Title: Unconditionality with respect to orthonormal systems in noncommutative $L_2$ spaces

Authors: Hun Hee Lee

Issue: Volume 56 (2007), Issue 6, 2763-2786

Abstract: Orthonormal systems in commutative $L_2$ spaces can be used to classify Banach spaces. When the system is complete and satisfies a certain norm condition, the unconditionality with respect to the system characterizes Hilbert spaces. As a noncommutative analogue, we introduce the notion of unconditionality of operator spaces with respect to orthonormal systems in noncommutative $L_2$ spaces and show that the unconditionality characterizes operator Hilbert spaces when the system is complete and satisfies a certain norm condition. The proof of the main result heavily depends on free probabilistic tools such as the contraction principle for $*$-free Haar unitaries; comparison of averages with respect to $*$-free Haar unitaries and $*$-free circular elements; and $K$-convexity, type 2 and cotype 2 with respect to $*$-free circular elements.