Module Arens regularity for semigroup algebras
The authors introduce and study the notion of module Arens regularity. A classical result by \textit{N. J. Young} [Q. J. Math., Oxf. II. Ser. 24, 59--62 (1973; Zbl 0252.43009)] says that the group algebra \(L^1(G)\) of a locally compact group \(G\) is Arens regular if and only if \(G\) is finite. The result is not true for discrete semigroups: for example, if \(S\) is a right zero semigroup of any cardinality, the semigroup algebra \(\ell^1(S)\) is Arens regular. The paper under review presents an extension of Young's theorem to inverse semigroups \(S\) such that the subsemigroup \(E\) of the idempotents in \(S\) is totally ordered. Namely, \(\ell^1(S)\) is module Arens regular with respect to suitable actions by \(\ell^1(E)\) if and only if a certain homomorphic image of \(S\) is finite. Reviewer's remark: There is a mistake in the proof of Theorem 2.2, and the definition of module Arens regularity needs to be modified in order to recover the results. The authors are aware of the problem and have written a correction.
scientific article; zbMATH DE number 5377133