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Dana P. Williams

Publications and source records attributed to Dana P. Williams.

At least 55 records · Page 3Linked to original sources
arXiv2008

Naturality of Rieffel's Morita equivalence for proper actions

Suppose that a locally compact group $G$ acts freely and properly on the right of a locally compact space $T$. Rieffel proved that if $α$ is an action of $G$ on a $C^*$-algebra $A$ and there is an equivariant embedding of $C_0(T)$ in $M(A)$, then the action $α$ of $G$ on $A$ is proper, and the crossed product $A\rtimes_{α,r}G$ is Morita equivalent to a generalised fixed-point algebra $\Fix(A,α)$ in $M(A)^α$. We show that the assignment $(A,α)\mapsto\Fix(A,α)$ extends to a functor $\Fix$ on a category of $C^*$-dynamical systems in which the isomorphisms are Morita equivalences, and that Rieffel's Morita equivalence implements a natural isomorphism between a crossed-product functor and $\Fix$. From this, we deduce naturality of Mansfield imprimitivity for crossed products by coactions, improving results of Echterhoff-Kaliszewski-Quigg-Raeburn and Kaliszewski-Quigg Raeburn, and naturality of a Morita equivalence for graph algebras due to Kumjian and Pask.

math.OA↗
arXiv2008

Renault's Equivalence Theorem for Groupoid Crossed Products

We provide an exposition and proof of Renault's equivalence theorem for crossed products by locally Hausdorff, locally compact groupoids. Our approach stresses the bundle approach, concrete imprimitivity bimodules and is a preamble to a detailed treatment of the Brauer semigroup for a locally Hausdorff, locally compact groupoid.

math.OA↗
arXiv2007

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.

math.OA↗
arXiv2006

Extension problems for representations of crossed-product C*-algebras

We consider the following problem. Suppose $α$ is an action of a locally compact group $G$ on a $C^*$-algebra $A$, $H$ is a closed subgroup of $G$, and $(π,U)$ is a covariant representation of $(A,H,α)$. For which closed subgroups $K$ containing $H$ is there a covariant representation $(π,V)$ of $(A,K,α)$ such that $V|_H=U$? We answer this problem by providing a criterion involving the induced representation of $(A,G,α)$. We then consider the dual problem for coactions of locally compact groups.

math.OA↗
arXiv2006

Inducing Primitive Ideals

We study conditions on a $C^*$-dynamical system $(A,G,α)$ under which induction of primitive ideals (resp. irreducible representations) from stabilizers for the action of $G$ on the primitive ideal space $\Prim(A)$ give primitive ideals (resp. irreducible representations) of the crossed product $A\rtimes_αG$. The results build on earlier results of Sauvageot and others, and will correct a (possibly overly optimistic) statement of the first author.

math.OA↗
arXiv2004

Classification theorems for the C*-algebras of graphs with sinks

We consider graphs E which have been obtained by adding one or more sinks to a fixed directed graph G. We classify the C*-algebra of E up to a very strong equivalence relation, which insists, loosely speaking, that C*(G) is kept fixed. The main invariants are vectors W_E : G^0 -> N which describe how the sinks are attached to G; more precisely, the invariants are the classes of the W_E in the cokernel of the map A-I, where A is the adjacency matrix of the graph G.

math.OA↗
arXiv2003

Tensor products with bounded continuous functions

We study that natural inclusions $C^b(X) \tensor A$ into $C^b(X,A)$ and $C^b(X, C^b(Y))$ into $C^b(X \times Y)$. In particular, excepting trivial cases, both these maps are isomorphisms only when $X$ and $Y$ are pseudocompact. This implies a result of Glicksberg showing that the Stone-Cech compactificiation $β(X \times Y)$ is naturally identified with $βX \times βY$ if and only if $X$ and $Y$ are pseudocompact.

math.OA↗
arXiv2002

The Dixmier-Douady Class of Groupoid Crossed Products

We give a formula for the Dixmier-Douady class of a continuous-trace groupoid crossed product that arises from an action of a locally trivial, proper, principal groupoid on a bundle of elementary $C^*$-algebras that satisfies Fell's condition.

math.OA↗
arXiv2002

Continuous-Trace Groupoid Crossed Products

Let $G$ be a second countable, locally compact groupoid with Haar system, and let $\mathcal{A}$ be a bundle of $C^{\ast}$-algebras defined over the unit space of $G$ on which $G$ acts continuously. We determine conditions under which the associated crossed product $C^{\ast}(G;\mathcal{A})$ is a continuous trace $C^{\ast}$-algebra.

math.OA↗
arXiv2002

Central twisted transformation groups and group $C^*$-algebras of central group extensions

