A note on isomorphisms of groups algebras

Author:
Geoffrey V. Wood

Journal:
Proc. Amer. Math. Soc. **25** (1970), 771-775

MSC:
Primary 42.56; Secondary 46.00

MathSciNet review:
0259503

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Abstract | References | Similar Articles | Additional Information

Abstract: In this note, it is shown that, if are compact groups, and are the (convolution) algebras of continuous, complex-valued functions on and respectively, then the existence of a norm-decreasing algebra-isomorphism of onto ensures that the groups are isomorphic. The corresponding theorem with and locally finite discrete groups is also proved.

**[1]**R. E. Edwards,*Bipositive and isometric isomorphisms of some convolution algebras*, Canad. J. Math.**17**(1965), 839–846. MR**0183807****[2]**Frederick P. Greenleaf,*Norm decreasing homomorphisms of group algebras*, Pacific J. Math.**15**(1965), 1187–1219. MR**0194911****[3]**I. N. Herstein,*Topics in algebra*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. MR**0171801****[4]**Lynn H. Loomis,*An introduction to abstract harmonic analysis*, D. Van Nostrand Company, Inc., Toronto-New York-London, 1953. MR**0054173****[5]**Daniel Rider,*Closed subalgebras of 𝐿¹(𝑇)*, Duke Math. J.**36**(1969), 105–115. MR**0243286****[6]**Roger Rigelhof,*Norm decreasing homomorphisms of measure algebras*, Trans. Amer. Math. Soc.**136**(1969), 361–371. MR**0239004**, 10.1090/S0002-9947-1969-0239004-8**[7]**J. G. Wendel,*Left centralizers and isomorphisms of group algebras*, Pacific J. Math.**2**(1952), 251–261. MR**0049911****[8]**G. V. Wood,*A generalization of the Peter-Weyl theorem*, Proc. Cambridge Philos. Soc.**63**(1967), 937–945. MR**0217615**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1970-0259503-6

Keywords:
Compact groups,
norm-decreasing algebra-isomorphism,
Peter-Weyl theorem,
group isomorphism and homeomorphism,
locally finite discrete groups

Article copyright:
© Copyright 1970
American Mathematical Society