Representations of B^∗-algebras on Banach spaces

Bruce Alan Barnes

Vol. 50 (1974), No. 1, 7–18

DOI: 10.2140/pjm.1974.50.7

Abstract

This paper deals with continuous irreducible representations of a B^∗-algebra on a Banach space. The main result is that if π is a continuous irreducible representation of a B^∗-algebra A on a reflexive Banach space X, and if there is a subset S of A such that the intersection of the mull spaces of the operators π(a) for all a ∈ S is a nonzero, finite dimensional subspace of X, then X is a Hilbert space in am equivalent norm and π is similar to a ∗-representativn of A on this Hilbert space.

Mathematical Subject Classification 2000

Primary: 46K10

Milestones

Received: 13 September 1972

Revised: 27 December 1972

Published: 1 January 1974