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Trace-class for an arbitrary $ H\sp{\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: 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 ({\kern 1pt} \,)$ which is related to the norm $ \vert\vert\;\vert\vert$ of $ A$ by the equality: $ \vert\vert a\vert{\vert^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

DOI: https://doi.org/10.1090/S0002-9939-1970-0267402-9
Keywords: Trace-class, $ {H^ \ast }$-algebra, Hilbert-Schmidt operator, trace, right centralizer, involution, mutually orthogonal projections
Article copyright: © Copyright 1970 American Mathematical Society

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