Multipliers on complemented Banach algebras

Abstract

Abstract.Let A be a semisimple right complemented Banach algebra, LA the left regular representation of A , and M¡(A) the left multiplier algebra of A . In this paper we are concerned with LA and its relationship to A and M/(A). We show that LA is an annihilator algebra and that it is a closed ideal of M¡(A). Moreover, LA and M¡{A) have the same socle. We also show that the left multiplier algebra of a minimal closed ideal of A is topologically algebra isomorphic to L(H), the algebra of bounded linear operators on a Hilbert space H . Conditions are given under which LA is right complemented.

Ideals and structure of operator algebras

“…1) This result is related to theorems of Tomiuk and Alexander (see e.g. [40,1] concerning 'complemented Banach algebras', but the proofs and conclusions are quite different.…”

Structure theory of homologically trivial and annihilator locally C*-algebras

“…Complemented Banach algebras were introduced by B. J. Tomiuk [95] and have been studied by many authors (see, e.g., [58,1,4,2,100,98,3,101,55,96,97]). Left (right) quasi-complemented topological algebras were defined by T. Husain and P. K. Wong [44].…”