Dana P. Williams

Pull-backs of *-algebras and crossed products by certain diagonal actions

Let G be a locally compact group and p: Q -T a principal G-bundle. If A is a C*-algebra with primitive ideal space T, the pull-back p*A of A along p is the balanced tensor product Co((Q) ?C(T) A. If ,3: G -Aut A consists of C(T)-module automorphisms, and -y: G -Aut Co (Q) is the natural action, then the automorphism group y 0X 3 of Co(Q) 0 A respects the balancing and induces the diagonal action p*,3 of G on p*A. We discuss some examples of such actions and study the crossed product p* A x p G. We suggest a substitute D for the fixed-point algebra, prove p*A x G is strongly Morita equivalent to D, and investigate the structure of D in various cases. In particular, we ask when D is strongly Morita equivalent to A-sometimes, but by no means always-and investigate the case where A has continuous trace. Let B be a C*-algebra and G a locally compact group acting on B as a strongly continuous automorphism group a. Our goal here is to study the crossed product C*-algebra B x , G for two classes of diagonal actions for which the induced action of G on B is free. The first class includes actions of the form -y 0X3 on B Co (Q) 0 A, where p: Q -* T is a principal G-bundle, -y is the dual action of G on Co (Q), and 3: G -* Aut A is an action of G on another C*-algebra A. We also consider diagonal actions on algebras which are the pull-backs of another algebra A along a principal bundle p: Q -* T: if A is a C*-algebra with primitive ideal space T, then the pull-back p*A is the balanced tensor product Co(Q) ?Cb(T) A. When 3: G -* Aut A consists of C(T)-module automorphisms, the product action ty 0 /3 preserves the balancing, and the diagonal action p*3 is, by definition, the induced action on p* A. In general, if f: X -* Y and q: PrimA -* Y are continuous, then Cb(Y) acts on Co(X) by composition with f, and on A by composition with q and the DaunsHofmann theorem. We can therefore define the pull-back f*A of A along f as the C*-algebraic tensor product Co(X) ?Cb(y) A. The reason for the name is that when A is the algebra of sections of some C*-bundle E over Y, there is a natural isomorphism of f*A onto the algebra of sections of the pull-back f*E. In ?1 we discuss this and other basic properties of pull-backs and give some evidence to show they are likely to be of interest. In particular, we show that if G is abelian and a: G -* Aut A is locally unitary in the sense of [18], then the crossed product Received by the editors October 20, 1983 and, in revised form, February 15, 1984. 1980 Mathematics Subject Classification. Primary 46L40, 46L55.

The Topology on the Primitive Ideal Space of Transformation Group C # - Algebras and C.C.R. Transformation Group C # -Algebras

If (G, 8) is a second countable transformation group and the stability groups are amenable then C*(G, 8) is C.C.R. if and only if the orbits are closed and the stability groups are C.C.R. In addition, partial results relating closed orbits to C.C.R. algebras are obtained in the nonseparable case. In several cases, the topology of the primitive ideal space is calculated explicitly. In particular, if the stability groups are all contained in a fixed abelian subgroup H, then the topology is computed in terms of H and the orbit structure, provided C*(G, 8) and C*(H, 8) are EH-regular. These conditions are automatically met if G is abelian and (G, 8) is second countable.

Transformation Group C*-Algebras with Continuous Trace

We obtain several results characterizing when transformation group C*-algebras have continuous trace. These results can be stated most succinctly when (G, L?) is second countable, and the stability groups are contained in a fixed abelian subgroup. In this case, C*(G, Q) has continuous trace if and only if the stability groups vary continuously on R and compact subsets of Q are wandering in an appropriate sense. In general, we must assume that the stability groups vary continuously, and if (G, J?) is not second countable, that the natural maps of G/S, onto G x are homeomorphisms for each x. Then C*(G, 0) has continuous trace if and only if compact subsets of 0 are wandering and an additionai C*-algebra, constructed from the stability groups and 0, has continuous trace.

Continuous-trace groupoid *-algebras. III

Suppose that G is a second countable locally compact groupoid with a Haar system and with abelian isotropy. We show that the groupoid C∗-algebra C∗(G, λ) has continuous trace if and only if there is a Haar system for the isotropy groupoid A and the action of the quotient groupoid G/A is proper on the unit space of G.

An equivariant Brauer semigroup and the symmetric imprimitivity theorem

Suppose that (X,G) is a second countable locally compact transformation group. We let SG(X) denote the set of Morita equivalence classes of separable dynamical systems (A,G,α) where A is a C0(X)-algebra and α is compatible with the given G-action on X. We prove that SG(X) is a commutative semigroup with identity with respect to the binary operation [A,G, α][B,G, β] = [A⊗X B,G,α⊗X β] for an appropriately defined balanced tensor product on C0(X)-algebras. If G and H act freely and properly on the left and right of a space X, then we prove that SG(X/H) and SH(G\X) are isomorphic as semigroups. If the isomorphism maps the class of (A,G,α) to the class of (B,H, β), then A oα G is Morita equivalent to B oβ H.

Dixmier-Douady classes of dynamical systems and crossed products

Continuous-trace C*-algebras A with spectrum T can be characterized as those algebras which are locally Morita equivalent to C 0 (T). The Dixmier-Douady class δ(A) is an element of the Cech cohomology group H 3 (T, Z) and is the obstruction to building a global equivalence from the local equivalences. Mere we shall be concerned with systems (A, G, α) which are locally Morita equivalent to their spectral system (C 0 (T), G, τ), in which G acts on the spectrum T of A via the action induced by α. Such systems include locally unitary actions as well as N-principal systems. Our new Dixmier-Douady class δ(A, G, α) will be the obstruction to piecing the local equivalences together to form a Morita equivalence of (A, G, α) with its spectral system

IRREDUCIBLE REPRESENTATIONS OF GROUPOID C*-ALGEBRAS

If G is a second countable locally compact Hausdorff groupoid with Haar system, we show that every representation induced from an irreducible representation of a stability group is irreducible.

Properties preserved under morita equivalence of C*-algebras

We show that important structural properties of C*-algebras and the multiplicity numbers of representations are preserved under Morita equivalence.

THE STRUCTURE OF CROSSED PRODUCTS BY SMOOTH ACTIONS

Abstract Let £ be a C*-bundle over T with {At} fibres l€A . Suppose that A is the C*-algebra of sectionsof £ which vanish at infinity, and that (A,G,a) is a C*-dymanical system such that, for each/ 6 T, the idea t =l I {f e A\f(t) = 0} is G-invariant. If in addition, the stabiliser group ofeach P € Prim(/i) is amenable, the Q Gn i As th xe section algebra of a C*-bundle with fibres{A, x Q G}, eT .The above theorem may be used to prove a structure theorem for crossed products builtfrom C*-dynamical system G, a)s wher (A, e the action of G on A is smooth. Assuming that thestabiliser groups are amenable, then A Q G x has a composition series such that each quotient isa section algebra of a C*-bundle where the fibres are o af G; the for moreoverm Ag, th »e Agcorrespond to locally closed subsets of Prim(A), and G acts transitivel 3 ).y I on Prim(A manycases, in particular when (G,A) is separable, th s xe A Q G have been computed explicitly by otherauthors.These results are actually proved for twisted C* -dynamical systems.1980 Mathematics subject classification (Amer. Math. Soc.) (198 Revision):5 46 L 05.

A Symmetric Imprimitivity Theorem for Commuting Proper Actions

We prove a symmetric imprimitivity theorem for commuting proper actions of locally com- pact groups H and K on a C � -algebra.