Shift invariant preduals of ?1
Speaker: M. Daws (Leeds)
Date: Tue 15th February 2011
Location: Mathematical Sciences Seminar Room
(Joint work with Richard Haydon, Thomas Schlumprecht and Stuart White)
The Banach space ?1 has many different preduals-- for example, if K is a locally compact, Hausdorff space which is countable, then the dual of C0(K) is ?1(K), which is isomorphic to ?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 ?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 ?1 with the additional property of making the bilateral shift weak*-continuous. For example, if a predual of the form C0(K) does this, then K must carry the discrete topology, so really we just get the canonical predual c0. 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.)