Completely 1-complemented subspaces of Schatten spaces

Le Merdy, Christian; Ricard, Éric; Roydor, Jean

doi:10.1090/S0002-9947-08-04594-7

Completely $1$-complemented subspaces of Schatten spaces
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by Christian Le Merdy, Éric Ricard and Jean Roydor PDF

Trans. Amer. Math. Soc. 361 (2009), 849-887 Request permission

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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