Trace-class and centralizers of an $H^{\ast }$-algebra

Let $A$ be a proper ${H^ \ast }$-algebra. Let $\tau (A) = \{ xy|x,y \in A\}$, let $R(A)$ be the set of all bounded linear operators $S$ on $A$ such that $S(xy) = (Sx)y$ for all $x,y \in A$ and let $C(A)$ be the closed subspace of $R(A)$ generated by the operators of the form $La:x \to ax,a \in A$. It is shown that $\tau (A)$ can be identified with the space of all bounded linear functionals on $C(A)$ and that $R(A)$ is the dual of $\tau (A)$. Also it is proved that $\tau (A)$ is a Banach algebra.