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

Abstract

Orthonormal systems in commutative $L_2$ spaces can be used to classify Banach spaces. When the system is complete and satisfies 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 satisfy certain norm condition. The proof of the main result heavily depends on free probabilistic tools such as contraction principle for $*$-free Haar unitaries, comparision of averages with respect to $*$-free Haar unitaries and $*$-free circular elements and $K$-covexity, type 2 and cotype 2 with respect to $*$-free circular elements.