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Le théorème des idempotents dans B(G). (The theorem of idempotents in B(G)). (French) Zbl 0606.43002

Let G be a locally compact group and B(G) the Fourier-Stieltjes algebra of G. This interesting paper gives a complete description of the integer valued functions in B(G). It turns out that these are exactly the linear combinations with integer coefficients of translates of characteristic functions of open subgroups of G. In particular, one thereby obtains a characterization of the idempotents in B(G). For discrete groups this theorem has been announced by M. Lefranc [C. R. Acad. Sci., Paris, Sér. A 274, 1882-1883 (1972; Zbl 0247.43014)].
It is worth mentioning that Cohen’s idempotent theorem for measures on abelian locally compact groups G can easily be deduced. In fact, \(B(G)=M(\Gamma)^{\wedge}\), where \({^{\wedge}}\) denotes Fourier transform and \(\Gamma\) the dual group of G. The author applies his result to describe homomorphisms from the Fourier algebra A(G) into B(H), where G and H are locally compact groups and, in addition, G is abelian.
Reviewer: E.Kaniuth

MSC:

43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups

Citations:

Zbl 0247.43014
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References:

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