Johnson-Schechtman and Khintchine inequalities in noncommutative probability theory (Q2822141)

The authors prove a number of deep results on independent random variables in the setting of noncommutative probability spaces, both for symmetric operator spaces and for modulars. In particular, they prove noncommmutative versions of the disjointification inequalities of \textit{W. B. Johnson} and \textit{G. Schechtman} [Ann. Probab. 17, No. 2, 789--808 (1989; Zbl 0674.60051)] and of Khinchine inequalities. The latter ones are easy enough to state:NEWLINENEWLINELet \(E=E(0,1)\) be a symmetric Banach function space and let \((\mathcal{M},\tau)\) be a noncommutative probability space (that is a von Neumann algebra with a faithful tracial state). Let \(x_k\in E(\mathcal{M})\), \(k\geq0\), be mean zero independent random variables. If \(E\) is an interpolation space between \(L_p\) and \(L_q\), \(1<p\leq q<\infty\), then NEWLINE\[NEWLINE \left\|\sum_{k\geq0}x_k\right\|_{E(\mathcal{M})} \approx_E\left\|\left(\sum_{k\geq0}x_k^2\right)^{1/2}\right\|_{E(\mathcal{M})} NEWLINE\]NEWLINE (here, \(\approx_E\) asserts an existence of a positive constant dependent only on \(E\) such that each side of the inequality is bounded by the other side multiplied by the constant). The theorem above significantly extends and strengthens a number of earlier results on noncommutative Khinchine inequalities.