The Relationship between Two Involutive Semigroups S and ST Is Defined by a Left Multiplier T

Abstract

Let S be a semigroup with a left multiplier T on S. There exists a new semigroup ST, which depends on S and T, which has the same underlying space as S. We study the question of involutions on ST and a Banach algebra AT. We find a condition of T under which ST and the second dual AT**** admit an involution. We will show that AT is C*-algebra if and only if T:AT→A is an isometry, under mild conditions. Also, A is C*-algebra if and only if so is AT, under other minor conditions.