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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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
MathSciNet review: 0267402
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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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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