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We consider the Schatten spaces S^p in the framework of operator space theoryand for any $1\leq p\not=2<\infty$, we characterize the completely1-complemented subspaces of S^p. They turn out to be the direct sums of spacesof the form S^p(H,K), where H,K are Hilbert spaces. This result is related tosome previous work of Arazy-Friedman giving a description of all 1-complementedsubspaces of S^p in terms of the Cartan factors of types 1-4. We use operatorspace structures on these Cartan factors regarded as subspaces of appropriatenoncommutative L^p-spaces. Also we show that for any $n\geq 2$, there is atriple isomorphism on some Cartan factor of type 4 and of dimension 2n which isnot completely isometric, and we investigate L^p-versions of such isomorphisms.

Completely 1-complemented subspaces of Schatten spaces