The Relationship between Two Involutive SemigroupsSandSTIs Defined by a Left MultiplierT

Abstract

LetSbe a semigroup with a left multiplierTonS. There exists a new semigroupST, which depends onSandT, which has the same underlying space asS. We study the question of involutions onSTand a Banach algebraAT. We find a condition ofTunder whichSTand the second dualAT****admit an involution. We will show thatATisC*-algebra if and only ifT:AT→Ais an isometry, under mild conditions. Also,AisC*-algebra if and only if so isAT, under other minor conditions.