Isomorphisms of locally compact groups and Banach algebras

Lau, Anthony To Ming; McKennon, Kelly

doi:10.1090/S0002-9939-1980-0560583-5

Isomorphisms of locally compact groups and Banach algebras

Authors: Anthony To Ming Lau and Kelly McKennon
Journal: Proc. Amer. Math. Soc. 79 (1980), 55-58
MSC: Primary 43A22; Secondary 43A15
DOI: https://doi.org/10.1090/S0002-9939-1980-0560583-5
MathSciNet review: 560583
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Abstract: If G is a locally compact group, then $\mathrm{UBC}_r (G)^*$ , the dual of the space of bounded right uniformly continuous complex-valued functions on G, with the Arens product is a Banach algebra. We prove in this paper a result that will have as a consequence the following: Let ${G_1},{G_2}$ be locally compact groups. Then the Banach algebras $\mathrm{UBC}_r{({G_1})^ \ast }$ and $\mathrm{UBC}_r{({G_2})^\ast}$ are isometric isomorphic if and only if ${G_1}$ and ${G_2}$ are topologically isomorphic.

[1] M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509-544. MR 0092128 (19:1067c)
[2] N. Dunford and J. T. Schwartz, Linear operators. I, Interscience, New York, 1958.
[3] B. E. Johnson, Isometric isomorphisms of measure algebras, Proc. Amer. Math. Soc. 15 (1964), 186-188. MR 0160846 (28:4056)
[4] E. Hewitt and K. Ross, Abstract harmonic analysis, vol. 1, Springer-Verlag, Berlin, 1964.
[5] -, Abstract harmonic analysis, vol. 2, Springer-Verlag, Berlin, 1970.
[6] K. McKennon, Multipliers, positive functionals, positive-definite functions, and Fourier-Stieltjes transforms, Mem. Amer. Math. Soc., no. 111, 1971. MR 0440299 (55:13174)
[7] J. G. Wendel, Left centralizers and isomorphisms of group algebras, Pacific J. Math. 2 (1952), 251-261. MR 0049911 (14:246c)