Weighted semigroup measure algebra as a WAP-algebra

TL;DR

This paper investigates conditions under which weighted measure algebras on semigroups are WAP-algebras, providing criteria related to the structure of the semigroup and the weight function, and improving existing results for discrete cases.

Contribution

It characterizes when weighted measure algebras are WAP-algebras or dual Banach algebras, extending previous results and applying them to discrete semigroups.

Findings

M_b(S,w) is a WAP-algebra iff wap(S) separates points of S

M_b(S,w) is a dual Banach algebra iff S is compactly cancellative

Results improve older theorems for discrete semigroups

Abstract

Banach algebra A for which the natural embedding x into x^ of A into WAP(A)* is bounded below; that is, for some m in R with m > 0 we have ||x^|| > m ||x||, is called a WAP-algebra. Through we mainly concern with weighted measure algebra M_b(S;w); where w is a weight on a semi-topological semigroup S. We study those con- ditions under which M_b(S;w) is a WAP-algebra (respectively dual Banach algebra). In particular, M_b(S) is a WAP-algebra (respectively dual Banach algebra) if and only if wap(S) separates the points of S (respectively S is compactly cancellative semigroup). We apply our results for improving some older results in the case where S is discrete.