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 √ 2 in the complex Gaussian case and for the sequence of functions {en}n=1 . The second case is Junge’s recent Khintchine-type inequality for subspaces of the operator space R ⊕ 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 √ 2 . As a consequence, it follows that any subspace of a quotient of (R⊕C)∗ is cb-isomorphic to a subspace of the predual of the hyperfinite factor of type III1 , with cb-isomorphism constant ≤ √ 2 . In particular, the operator Hilbert space OH has this property.