A bicommutant theorem for dual Banach algebras

TL;DR

This paper proves a bicommutant theorem for unital dual Banach algebras, showing they can be represented on reflexive Banach spaces with their images equal to their bicommutants.

Contribution

It establishes a bicommutant theorem for dual Banach algebras, extending classical results to this broader context.

Findings

Existence of a reflexive Banach space representation for unital dual Banach algebras.

Representation is isometric and weak*-weak* continuous.

The image of the algebra equals its bicommutant under this representation.

Abstract

A dual Banach algebra is a Banach algebra which is a dual space, with the multiplication being separately weak$^*$-continuous. We show that given a unital dual Banach algebra $\mc A$, we can find a reflexive Banach space $E$, and an isometric, weak$^*$-weak$^*$-continuous homomorphism $\pi:\mc A\to\mc B(E)$ such that $\pi(\mc A)$ equals its own bicommutant.