Vol. 58, No. 1, 1975

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On the structure of the Fourier-Stieltjes algebra

Martin E. Walter

Vol. 58 (1975), No. 1, 267–281

If G is a locally compact group, denote its Fourier-Stieltjes algebra by B(G) and its Fourier algebra by A(G). If G is compact, then B(G) = A(G) and σ(B(G)), the spectrum of B(G), is G. If G is not compact then σ(B(G)) contains partial isometries and projections different from e, the identity of G. More generally, σ(B(G)) is closed under operations that commute with “representing” and the “taking of tensor products”. It is shown that σ(B(G)) contains a smallest positive element, zF; and that g G σ(B(G))zFg σ(B(G))zF is an epimorphism of G into G, the almost periodic compactification of G.

A structure theorem is given for the closed, bi-translation, invariant subspaces of B(G). In so doing we introduce the concepts of inverse Fourier transform localized at π, and the standardization of π, where π is a continuous, unitary representation of G.

Mathematical Subject Classification 2000
Primary: 22D15
Secondary: 46L25
Received: 5 March 1974
Published: 1 May 1975
Martin E. Walter