Multipliers of 𝐴*-algebras

Abstract

Let A be an A ∗ {A^\ast } -algebra of the first kind with C ∗ {C^\ast } -algebra completion A \mathfrak {A} . It is known that if A is dual then A 2 {A^2} is dense in A and the Banach algebras M L ( A ) {M_L}(A) and M L ( A ) {M_L}(\mathfrak {A}) of left multipliers of A and A \mathfrak {A} are algebra isomorphic. In this note it is proved that M L ( A ) {M_L}(A) and M L ( A ) {M_L}(\mathfrak {A}) are topologically algebra isomorphic when A is an arbitrary A ∗ {A^\ast } -algebra of the first kind such that A 2 {A^2} is dense in A. As a consequence, it follows that every left multiplier of a replete Hilbert algebra A is automatically continuous.