Alan L. T. Paterson

Inner amenable locally compact groups

In this paper we study the relationship between amenability, inner amenability and property P of a von Neumann algebra. We give necessary conditions on a locally compact group G to have an inner invariant mean m such that m(V) = 0 for some compact neighborhood V of G invariant under the inner automorphisms. We also give a sufficient condition on G (satisfied by the free group on two generators or an I.C.C. discrete group with Kazhdans property T, e.g., SL(n, Z), n > 3) such that each linear form on L 2(G) which is invariant under the inner automorphisms is continuous. A characterization of inner amenability in terms of a fixed point property for left Banach G-modules is also obtained.

The exact cardinality of the set of topological left invariant means on an amenable locally compact group

The purpose of this note is to prove that if G is an amenable locally compact noncompact group, then the set of topological left invariant means on Lo(G) has cardinality 22, where d is the smallest cardinality of the covering of G by compact sets. We also prove that in this case the spectrum of the bounded left uniformly continuous complex-valued functions contains exactly 22d minimal closed invariant subsets (or left ideals)

TYCHONOFF'S THEOREM FOR LOCALLY COMPACT SPACES AND AN ELEMENTARY APPROACH TO THE TOPOLOGY OF PATH SPACES

The path spaces of a directed graph play an important role in the study of graph C*-algebras. These are topological spaces that were originally constructed using groupoid and inverse semigroup techniques. In this paper, we develop a simple, purely topological, approach to this construction, based on Tychonoffs theorem. In fact, the approach is shown to work even for higher dimensional graphs satisfying the finitely aligned condition, and we construct the groupoid of the graph. Motivated by these path space results, we prove a Tychonoff theorem for an infinite, countable product of locally compact spaces. The main idea is to include certain finite products of the spaces along with the infinite product. We show that the topology is, in a reasonable sense, a pointwise topology.

Amenability for twisted covariance algebras and group C∗-algebras

Abstract The basic question studied in this paper is, When is a twisted covariance C ∗ -algebra C ∗ ( G, A , F ) amenable? This includes as a special case the question, When is a group C ∗ -algebra C ∗ (G) amenable? We show, in particular, that C ∗ ( H, A , F is amenable if C ∗ ( G, A , F is amenable and H is a closed normal subgroup of G . Other results that we prove use the derivation and injective characterizations for amenable C ∗ -algebras.

Higher-dimensional virtual diagonals and ideal cohomology for triangular algebras

We investigate the cohomology of non-self-adjoint algebras using virtual diagonals and their higher-dimensional generalizations. We show that infinite dimensional nest algebras always have non-zero second cohomology by showing that they cannot possess 2-virtual diagonals. In the case of the upper triangular atomic nest algebra we exhibit concrete modules for non-vanishing cohomology.

Inverse Semigroups, Groupoids and A Problem of J. Renault

In [8], J. Renault showed that a topological groupoid G relates to inverse semigroups through its ample semigroup G a.In the case when G is r-discrete, the semigroup G a is “large” and determines the topology of G. The main theme of this paper is the following natural problem which was raised by Renault: given an inverse semigroup S (assumed, for convenience, unital) does there exist a (Hausdorff) r-discrete groupoid G with S “determining” G as an inverse subsemigroup of G a? What kind of uniqueness can we expect? Renault shows that there exists such a groupoid in the case where S is the Cuntz inverse semigroup O n, the groupoid G in that case being the Cuntz groupoid O n.

Groupoid C *-Algebras and Their Relation to Inverse Semigroup Covariance C *-Algebras

Let G be a locally compact groupoid. We recall (Definition 2.2.2) that G is equipped with a left Haar system {λ u }.

Inverse Semigroups and Locally Compact Groupoids

In this section we give a brief and largely self-contained account of the results on inverse semigroups that will be required in the sequel. The results we need from the algebraic theory of semigroups are well-known, and are contained in the standard textbooks (such as [50, 51, 133, 202]). However, for the convenience of the reader and for the purpose of establishing notation, an account concentrating on what we will need later from that theory is desirable.

The Groupoid C*-Algebras of Inverse Semigroups

In Chapter 3, we investigated the relationship between r-discrete groupoids and inverse semigroup actions (in the form of localizations (X, S)). We showed in particular (Theorem 3.3.1) that for every r-discrete groupoid G, there is an inverse subsemigroup S of G op such that C*(G) is isomorphic to the covariance algebra C0(G0) × β S. Conversely, under reasonable conditions, covariance C*-algebras are r-discrete groupoid C*-algebras (Corollary 3.3.2). So r-discrete groupoids are essentially the same as localizations.