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
${L}^{1}\left({G}_{i}\right)\subseteq {A}_{i}\subseteq \mathcal{\mathcal{M}}\left({G}_{i}\right)$ 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.
