Uniformly continuous functionals on the Fourier algebra of any locally compact group
Author:
Anthony To Ming Lau
Journal:
Trans. Amer. Math. Soc. 251 (1979), 39-59
MSC:
Primary 43A60; Secondary 22D25
DOI:
https://doi.org/10.1090/S0002-9947-1979-0531968-4
MathSciNet review:
531968
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Abstract | References | Similar Articles | Additional Information
Abstract: Let G be any locally compact group. Let be the von Neumann algebra generated by the left regular representation of G. We study in this paper the closed subspace
of
consisting of the uniformly continuous functionals as defined by E. Granirer. When G is abelian,
is precisely the bounded uniformly continuous functions on the dual group Ĝ. We prove among other things that if G is amenable, then the Banach algebra
(with the Arens product) contains a copy of the Fourier-Stieltjes algebra in its centre. Furthermore,
is commutative if and only if G is discrete. We characterize
, the weakly almost periodic functionals, as the largest subspace X of
for which the Arens product makes sense on
and
is commutative. We also show that if G is amenable, then for certain subspaces Y of
which are invariant under the action of the Fourier algebra
, the algebra of bounded linear operators on Y commuting with the action of
is isometric and algebra isomorphic to
for some
.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1979-0531968-4
Keywords:
Locally compact group,
amenable group,
regular representation,
Fourier algebra,
Fourier-Stieltjes algebra,
positive definite function,
-group algebra,
almost periodic functionals,
uniformly continuous functionals,
multipliers,
invariant mean,
second conjugate algebra,
Arens product
Article copyright:
© Copyright 1979
American Mathematical Society