Advances in Operator Theory

Complex interpolation and non-commutative integration

Klaus Werner

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Abstract

We show that under suitable conditions interpolation between a Banach space and its dual yields a Hilbert space at $\theta =\frac{1}{2}$. By application of this result to the special case of the non-commutative $L^p$-spaces of Leinert [Int. J. Math. 2 (1991), no. 2, 177-182] and  Terp [J. Operator Theory 8 (1982), 327-360] we conclude that $L^2$ is a Hilbert space and that $L^p$ is isometrically isomorphic to the dual of $L^q$ without using the isomorphisms of these spaces to $L^p$-spaces of Hilsum [J. Funct. Anal. 40 (1981), 151-169.] and Haagerup [Colloq. Internat. CNRS, 274, CNRS, Paris, 1979].\Haagerup and Pisier [Canad. J. Math. 41 (1989), no. 5, 882-906.], Pisier [Mem. Amer. Math. Soc. 122 (1996), no. 585, viii+103 pp] and  Watbled [C. R. Acad. Sci. Paris, t. 321, Série I, p. 1437-1440, 1995] gave conditions under which interpolation between a Banach space and its conjugate dual yields a Hilbert space at $\frac{1}{2}$. The result mentioned above when put in “conjugate form” extends their results.

Article information

Source
Adv. Oper. Theory, Volume 3, Number 1 (2018), 1-16.

Dates
Received: 22 November 2016
Accepted: 27 January 2017
First available in Project Euclid: 5 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aot/1512497949

Digital Object Identifier
doi:10.22034/aot.1611-1061

Mathematical Reviews number (MathSciNet)
MR3730336

Zentralblatt MATH identifier
06804313

Subjects
Primary: 46B70: Interpolation between normed linear spaces [See also 46M35]
Secondary: 46L51: Noncommutative measure and integration 46L10: General theory of von Neumann algebras

Keywords
complex interpolation noncommutative integration von Neumann algebra

Citation

Werner, Klaus. Complex interpolation and non-commutative integration. Adv. Oper. Theory 3 (2018), no. 1, 1--16. doi:10.22034/aot.1611-1061. https://projecteuclid.org/euclid.aot/1512497949


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References

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