Some remarks on noncommutative Khintchine inequalities

TL;DR

This paper explores how the upper Khintchine inequality for free semi-circular variables extends to noncommutative Banach spaces, providing new proofs and resolving open questions in noncommutative moment inequalities.

Contribution

It demonstrates the extension of Khintchine inequalities to noncommutative Banach spaces and addresses an open problem in noncommutative moment inequalities.

Findings

Extended Khintchine inequalities to noncommutative Banach spaces.

Provided a simple proof for interpolation of row and column operator spaces.

Resolved an open question on noncommutative moment inequalities.

Abstract

Normalized free semi-circular random variables satisfy an upper Khintchine inequality in $L_\infty$. We show that this implies the corresponding upper Khintchine inequality in any noncommutative Banach function space. As applications, we obtain a very simple proof of a well-known interpolation result for row and column operator spaces and, moreover, answer an open question on noncommutative moment inequalities concerning a paper by Bekjan and Chen.