A grothendieck factorization theorem on 2-convex schatten spaces

Abstract

We prove that for every bounded linear operatorT:C2p→H(1≤p<∞,H is a Hilbert space,C2pp is the Schatten space) there exists a continuous linear formf onCp such thatf≥0, ‖f‖(CCp)*=1 and $$\forall x \in C^{2p} , \left\| {T(x)} \right\| \leqslant 2\sqrt 2 \left\| T \right\| 1/2$$ . Forp=∞ this non-commutative analogue of Grothendieck’s theorem was first proved by G. Pisier. In the above statement the Schatten spaceC2p can be replaced byEE2 whereE(2) is the 2-convexification of the symmetric sequence spaceE, andf is a continuous linear form onCE. The statement can also be extended toLE{(su2)}(M, τ) whereM is a Von Neumann algebra,τ a trace onM, E a symmetric function space.