On subspaces spanned by freely independent random variables in noncommutative L p -spaces
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ON SUBSPACES SPANNED BY FREELY INDEPENDENT RANDOM VARIABLES IN NONCOMMUTATIVE Lp -SPACES
BY
Yong Jiao∗
School of Mathematics and Statistics, Central South University Changsha 410075, China e-mail: [email protected]
AND
Fedor Sukochev and Dmitriy Zanin∗∗
School of Mathematics and Statistics, University of New South Wales Sydney, NSW 2052, Australia e-mail: [email protected], [email protected]
∗ Yong Jiao is supported by NSFC(11471337, 11722114). ∗∗ Fedor Sukochev and Dmitriy Zanin are supported by the ARC.
Received January 30, 2019
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Y. JIAO, F. SUKOCHEV AND D. ZANIN
Isr. J. Math.
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 2 (M⊗ ¯ ∞ ). We obtain noncomin a certain symmetric operator space ZE mutative (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 2 , 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 Φ 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 2 .
1. Introduction The main motivation of this paper is to study the geometrical properties of subspaces of noncommutative Lp -spaces spanned by sequences of freely independent random variables. Our objective is twofold: to describe such subspaces in familiar terms of symmetric sequence spaces and to investigate when such subspaces are complemented. In order to better explain our motivation, we now recall some classical results from Banach space theory which have inspired our research. Pelczynski [43] proved that if the sequence (fi ) in Lp (0, 1), 1 ≤ p < ∞, is isometrically equivalent to the usual basis of p , that is, the map T sending the usual basis (ei ) of p to the sequence (fi ) is an isometry, then the functions (fi ) have disjoint supports and consequently there is a norm 1 projection of Lp (0, 1) onto the closed linear span [fi ] of (fi ). Dor’s remarkable result [23, Theorems A and B] gives an isomorphic analogue of this result for the case when the isomorphism constant T · T −1 is small enough. On the other hand, in the case when 1 < p = 2 < ∞, there are uncomp
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