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Zbl 0911.43003
Font, Juan J.
Disjointness preserving mappings between Fourier algebras. (English)
[J] Colloq. Math. 77, No.2, 179-187 (1998). ISSN 0010-1354; ISSN 1730-6302/e

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 $G_2$?
[H.-C.Lai (Taiwan)]

MSC 2000:: *43A30 Fourier type transforms on nonabelian groups, etc.
43A15 Lp-spaces and other function spaces on groups, etc.
47B48 Operators on Banach algebras

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

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