Multipliers on dual 𝐴*-algebras

Abstract

Let A be an A ∗ {A^\ast } -algebra which is a dense ∗ ^\ast -ideal of a B ∗ {B^\ast } -algebra A \mathfrak {A} . We use tensor products and the algebra M l ( A ) {M_l}(A) of left multipliers on A to obtain a characterization of duality in A. We show, moreover, that if A is dual then M l ( A ) {M_l}(A) is algebra isomorphic to the second conjugate space A ∗ ∗ {\mathfrak {A}^{\ast \ast }} of A \mathfrak {A} when A ∗ ∗ {\mathfrak {A}^{\ast \ast }} is given Arens product.