Pseudomeasures and pseudofunctions on inverse semigroups (Q744850)

Let \(S\) be an inverse semigroup. \textit{A. R. Medghalchi} and \textit{H. Pourmahmood-Aghababa} [J. Math. Anal. Appl. 395, No. 2, 473--485 (2012; Zbl 1257.43004)] introduced the space of \(p\)-pseudomeasures \(\text{PM}_p(S)\). The von Neumann algebra of \(S\) is defined as \(\text{VN}(S):=\text{PM}_2(S)\) and the Figá-Talamanca-Herz algebra is denoted by \(\text{A}_p(S)\). The paper under review is a continuation of the above paper. The author defines \(\text{Cv}_p(S)\) for \(1<p<\infty\) as the space of convolutors of \(l^p(S)\) and proves that if \(S\) is amenable with finitely many idempotents, then \(\text{Cv}_p(S)\) can be identified with \(\text{PM}_p(S)\) and the dual \(\text{W}_p(S)\) of the space of \(p\)-pseudofunctions \(\text{PF}_p(S)\) can be embedded into the pointwise multipliers of \(\text{A}_p(S)\). In addition, he shows that when \(\text{A}(S)\) has a bounded approximate identity with bound \(1\), the space of \(\text{A}(S)\)-multipliers from \(\text{VN}(S)\) into \(\text{VN}(S)\) can be identified with the dual space of \(\text{UC}(\widehat{S})\), which is the closure of \(\text{A}(S)\text{VN}(S)\) in \(\text{VN}(S)\).