On the best constants in noncommutative Khintchine-type inequalities

Abstract

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 $\frac{1}{\sqrt{2}}$ in the complex Gaussian case and for the sequence of functions $\{e^{i{2}^{n}t}{\}}_{n=1}^{\infty }$ . The second case is Junge's recent Khintchine-type inequality for subspaces of the operator space $R\oplus C$, 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 $\frac{1}{\sqrt{2}}$ . As a consequence, it follows that any subspace of a quotient of $(R\oplus C{)}^{\ast }$ is cb-isomorphic to a subspace of the predual of the hyperfinite factor of type $II{I}_1$, with cb-isomorphism constant $\le \sqrt{2}$ . In particular, the operator Hilbert space $OH$ has this property.

30 pages