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The $ p$-classes of a Hilbert module


Author: James F. Smith
Journal: Proc. Amer. Math. Soc. 36 (1972), 428-434
MSC: Primary 46K15
DOI: https://doi.org/10.1090/S0002-9939-1972-0310655-0
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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  • [1] G. R. Giellis, Trace-class for a Hilbert module, Proc. Amer. Math. Soc. 29 (1971), 63-68. MR 43 #2523. MR 0276783 (43:2523)
  • [2] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups analysis on locally compact Abelian groups, Die Grundlehren der math. Wissenschaften, Band 152, Springer-Verlag, New York and Berlin, 1970. MR 41 #7378. MR 0262773 (41:7378)
  • [3] C. A. McCarthy, $ {c_p}$, Israel J. Math. 5 (1967), 249-271. MR 37 #735. MR 0225140 (37:735)
  • [4] P. P. Saworotnow, A generalized Hilbert space, Duke Math. J. 35 (1968), 191-197. MR 37 #3333. MR 0227749 (37:3333)
  • [5] -, Trace-class and centralizers of an $ {H^ \ast }$-algebra, Proc. Amer. Math. Soc. 26 (1970), 101-104. MR 42 #2305. MR 0267403 (42:2305)
  • [6] P. P. Saworotnow and J. C. Friedell, Trace-class for an arbitrary $ {H^\ast}$-algebra, Proc. Amer. Math. Soc. 26 (1970), 95-100. MR 42 #2304. MR 0267402 (42:2304)
  • [7] J. F. Smith, The p-classes of an $ {H^\ast}$-algebra, Pacific J. Math. (to appear). MR 0322517 (48:879)

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Additional Information

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

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