A bicommutant theorem for dual Banach algebras

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.Comment: 6 page