1045 – 1145  David Blecher  Noncommutative peak interpolation: prescribing behaviour of noncommutative functions on noncommutative subsets  

We describe recent work on noncommutative peak interpolation and noncommutative Urysohn lemmas. That is, finding noncommutative versions of functions (usually from a fixed algebra of operators) that have certain behaviours on certain noncommutative sets. In fact we have recently been able to essentially complete this theory. This is mostly joint work, with Damon Hay, Matt Neal, and Charles Read.

1200 – 1210  Welcome from the ViceChancellor, Professor Mark Smith, and Head of Department, Professor John Whitehead 
1215 – 1315  Tomasz Kochanek  Some open questions arising from the stability problem for vector measures  

In 1983, motivated by the question whether \(c_0\) and \(\ell_\infty\) are \(K\)spaces ( i.e. their only extensions by the onedimensional space \(\mathbb{R}\) are locally convex), N.J. Kalton and J.W. Roberts proved that whenever \(\mathscr{F}\) is a set algebra and a function \(\nu \colon \mathscr{F} \to \mathbb{R}\) satisfies
\[
 \nu(A\cup B)\nu(A)\nu(B)  \leq 1 \quad \mbox{for } A, B \in \mathscr{F} \mbox{ with } A \cap B = \varnothing,
\]
there exists an additive set function \(\mu \colon \mathscr{F} \to \mathbb{R}\) such that \( \nu(A)\mu(A)  \leq 45\) for every \(A \in \mathscr{F}\). In a recent paper, I dealt with the vector analogue of this statement; a Banach space \(X\) is said to have the \(\mathsf{SVM}\) property [the \(\kappa\)\(\mathsf{SVM}\) property, for some cardinal \(\kappa\)], provided that there exists a constant \(v( X ) < \infty\) such that for every set algebra \(\mathscr{F}\) [with cardinality less than \(\kappa\)] and any function \(\nu \colon \mathscr{F} \to X\) satisfying
\[
\ \nu(A\cup B)\nu(A)\nu(B) \ \leq 1 \quad \mbox{for } A, B \in \mathscr{F} \mbox{ with } A \cap B = \varnothing
\]
there exists a vector measure \(\mu \colon \mathscr{F} \to X\) such that \(\ \nu( A )  \mu( A ) \ \leq v( X )\) for every \(A \in \mathscr{F}\). During the talk, I shall discuss several open questions stemming from the study of the \(\mathsf{SVM}\) and \(\kappa\)\(\mathsf{SVM}\) properties such as:
 To what extent the Lindenstrauss lifting principle characterises \(\mathscr{L}_1\)spaces?
 Does the \(\mathsf{SVM}\) property imply injectivity?
 Must every infinitedimensional Banach space having the \(\omega_1\)\(\mathsf{SVM}\) property contain an isomorphic copy of \(c_0\)?
I shall also explain what are possible consequences of positive/negative answers and present some partial results concerning these problems.

Lunch 
1430 – 1530  Richard Smith  Topological properties associated with strict convexity  

In the past few years several topological properties associated with strictly convex norms have come to light, e.g., Gruenhage spaces and spaces having the socalled \((*)\) property. We survey these properties and see how they fit with existing notions, such as spaces having \(G_\delta\)diagonals. In addition, we outline a general method, based on walks on trees and ideas from topological dynamics, that can be used to construct examples of locally compact scattered nonGruenhage spaces having \(G_\delta\)diagonals.

1545 – 1645  Charles Read  Proximinality, operator convolution algebras and other topics  

We will discuss several topics in operator algebras, in all of which contractive approximate identities play a key role. Among these are included operator algebras obtained from actions of \(L^1( R^+, \ \omega )\) on \(L^2( R^+, \ \omega ),\) for a radical weight omega. These radical "operator convolution algebras" can be arranged to have compact or weakly compact multiplication according to which radical weight one is using. We will also discuss a basic counterexample in the theory of proximinality for operator algebras, which gives limits on the extent to which commutative results, like Glicksberg's peak set theorem, can be extended to noncommutative situations.

Refreshments 
1715 – 1815  Gilles Godefroy  Uniqueness of preduals: a survey  

Easy examples show that two nonisomorphic Banach spaces can have isometric preduals. However, a number of conditions on a Banach space \(X\), or on its dual \(X^*\), imply that \(X\) has a unique isometric predual: if \(Y^*\) is isometric to \(X^*\) then \(Y\)is isometric to \(X\). We will survey this topic and formulate some applications, e.g., to automatic weakstar continuity, or to uniqueness of preduals for some operator algebras. We will also recall some relevant open problems.

1845 – 2015  Buffet dinner – please let Garth Dales or Alex Belton know if you wish to attend 