Vol. 15, No. 1, 1965

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Isometric isomorphisms of measure algebras

Robert S. Strichartz

Vol. 15 (1965), No. 1, 315–317

The following theorem is proved:

If G1 and G2 are locally compact groups, Ai are algebras of finite regular Borel measures such that L1(Gi) Ai (Gi) for i = 1,2, and T is an isometric algebra isomorphism of A1 onto A2, then there exists a homeomorphic isomorphism α of G1 onto G2 and a continuous character χ on G1 such that Tμ(f) = μ(χ(f α)) for μ A1 and f C0(G2).

This result was previously known for abelian groups and compact groups (Glicksberg) and when Ai = L1(Gi) (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
Received: 8 March 1964
Published: 1 March 1965
Robert S. Strichartz
Department of Mathematics
Cornell University
310 Malott Hall
Ithaca NY 14853-4201
United States