Unconditionality with respect to orthonormal systems in noncommutative $L_2$ spaces
- Autores: Hun Hee Lee
- Localización: Indiana university mathematics journal, ISSN 0022-2518, Vol. 56, Nº 6, 2007, págs. 2763-2786
- Idioma: inglés
- DOI: 10.1512/iumj.2007.56.3118
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Resumen
Orthonormal systems in commutative L2 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 L2 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.
