, Volume 79, Issue 2-3, pp 331-365

A grothendieck factorization theorem on 2-convex schatten spaces

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Abstract

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