M. Daws (Leeds)

will speak on

Shift invariant preduals of $\ell^1$

Time: 3:00PM
Date: Tue 15th February 2011
Location: Mathematical Sciences Seminar Room [map]

Abstract: (Joint work with Richard Haydon, Thomas Schlumprecht and Stuart
White)


The Banach space $\ell^1$ has many different preduals-- for
example, if $K$ is a locally compact, Hausdorff space which is
countable, then the dual of $C_0(K)$ is $\ell^1(K)$, which is
isomorphic to $\ell^1$ through picking an enumeration of $K$. There
are also more "exotic" preduals-- the recent solution to the
Scalar-Compact problem, by Argyros and Haydon, is a Banach space with
is an $\ell^1$ predual.

In this talk, I will take as my indexing set the integers, and so we
have the bilateral shift operator. We shall investigate if there
exist preduals of $\ell^1$ with the additional property of making the
bilateral shift weak*-continuous. For example, if a predual of the
form $C_0(K)$ does this, then K must carry the discrete topology, so
really we just get the canonical predual $c_0$. However, we give an
explicit construction of a different predual which does make the
bilaterial shift weak*-continuous.

Time allowing, I will show how Banach algebraic tools become useful
(indeed, my original motivation came from Banach algebra theory).
Indeed, some sort of classification is possible, and a more abstract
construction leads to a wealth of examples.

(This talk is part of the Analysis series.)

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