Isometric isomorphisms of measure algebras

Robert S. Strichartz

Vol. 15 (1965), No. 1, 315–317

DOI: 10.2140/pjm.1965.15.315

Abstract

The following theorem is proved:

If $G_1$ and $G_2$ are locally compact groups, $A_i$ are algebras of finite regular Borel measures such that $L^1\left(G_i\right)\subseteq A_i\subseteq \mathcal{ℳ}\left(G_i\right)$ for $i=1,2$, and $T$ is an isometric algebra isomorphism of $A_1$ onto $A_2$, then there exists a homeomorphic isomorphism $\alpha$ of $G_1$ onto $G_2$ and a continuous character $\chi$ on $G_1$ such that $T\mu \left(f\right)=\mu \left(\chi \left(f\circ \alpha \right)\right)$ for $\mu \in A_1$ and $f\in C_0\left(G_2\right)$.

This result was previously known for abelian groups and compact groups (Glicksberg) and when $A_i=L^1\left(G_i\right)$ (Wendel) where $T$ is only assumed to be a norm decreasing algebra isomorphism.

A corollary is that a locally compact group is determined by its measure algebra.

Mathematical Subject Classification

Primary: 22.20

Secondary: 42.56

Milestones

Received: 8 March 1964

Published: 1 March 1965