Trace-class for an arbitrary $H^{\ast }$-algebra

Let $A$ be a proper ${H^ \ast }$-algebra and let $\tau (A)$ be the set of all products $xy$ of members $x,y$ of $A$. Then $\tau (A)$ is a normed algebra with respect to some norm $\tau (\;)$ which is related to the norm $||\;||$ of $A$ by the equality: $||a|{|^2} = \tau (a^ \ast a),a \in A$. There is a trace tr defined on $\tau (A)$ such that $\operatorname {tr} (a) = \sum \nolimits _\alpha {(a{e_\alpha },{e_\alpha })}$ for each $a \in \tau (A)$ and each maximal family $\{ {e_\alpha }\}$ of mutually orthogonal projections in $A$. The trace is related to the scalar product of $A$ by the equality: $\operatorname {tr} (xy) = (x,{y^ \ast }) = (y,{x^ \ast })$ for all $x,y \in A$.