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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.
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