imprimitivity Theorem for induced algebras, Green's
Imprimitivity Theorem for actions of groups, and Mansfield's Imprimitivity
Theorem for coactions of groups can all be viewed as natural equivalences
between various crossed-product functors among certain equivariant categories.
The categories involved have C*-algebras with actions or coactions (or both)
of a fixed locally compact group G as their objects, and equivariant
equivalence classes of right-Hilbert bimodules as their morphisms. Composition
is given by the balanced tensor product of bimodules.
The functors involved arise from taking crossed products; restricting,
inflating, and decomposing actions and coactions; inducing actions; and various
combinations of these.
Several applications of this categorical approach are also presented,
including some intriguing relationships between the Green and Mansfield
bimodules, and between restriction and induction of representations.Comment: LaTeX2e, 152 pages, uses class memo-l and packages amscd, xy, and
amssymb; fixed several typos and updated bibliograph
Induction in stages for crossed products of C*-algebras by maximal coactions
Let B be a C*-algebra with a maximal coaction of a locally compact group G,
and let N and H be closed normal subgroups of G with N contained in H. We show
that the process Ind_(G/H)^G which uses Mansfield's bimodule to induce
representations of the crossed product of B by G from those of the restricted
crossed product of B by (G/H) is equivalent to the two-stage induction process:
Ind_(G/N)^G composed with Ind_(G/H)^(G/N). The proof involves a calculus of
symmetric imprimitivity bimodules which relates the bimodule tensor product to
the fibred product of the underlying spaces.Comment: 38 pages, LaTeX, uses Xy-pic; significant reorganization of previous
version; short section on regularity of induced representations adde
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