A special retirement meeting for H. Garth Dales

Talks

Matthew Daws

Title: Ring-theoretical infiniteness and ultrapowers

Abstract: For any ring, in particular, for a Banach algebra, there are notions of being "infinite", stated in terms of idempotents and equivalence. We investigate when such notions are preserved by the ultrapower (or ultraproduct) construction for Banach algebras. It turns out that having (or not) certain forms of norm control are central to answering this question, and we show by various examples that operator algebras and general Banach algebras can behave very differently. This is joint work with Bence Horvath.

Jared White

Title: Invariant means and Arens products

Abstract: Given a Banach algebra A its second dual A^** is a Banach algebra with the so-called Arens product. Let G be an amenable group. We introduce a subalgebra A_sn(G) of ℓ¹(G)^**, which is related to invariant means on the subnormal subgroups of G. We explain how the nilpotence of A_sn(G) reflects the subnormal structure of G in the case that G is a just infinite group. Finally we discuss how this sheds some light on the nilpotence of rad ℓ¹(G)^**.

Niels J. Laustsen

Title: A C(K)-space with few operators

Abstract: I shall report on joint work with Piotr Koszmider (IMPAN, Warsaw), published earlier this year in Advances in Mathematics, concerning the closed subspace of ℓ^∞ generated by c₀ and the characteristic functions of elements of an uncountable, almost disjoint family A of infinite subsets of the natural numbers. This Banach space has the form C₀(K_A) for a locally compact Hausdorff space K_A that is known under many names, including Ψ-space and Isbell-Mrówka space.

Koszmider and I construct an uncountable, almost disjoint family _A such that the algebra of all bounded operators on C₀(K_A) is as small as possible in the precise sense that every bounded operator on C₀(K_A) is the sum of a scalar multiple of the identity and an operator that factors through c₀ (which in this case is equivalent to having separable range). This construction improves a previous construction of Koszmider, which required the assumption of the Continuum Hypothesis.