Generalised F-algebras as predual algebras of JBW⁎-triples and characterisations of the Fourier algebras on locally compact groups

Abstract

We introduce the notion of a generalised F-algebra as a Banach algebra A whose Banach space dual is a JBW⁎-triple with a certain additional condition on a distinguished character of A. This generalises the well-known notion of F-algebras, which in turn, includes as special cases many important classes of Banach algebras on locally compact groups and semigroups. Our main use of this new notion is to give some axiomatic characterisations of the Fourier algebras on locally compact groups, strengthening our previous work [37]. But we also quickly review some well-known results in F-algebras that can be extended to the context of generalised F-algebras.