Characterisations of Fourier and Fourier–Stieltjes algebras on locally compact groups

Abstract

Motivated by the beautiful work of M. A. Rieffel (1965) and of M. E. Walter (1974), we obtain characterisations of the Fourier algebra A(G) of a locally compact group G in terms of the class of F -algebras (i.e. a Banach algebra A such that its dual A ′ is a W * -algebra whose identity is multiplicative on A). For example, we show that the Fourier algebras are precisely those commutative semisimple F -algebras that are Tauberian, contain a nonzero real element, and possess a dual semigroup that acts transitively on their spectrums. Our characterisations fall into three flavours, where the first one will be the basis of the other two. The first flavour also implies a simple characterisation of when the predual of a Hopf-von Neumann algebra is the Fourier algebra of a locally compact group. We also obtain similar characterisations of the Fourier-Stieltjes algebras of G. En route, we prove some new results on the problem of when a subalgebra of A(G) is the whole algebra and on representations of discrete groups.2010 Mathematics Subject Classification. Primary 43A30, 22D25; Secondary 46J05, 22D10.

Abstract Structure of Measure Algebras on Coset Spaces of Compact Subgroups in Locally Compact Groups

“…The following paper extends our recent results concerning the abstract structure of involutions on some Banach algebras on homogeneous spaces [7,9,17] to more general settings. The mathematical theory of Banach convolution algebras plays significant and classical roles in abstract harmonic analysis, representation theory, functional analysis, operator theory, and C * -algebras, see [2,3,21,22,25,30,31,32,33,34] and references therein. Over the last decades, some new aspects and applications of Banach convolution algebras have achieved significant popularity in different areas such as constructive approximation [4,5,6], and theoretical aspects of coherent state (covariant) analysis, see [26] and references therein.…”