## 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 $$G_2$$?
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
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