On the best constants in noncommutative Khintchine-type inequalities

We obtain new proofs with improved constants of the Khintchine-type inequality with matrix coefficients in two cases. The first case is the Pisier and Lust-Piquard noncommutative Khintchine inequality for p = 1, where we obtain the sharp lower bound of 1/root 2 in the complex Gaussian case and for the sequence of functions {e(i2n t)}(n=1)(infinity). The second case is Junge's recent Khintchine-type inequality for subspaces of the operator space R circle plus C, which he used to construct a cb-embedding of the operator Hilbert space OH into the predual of a hyperfinite factor. Also in this case, we obtain a sharp lower bound of 1/root 2. As a consequence, it follows that any subspace of a quotient of (R circle plus C)* is cb-isomorphic to a subspace of the predual of the hyperfinite factor of type III1, with cb-isomorphism constant <= root 2. In particular, the operator Hilbert space OH has this property. (C) 2007 Elsevier Inc. All rights reserved.