Multipliers on dual $A^*$-algebras

Abstract:

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