Herz-Schur multipliers and weakly almost periodic functions on locally compact groups

Abstract

For a locally compact group G G and 1 > p > ∞ 1>p>\infty , let A p ( G ) A_{p}(G) be the Herz–Figà-Talamanca algebra and B p ( G ) B_{p}(G) the Herz-Schur multipliers of G G , and M A p ( G ) MA_{p}(G) the multipliers of A p ( G ) A_{p}(G) . Let W ( G ) W(G) be the algebra of continuous weakly almost periodic functions on G G . In this paper, we show that (1), if G G is a noncompact nilpotent group or a noncompact [IN]-group, then W ( G ) / B p ( G ) − W(G)/B_{p}(G)^{-} contains a linear isometric copy of l ∞ ( N ) l^{\infty }({\mathbb {N}}) ; (2), for a noncommutative free group F , B p ( F ) F, B_{p}(F) is a proper subset of M A p ( F ) ∩ W ( F ) {MA_{p}(F)\cap {W(F)}} .