We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of one--sided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions their C --algebras are Morita equivalent; the groupoid C --algebra C (G) is the Cuntz--Krieger algebra of an infinite f0; 1g matrix defined by G, and that the algebras C (G(?)) contain the C --algebras used by Doplicher and Roberts in their duality theory for compact groups. We then analyse the ideal structure of these groupoid C --algebras using the general theory of Renault, and calculate their K-theory. 1 Introduction Over the past fifteen years many C -algebras and classes of C -algebras have been given groupoid models. Here we consider locally finite directed graphs, which may have infinitely many vertices, but only finitely many edges in and out of each vertex. We associate ...

We study operator spaces, operator algebras, and operator modules, from the point of view of the ‘noncommutative Shilov boundary’. In this attempt to utilize some ‘noncommutative Choquet theory’, we find that Hilbert C ∗-modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We introduce certain multiplier operator algebras and C ∗-algebras of an operator space, which generalize the algebras of adjointable operators on a C ∗-module, and the ‘imprimitivity C ∗-algebra’. It also generalizes a classical Banach space notion. This multiplier algebra plays a key role here. As applications of this perspective, we unify and strengthen several theorems characterizing operator algebras and modules. We also include some general notes on the ‘commutative case ’ of some of the topics we discuss, coming in part from joint work with Christian Le Merdy, about ‘function modules’.

Abstract. 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. Coactions of groups on C∗-algebras, and their crossed products, were introduced to make duality arguments available for the study of dynamical systems involving actions of nonabelian groups. For these to be effective, one needs to understand the representation theory of crossed products by coactions. The most powerful tool we have was provided by Mansfield [12]: he showed how to induce representations from crossed products by quotient groups, and proved an imprimitivity theorem which characterises these induced representations. Unfortunately, Mansfield’s construction is complicated and technical.

Abstract. 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 × H, then Br(G) is the equivariant Brauer group BrH(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 BrG(X/H) and BrH(G\X) are isomorphic. Secondly we show that the subgroup Br0(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. 1.

Abstract. We give a condition on a full coaction (A, G, δ) of a (possibly) nonamenable group G and a closed normal subgroup N of G which ensures that Mansfield imprimitivity works; i.e. that A × δ | G/N is Morita equivalent to A ×δ G × ˆ δ,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. 1.

Abstract. We continue our study of the general theory of possibly nonselfadjoint algebras of operators on a Hilbert space, and modules over such algebras, developing a little more technology to connect ‘nonselfadjoint operator algebra ’ with the C ∗ −algebraic framework. More particularly, we make use of the universal, or maximal, C ∗ −algebra generated by an operator algebra, and C ∗ −dilations. This technology is quite general, however it was developed to solve some problems arising in the theory of Morita equivalence of operator algebras, and as a result most of the applications given here (and in a companion paper) are to that subject. Other applications given here are to extension problems for module maps, and characterizations of C ∗ −algebras. * Supported by a grant from the NSF. The contents of this paper were announced at the January 1999 meeting of the American Mathematical Socety. 1 2 DAVID P. BLECHER 1. Introduction- Modules

Abstract. 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 × ˆ δ,r N implements equivalences between Mansfield induction of representations from A×G/N to A×G and restriction of representations from A×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. 1.

. We give a condition on a full coaction (A; G; ffi ) 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 \Theta ffij G=N is Morita equivalent to A \Theta ffi G \Theta ffi ;r N . This condition obtains if N is amenable or ffi is normal. It is preserved under Morita equivalence, inflation of coactions, the stabilization trick of Echterhoff and Raeburn, and on passing to twisted coactions. 1. Introduction One of Mackey's great gifts to mathematics is the imprimitivity theorem for locally compact groups, which Takesaki generalized in [Tak67] to C --covariant systems. The modern (i.e., post-Rieffel) way to prove an imprimitivity theorem is to set up a (strong) Morita equivalence, and Green did this for Takesaki's theorem in [Gre78]. In the special case of abelian groups, Green's theorem can be viewed as saying that if ffi is an action of G on A, ffi is the dual action of G on the crossed product...

Abstract. We show that two C ∗-algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K0-groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C ∗-algebras of arbitrary finitely generated abelian groups. 1.

Abstract. We give a Clifford correspondence for an algebra A over an algebraically closed field, that is an algorithm for constructing some finite-dimensional simple A-modules from simple modules for a subalgebra and endomorphism algebras. This applies to all finite-dimensional simple A-modules in the case that A is finite-dimensional and semisimple with a given semisimple subalgebra. We discuss connections between our work and earlier results, with a view towards applications particularly to finite-dimensional semisimple Hopf algebras. 1.