The $p$-classes of a Hilbert module
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- by James F. Smith PDF
- Proc. Amer. Math. Soc. 36 (1972), 428-434 Request permission
Abstract:
Let H be a right Hilbert module over a proper ${H^ \ast }$-algebra A. For $0 < p \leqq \infty$, an extended-real value ${\left \| f \right \|_p}$ is associated with each $f \in H$, and the p-class ${H_p}$ is defined to be $\{ f \in H:{\left \| f \right \|_p} < \infty \}$. For $1 \leqq p \leqq \infty ,({H_p},{\left \| \cdot \right \|_p})$ is a right normed A-module. If $1 \leqq p \leqq 2$, there is a conjugate-linear isometry of $({H_p},{\left \| \cdot \right \|_p})$ onto the dual of $({H_q},{\left \| \cdot \right \|_q})$, where $(1/p) + (1/q) = 1$; hence ${H_p}$ is complete in its norm.References
- George R. Giellis, Trace-class for a Hilbert module, Proc. Amer. Math. Soc. 29 (1971), 63–68. MR 276783, DOI 10.1090/S0002-9939-1971-0276783-2
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
- Charles A. McCarthy, $c_{p}$, Israel J. Math. 5 (1967), 249–271. MR 225140, DOI 10.1007/BF02771613
- Parfeny P. Saworotnow, A generalized Hilbert space, Duke Math. J. 35 (1968), 191–197. MR 227749
- Parfeny P. Saworotnow, Trace-class and centralizers of an $H^{\ast }$-algebra, Proc. Amer. Math. Soc. 26 (1970), 101–104. MR 267403, DOI 10.1090/S0002-9939-1970-0267403-0
- Parfeny P. Saworotnow and John C. Friedell, Trace-class for an arbitrary $H^{\ast }$-algebra, Proc. Amer. Math. Soc. 26 (1970), 95–100. MR 267402, DOI 10.1090/S0002-9939-1970-0267402-9
- James F. Smith, The $p$-classes of an $H^{\ast }$-algebra, Pacific J. Math. 42 (1972), 777–793. MR 322517
Additional Information
- © Copyright 1972 American Mathematical Society
- 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