We examine the structure of central twisted transformation group \cs-algebras $C_{0}(X)\rtimes_{\id,u}G$, and apply our results to the group \cs-algebras of central group extensions. Our methods require that we study Moore's cohomology group $H^{2}\bigl(G,C(X,\T)\bigr)$, and, in particular, we prove an inflation result for pointwise trivial cocyles which may be of use elsewhere.

math.OA↗
arXiv2001

Proper actions on imprimitivity bimodules and decompositions of Morita equivalences

We consider a class of proper actions of locally compact groups on imprimitivity bimodules over C*-algebras which behave like the proper actions on C*-algebras introduced by Rieffel in 1988. We prove that every such action gives rise to a Morita equivalence between a crossed product and a generalized fixed-point algebra, and in doing so make several innovations which improve the applicability of Rieffel's theory. We then show how our construction can be used to obtain canonical tensor-product decompositions of important Morita equivalences. Our results show, for example, that the different proofs of the symmetric imprimitivity theorem for actions on induced algebras yield isomorphic equivalences. A similar analysis of the symmetric imprimitivity theorem for graph algebras gives new information about the amenability of actions on graph algebras.

math.OA↗
arXiv2000

Ideals in transformation-group $C^*$-algebras

We characterize the ideal of continuous-trace elements in a separable transformation-group $C^{*}$-algebra $C_0(X)\times G$. In addition, we identify the largest Fell ideal, the largest liminal ideal and the largest postliminal ideal.

math.OA↗
arXiv1997

Locally Inner Actions on $C_0(X)$-Algebras

We make a detailed study of locally inner actions on C*-algebras whose primitive ideal spaces have locally compact Hausdorff complete regularizations. We suppose that $G$ has a representation group and compactly generated abelianization $G_{ab}$. Then if the complete regularization of $\Prim(A)$ is $X$, we show that the collection of exterior equivalence classes of locally inner actions of $G$ on $A$ is parameterized by the group $\E_G(X)$ of exterior equivalence classes of $C_0(X)-actions of $G$ on $C_0(X,\K)$. Furthermore, we exhibit a group isomorphism of $\E_G(X)$ with the direct sum $H^1(X,\sheaf \hat{G_{ab}}) \oplus C(X,H^2(G,\T))$. As a consequence, we can compute the equivariant Brauer group $\Br_G(X)$ for $G$ acting trivially on $X$.

funct-an↗
arXiv1997

Crossed products by $C_0(X)$-actions

Suppose that $G$ has a representation group $H$, that $G_{ab}:= G/\bar{[G,G]}$ is compactly generated, and that $A$ is a \cs-algebra for which the complete regularization of $\Prim(A)$ is a locally compact Hausdorff space $X$. In a previous article, we showed that there is a bijection $α\mapsto (Z_α,f_α)$ between the collection of exterior equivalence classes of locally inner actions $α:G\to\Aut(A)$, and the collection of principal $\hgab$-bundles $Z_α$ together with continuous functions $f_α:X\to H^2(G,\T)$. In this paper, we compute the crossed products $A\rtimes_αG$ in terms of the data $Z_α$, $f_α$, and~$\cs(H)$.

funct-an↗
arXiv1997

The Brauer Group of a Locally Compact Groupoid

We define the Brauer group $\Br(G)$ of a locally compact groupoid $G$ to be the set of Morita equivalence classes of pairs $(\A,α)$ consisting of an elementary C*-bundle $\A$ over $G^{(0)}$ satisfying Fell's condition and an action $α$ of $G$ on $\A$ by $*$-isomorphisms. When $G$ is the transformation groupoid $X\times H$, then $\Br(G)$ is the equivariant Brauer group $\Br_H(X)$. In addition to proving that $\Br(G)$ is a group, we prove three isomorphism results. First we show that if $G$ and $H$ are equivalent groupoids, then $\Br(G)$ and $\Br(H)$ are isomorphic. This generalizes the result that if $G$ and $H$ are groups acting freely and properly on a space $X$, say $G$ on the left and $H$ on the right then $\Br_G(X/H)$ and $\Br_H(G/ X)$ are isomorphic. Secondly we show that the subgroup $\Br_0(G)$ of $\Br(G)$ consisting of classes $[\A,α]$ with $\A$ having trivial Dixmier-Douady invariant is isomorphic to a quotient $\E(G)$ of the collection $\Tw(G)$ of twists over $G$. Finally we prove that $\Br(G)$ is isomorphic to the inductive limit $\Ext(G,T)$ of the groups $\E(G^X)$ where $X$ varies over all principal $G$ spaces $X$ and $G^X$ is the imprimitivity groupoid associated to $X$.

funct-an↗