Click here to view fulltext PDF

Permanent link:
https://www.ias.ac.in/article/fulltext/pmsc/127/04/0689-0705

Arens multiplication; foundation semigroup; second dual; compactly cancellative; involution; trivolution; Fourier algebra

We investigate involutions and trivolutions in the second dual of algebras related to a locally compact topological semigroup and the Fourier algebra of a locally compact group. We prove, among the other things, that for a large class of topological semigroups namely, compactly cancellative foundation $\ast$-semigroup $S$ when it is infinite non-discrete cancellative, $M_{a}(S)^{\ast\ast}$ does not admit an involution, and $M_{a}(S)^{\ast\ast}$ has atrivolution with range $M_{a}(S)$ if and only if $S$ is discrete. We also show that when $G$ isan amenable group, the second dual of the Fourier algebra of $G$ admits an involutionextending one of the natural involutions of $A(G)$ if and only if $G$ is finite. However,$A(G)^{\ast\ast}$ always admits trivolution.

A ALINEJAD¹ A GHAFFARI¹

1. Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

Published

1 September 2017

Manuscript received

5 April 2015

Manuscript revised

5 November 2015

Accepted

3 December 2015

Early published

Unedited version published online

Final version published online

2 August 2017