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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The $ p$-classes of a Hilbert module


Author: James F. Smith
Journal: Proc. Amer. Math. Soc. 36 (1972), 428-434
MSC: Primary 46K15
MathSciNet review: 0310655
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Abstract: Let H be a right Hilbert module over a proper $ {H^ \ast }$-algebra A. For $ 0 < p \leqq \infty $, an extended-real value $ {\left\Vert f \right\Vert _p}$ is associated with each $ f \in H$, and the p-class $ {H_p}$ is defined to be $ \{ f \in H:{\left\Vert f \right\Vert _p} < \infty \} $. For $ 1 \leqq p \leqq \infty ,({H_p},{\left\Vert \cdot \right\Vert _p})$ is a right normed A-module. If $ 1 \leqq p \leqq 2$, there is a conjugate-linear isometry of $ ({H_p},{\left\Vert \cdot \right\Vert _p})$ onto the dual of $ ({H_q},{\left\Vert \cdot \right\Vert _q})$, where $ (1/p) + (1/q) = 1$; hence $ {H_p}$ is complete in its norm.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0310655-0
PII: S 0002-9939(1972)0310655-0
Keywords: Hubert module, $ {H^ \ast }$//"-algebra, trace class, p-classes, normed module
Article copyright: © Copyright 1972 American Mathematical Society