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Disjointness preserving mappings between Fourier algebras. (English) Zbl 0911.43003
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 ?
43A30Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A15L p -spaces and other function spaces on groups, semigroups, etc.
47B48Operators on Banach algebras