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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 $B\left(G\right)$ and the Fourier algebra $A\left(G\right)$ 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 $A\left({G}_{1}\right)$ and $A\left({G}_{2}\right)$ are algebra isomorphic if and only if there exists a disjointness preserving bijection between them (Theorem 4), and such disjointness preserving bijection of $A\left({G}_{1}\right)$ onto $A\left({G}_{2}\right)$ can be extended, in a unique way, to a weighted composition bijection of $B\left({G}_{1}\right)$ onto $B\left({G}_{2}\right)$ (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 $A\left({G}_{1}\right)$ and $A\left({G}_{2}\right)$ (or $B\left({G}_{1}\right)$ and $B\left({G}_{2}\right)\right)$ be deduced to a topological isomorphism of ${G}_{1}$ onto ${G}_{2}$?
##### 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 Operators on Banach algebras