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On subspaces spanned by freely independent random variables in noncommutative Lp-spaces

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Abstract

This paper deals with new phenomena appearing in the structure of Banach spaces arising in noncommutative integration theory. Let E be a fully symmetric Banach function space on [0, 1] and M be a finite von Neumann algebra. Let [xk] be the closed subspace spanned by a sequence (xk) of freely independent mean zero random variables from E(ℳ). The subspace [xk] is complemented in E(ℳ) if and only if the closed subspace spanned by the pairwise orthogonal sequence (xk ⊗ ek) is complemented in a certain symmetric operator space \(Z_E^2\left({{\cal M}\overline \otimes {\ell_\infty}} \right)\). We obtain noncommutative (free) analogues of classical results of Dor and Starbird as well as those of Kadec and Pelczynski. We show that [xk] is complemented in L1(ℳ)provided (xk) is equivalent in L1(ℳ) to the standard basis of ℓ2, while this never happens in the classical case. We prove that a sequence of freely independent copies of a mean zero random variable x in Lp(ℳ), 1 ≤ p ≤ 2, is equivalent to the standard basis in some Orlicz sequence space ℳΦ and give a precise description of the connection between the Orlicz function Φ and the distribution of the given random variable x. Finally, we prove that [xk] spanned by a sequence of freely independent copies of a mean zero random variable is complemented in E(ℳ) if and only if (xk) is equivalent in E(ℳ) to the standard basis of ℓ2.

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Acknowledgements

The authors thank Sergei Astashkin for helpful discussions. Yong Jiao is supported by NSFC(11471337, 11722114); Fedor Sukochev and Dmitriy Zanin are supported by the Australian Research Council.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410075, China

    Yong Jiao

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Fedor Sukochev & Dmitriy Zanin

Authors
  1. Yong Jiao
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  2. Fedor Sukochev
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  3. Dmitriy Zanin
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Correspondence to Dmitriy Zanin.

Additional information

Yong Jiao is supported by NSFC(11471337, 11722114).

Fedor Sukochev and Dmitriy Zanin are supported by the ARC.

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Jiao, Y., Sukochev, F. & Zanin, D. On subspaces spanned by freely independent random variables in noncommutative Lp-spaces. Isr. J. Math. 238, 431–477 (2020). https://doi.org/10.1007/s11856-020-2047-3

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  • DOI: https://doi.org/10.1007/s11856-020-2047-3

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