Monday, 16th August 2021, Lancaster, UK
all times British Summer Time
10-10.15am: Welcome,
10.15-10.45: Jared White,
10.50-11.20: Niels Laustsen,
11.25-11.55: Matt Daws.
Titles and abstractsis dedicated to celebrating the contributions to mathematical analysis of Garth Dales on the occasion of his retirement.
• Matthew Daws (UCLan, Preston)
• Jared White (Lancaster and Open University)
• Niels J. Laustsen (Lancaster)
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 Asn(G) of ℓ1(G)**, which is related to invariant means on the subnormal subgroups of G. We explain how the nilpotence of Asn(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(KA) for a locally compact Hausdorff space KA 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(KA) is as small as possible in the precise sense that every bounded operator on C0(KA) 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.
Please contact the organisers with any questions you may have:
Kevin Beanland,
Tomasz Kania
Niels Jakob Laustsen.