Disjointness preserving mappings between Fourier algebras. (English) Zbl 0911.43003

Font, Juan J.

Colloq. Math. 77, No. 2, 179-187 (1998).

Let \(G\) be a locally compact group. The Fourier-Stieltjes algebra \(B(G)\) and the Fourier algebra \(A(G)\) were for the first time investigated by P. Eymard in 1964. The paper proves mainly that for two locally compact amenable groups \(G_1\) and \(G_2\), the Fourier algebras \(A(G_1)\) and \(A(G_2)\) are algebra isomorphic if and only if there exists a disjointness preserving bijection between them (Theorem 4), and such disjointness preserving bijection of \(A(G_1)\) onto \(A(G_2)\) can be extended, in a unique way, to a weighted composition bijection of \(B(G_1)\) onto \(B(G_2)\) (Theorem 5). The author notices that if the amenability of the group \(G\) is dropped, these results may fail to be true. The following may be an interesting question (like the inverse problem of the paper): can an isometric isomorphism or a bipositive algebra isomorphism between \(A(G_1)\) and \(A(G_2)\) (or \(B(G_1)\) and \(B(G_2))\) be deduced to a topological isomorphism of \(G_1\) onto \(G_2\)?

Reviewer: H.-C.Lai (Taiwan)

MSC:

43A30	Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A15	\(L^p\)-spaces and other function spaces on groups, semigroups, etc.
47B48	Linear operators on Banach algebras

Keywords:

multiplier; locally compact group; Fourier-Stieltjes algebra; Fourier algebra; amenable groups; disjointness preserving bijection