Unconditionality with respect to orthonormal systems in noncommutative $L_2$ spaces Hun Hee Lee 47L2546L53orthonormal systemoperator spaceoperator Hilbert spaceHaar unitarycircular element 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. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.3118 10.1512/iumj.2007.56.3118 en Indiana Univ. Math. J. 56 (2007) 2763 - 2786 state-of-the-art mathematics http://iumj.org/access/