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Completely $ 1$-complemented subspaces of Schatten spaces

Author(s): Christian Le Merdy; Éric Ricard; Jean Roydor
Journal: Trans. Amer. Math. Soc. 361 (2009), 849-887.
MSC (2000): Primary 46L07, 46L89, 17C65
Posted: August 14, 2008
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Abstract: We consider the Schatten spaces $ S^p$ in the framework of operator space theory and for any $ 1\leq p\not=2<\infty$, we characterize the completely $ 1$-complemented subspaces of $ S^p$. They turn out to be the direct sums of spaces of the form $ S^p(H,K)$, where $ H,K$ are Hilbert spaces. This result is related to some previous work of Arazy and Friedman giving a description of all $ 1$-complemented subspaces of $ S^p$ in terms of the Cartan factors of types 1-4. We use operator space structures on these Cartan factors regarded as subspaces of appropriate noncommutative $ L^p$-spaces. Also we show that for any $ n\geq 2$, there is a triple isomorphism on some Cartan factor of type 4 and of dimension $ 2n$ which is not completely isometric, and we investigate $ L^p$-versions of such isomorphisms.

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Additional Information:

Christian Le Merdy
Affiliation: Laboratoire de Mathématiques, Université de Franche-Comté, 25030 Besançon Cedex, France
Email: clemerdy@univ-fcomte.fr

Éric Ricard
Affiliation: Laboratoire de Mathématiques, Université de Franche-Comté, 25030 Besançon Cedex, France
Email: eric.ricard@univ-fcomte.fr

Jean Roydor
Affiliation: Laboratoire de Mathématiques, Université de Franche-Comté, 25030 Besançon Cedex, France
Email: jean.roydor@univ-fcomte.fr

DOI: 10.1090/S0002-9947-08-04594-7
PII: S 0002-9947(08)04594-7
Received by editor(s): March 27, 2007
Posted: August 14, 2008
Copyright of article: Copyright 2008, American Mathematical Society

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