Advances in Operator Theory

Complex interpolation and non-commutative integration

Klaus Werner

Full-text: Access denied (subscription has expired)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

complex interpolation noncommutative integration von Neumann algebra


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

Export citation


  • J. Bergh, On the relation between the two complex methods of interpolation, Indiana Univ. Math. J. 28 (1979), no. 5, 775–778.
  • J. Bergh and J. Löfström, An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.
  • A. P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190.
  • F. Cobos and T. Schonbek, On a theorem by Lions and Peetre about interpolation between a Banach space and its dual, Houston J. Math. 24 (1998), 325–344.
  • M. Cwikel, Complex interpolation spaces, a discrete definition and reiteration, Indiana Univ. Math. J. 27 (1978), no. 6, 1005–1009.
  • U. Haagerup, $L\sp{p}$-spaces associated with an arbitrary von Neumann algebra, Algèbres d'opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977), pp. 175–184, Colloq. Internat. CNRS, 274, CNRS, Paris, 1979.
  • U. Haagerup and G. Pisier, Factorization of analytic functions with values in noncommutative $L_1$-spaces and applications, Canad. J. Math. 41 (1989), no. 5, 882–906.
  • M. Hilsum, Les espaces $\Lp$ d'une algèbre de von Neumann définis par la dérivée spatiale, J. Funct. Anal. 40 (1981), 151–169.
  • M. Leinert, Integration with respect to a weight, Int. J. Math. 2 (1991), no. 2, 177–182.
  • G. Pisier, The operator Hilbert space ${\rm OH}$, complex interpolation and tensor norms, Mem. Amer. Math. Soc. 122 (1996), no. 585, viii+103 pp.
  • M. Terp, Interpolation spaces between a von Neumann-algebra and its predual, J. Operator Theory 8 (1982), 327–360.
  • F. Watbled, Interpolation complexe d'un espace de Banach et de son antidual, C. R. Acad. Sci. Paris, t. 321, Série I, p. 1437–1440, 1995.
  • F. Watbled, Complex interpolation of a Banach space with its dual, Math. Scand. 87 (2000), no. 2, 200–210.