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
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by Anthony To Ming Lau and Kelly McKennon PDF

Proc. Amer. Math. Soc. 79 (1980), 55-58 Request permission

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.

References

Mahlon M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–544. MR 92128

Linear operators

B. E. Johnson, Isometric isomorphisms of measure algebras, Proc. Amer. Math. Soc. 15 (1964), 186–188. MR 160846, DOI 10.1090/S0002-9939-1964-0160846-2

Abstract harmonic analysis

Kelly McKennon, Multipliers, positive functionals, positive-definite functions, and Fourier-Stieltjes transforms, Memoirs of the American Mathematical Society, No. 111, American Mathematical Society, Providence, R.I., 1971. MR 0440299
J. G. Wendel, Left centralizers and isomorphisms of group algebras, Pacific J. Math. 2 (1952), 251–261. MR 49911