Trace-class for an arbitrary $H^{\ast }$-algebra
Authors:
Parfeny P. Saworotnow and John C. Friedell
Journal:
Proc. Amer. Math. Soc. 26 (1970), 95-100
MSC:
Primary 46.60
DOI:
https://doi.org/10.1090/S0002-9939-1970-0267402-9
MathSciNet review:
0267402
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Abstract | References | Similar Articles | Additional Information
Abstract: 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 $x,y \in A$.
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Additional Information
Keywords:
Trace-class,
<!– MATH ${H^ \ast }$ –> <IMG WIDTH="33" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${H^ \ast }$">-algebra,
Hilbert-Schmidt operator,
trace,
right centralizer,
involution,
mutually orthogonal projections
Article copyright:
© Copyright 1970
American Mathematical Society