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

Host, B.

Bull. Soc. Math. Fr. 114, 215-223 (1986).

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

Keywords:

locally compact group; Fourier-Stieltjes algebra; integer valued functions; idempotents; homomorphisms

Citations:

Zbl 0247.43014

References:

[1]	COHEN (P. J.) . - On a conjecture of Littlewood and idempotents measures , Amer. J. Math., vol. 82, 1960 , p. 191-212. MR 24 #A3231 | Zbl 0099.25504 · Zbl 0099.25504
[2]	COHEN (P. J.) . - On homomorphisms of groups algebras , Amer. J. Math., vol. 82, 1960 , p. 213-226. MR 24 #A3232 | Zbl 0099.25601 · Zbl 0099.25601
[3]	DIXMIER (J.) . - Les C*-algèbres et leurs représentations , Gauthier-Villars, Paris, 1969 . MR 39 #7442 | Zbl 0174.18601 · Zbl 0174.18601
[4]	EYMARD (P.) . - L’algèbre de Fourier d’un groupe localement compact , Bull. Soc. Math. Fr., vol. 92, 1964 , p. 181-236. Numdam | MR 37 #4208 | Zbl 0169.46403 · Zbl 0169.46403
[5]	LEFRANC (M.) . - Sur certaines algèbres de fonctions sur un groupe , C.R. Acad. Sc., t. 274, 1972 , p. 1882-1883. MR 47 #5523 | Zbl 0247.43014 · Zbl 0247.43014
[6]	RUDIN (W.) . - Fourier analysis on groups , Interscience tracts in Math. n^\circ 12, Interscience, New York, 1962 . MR 27 #2808 | Zbl 0107.09603 · Zbl 0107.09603