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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The trace-class of a full Hilbert algebra


Author: Michael R. W. Kervin
Journal: Trans. Amer. Math. Soc. 178 (1973), 259-270
MSC: Primary 46K15
DOI: https://doi.org/10.1090/S0002-9947-1973-0318900-8
MathSciNet review: 0318900
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Abstract: The trace-class of a full Hilbert algebra A is the set $ \tau (A) = \{ xy\vert x \in A,y \in A\} $. This set is shown to be a $ \ast $-ideal of A, and possesses a norm $ \tau $ defined in terms of a positive hermitian linear functional on $ \tau (A)$. The norm $ \tau $ is in general both incomplete and not an algebra norm, and is also not comparable with the Hilbert space norm $ \left\Vert\right\Vert$ on $ \tau (A)$. However, a one-sided ideal of $ \tau (A)$ is closed with respect to one norm if and only if it is closed with respect to the other. The topological dual of $ \tau (A)$ with respect to the norm $ \tau $ is isometrically isomorphic to the set of left centralizers on A.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0318900-8
Keywords: Full Hilbert algebra, projection base, left (right) centralizer, trace-class, orthogonal complement
Article copyright: © Copyright 1973 American Mathematical Society