On some group algebra modules related to Wiener's algebraM1

Teng-Sun Liu, Arnoud van Rooij, Ju-Kwei Wang
1974 Pacific Journal of Mathematics  
TENG-SUN LIU, ARNOUD VAN ROOIJ AND JU-KWEI WANG Along with his study of the general Tauberian theorem in Li, N. Wiener introduced the algebra Mi which consists of all those continuous functions / on the real line R for which Σ max |/(aOI<oo. He proved that many features of L 19 including the general Tauberian theorem, are shared by M x . In this paper to generalize M x to an arbitrary locally compact group G. While doing this, a host of L X {G)-modules mutually related by conjugation and the
more » ... ration of forming multiplier modules. -<[((?) is among them. In case G is abelian, -<[(£) is a Segal algebra, so that it has the same ideal-theoretical structure as LΛG). If further G = R, -<(G) reduces to the Wiener algebra Mi with an equivalent norm. ON SOME GROUP ALGEBRA MODULES 509 exists a unique |μ\ e R such that ξ A \μ\ = \ξ Λ μ| for every compact set A (see [1; Ch. 13]). By the Radon-Nikodym Theorem [3; 12.17] we may identify L ιΛθQ with {μeR: μ < λ}. THEOREM 1.2. If f eL γ has compact support and if μeR, then f*μ and μ*f lie in L lt i oc . If, in addition, f is bounded, and μe -ΣΊ.IOO then f*μ and μ*f are continuous.
doi:10.2140/pjm.1974.55.507 fatcat:etq6bl245fgzxnv2tcmpvltr5u