Completely 1-complemented subspaces of Schatten spaces

TL;DR

This paper characterizes the completely 1-complemented subspaces of Schatten spaces S^p for p not equal to 2, showing they are direct sums of spaces of the form S^p(H,K), and explores related operator space structures and isomorphisms.

Contribution

It provides a complete characterization of completely 1-complemented subspaces of Schatten spaces S^p for p ≠ 2, linking operator space theory with Cartan factors and noncommutative L^p-spaces.

Findings

Completely 1-complemented subspaces are direct sums of S^p(H,K) spaces.

Identifies non-completely isometric triple isomorphisms on Cartan factors of type 4.

Connects operator space structures with Cartan factors and noncommutative L^p-spaces.

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-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.