Multipliers of $A^{\ast }$-algebras

Abstract:

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