Completely 1-complemented subspaces of Schatten spaces

Abstract

We consider the Schatten spaces S p S^p in the framework of operator space theory and for any 1 ≤ p ≠ 2 > ∞ 1\leq p\not =2>\infty , we characterize the completely 1 1 -complemented subspaces of S p S^p . They turn out to be the direct sums of spaces of the form S p ( H , K ) S^p(H,K) , where H , K H,K are Hilbert spaces. This result is related to some previous work of Arazy and Friedman giving a description of all 1 1 -complemented subspaces of S p 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 L^p -spaces. Also we show that for any n ≥ 2 n\geq 2 , there is a triple isomorphism on some Cartan factor of type 4 and of dimension 2 n 2n which is not completely isometric, and we investigate L p L^p -versions of such isomorphisms.