Fereidoun Ghahramani

The Amenability of Measure Algebras

In this paper we shall prove that the measure algebra M(G) of a locally compact group G is amenable as a Banach algebra if and only if G is discrete and amenable as a group. Our contribution is to resolve a conjecture by proving that M(G) is not amenable in the case where the group G is not discrete. Indeed, we shall prove a much stronger result: the measure algebra of a non-discrete, locally compact group has a non-zero, continuous point derivation at a certain character on the algebra.

Approximate and pseudo-amenability of various classes of Banach algebras

We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors together with R.J. Loy. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity, and that the Fourier algebra of the free group on two generators is not operator approximately amenable. Further examples are obtained of l1-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate contractibility need not imply sequential approximate amenability. Results are also given for Segal algebras on locally compact groups, and algebras of p-pseudo-functions on discrete groups.

APPROXIMATE AMENABILITY OF SCHATTEN CLASSES, LIPSCHITZ ALGEBRAS AND SECOND DUALS OF FOURIER ALGEBRAS

Amenability of any of the algebras described in the title is known to force them to be finite-dimensional. The analogous problems for approximate amenability have been open for some years now. In this article we give a complete solution for the first two classes, using a new criterion for showing that certain Banach algebras without bounded approximate identities cannot be approximately amenable. The method also provides a unified approach to existing non-approximate amenability results, and is applied to the study of certain commutative Segal algebras. Using different techniques, we prove that bounded approximate amenability of the second dual of a Fourier algebra implies that it is finite-dimensional. Some other results for related algebras are obtained.

Standard homomorphisms and regulated weights on weighted convolution algebras

Abstract In this paper we introduce a class of homomorphisms between weighted convolution algebras which we call standard homomorphisms and derive various equivalents of standardness. We also introduce the convergence ideal of a homomorphism. We find various descriptions of the convergence ideal together with its relation to standardness. We show that every continuous homomorphism from a weighted convolution algebra into another weighted convolution algebra, with a regulated weight, is standard, and when the algebras have both regulated weights the extension of a homomorphism to the weighted measure algebras satisfies additional continuity properties.