#### Vol. 15, No. 1, 1965

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Isometric isomorphisms of measure algebras

### Robert S. Strichartz

Vol. 15 (1965), No. 1, 315–317
##### 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 ${L}^{1}\left({G}_{i}\right)\subseteq {A}_{i}\subseteq \mathsc{ℳ}\left({G}_{i}\right)$ 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.

Primary: 22.20
Secondary: 42.56