@article {
author = {Werner, Klaus},
title = {Complex interpolation and non-commutative integration},
journal = {Advances in Operator Theory},
volume = {3},
number = {1},
pages = {1-16},
year = {2018},
publisher = {Tusi Mathematical Research Group (TMRG)},
issn = {2538-225X},
eissn = {2538-225X},
doi = {10.22034/aot.1611-1061},
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. \textbf{2} (1991), no. 2, 177--182] andÂ Terp [J. Operator Theory \textbf{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. \textbf{40} (1981), 151--169.] and Haagerup [Colloq. Internat. CNRS, 274, CNRS, Paris, 1979].\\Haagerup and Pisier [Canad. J. Math. \textbf{41} (1989), no. 5, 882--906.], Pisier [Mem. Amer. Math. Soc. \textbf{122} (1996), no. 585, viii+103 pp] andÂ Watbled [C. R. Acad. Sci. Paris, t. 321, S\'erie 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.},
keywords = {Hilbert space,Interpolation,Banach space},
url = {http://www.aot-math.org/article_42356.html},
eprint = {http://www.aot-math.org/article_42356_71777b147346763bfff24fc7d39d965f.pdf}
}