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The unconditional pointwise convergence of orthogonal series in \(L_ 2\) over a von Neumann algebra. (English) Zbl 0856.46034
Let \(\mathcal M\) be a \(\sigma\)-finite von Neumann algebra with a faithful normal state \(\Phi\) and let \(H= L_2({\mathcal M}, \Phi)\) be the completion of \(\mathcal M\) under the norm \(|x|:= \Phi(x^* x)^{1/2}\). A sequence \((\xi_n)^\infty_{n= 1}\subseteq H\) is said to be almost surely convergent to \(\xi\in H\) if, for every \(\varepsilon\in 0\), there exists a projection \(p\in {\mathcal M}\) such that \(\Phi(p^\perp)< \varepsilon\) and \(|\xi_n- \xi|_p\to 0\) as \(n\to \infty\). Here, \(|\xi|_p:= \inf\{|\sum^\infty_{k= 1} x_k p|_\infty: (x_k)\in S_{\xi, p}\}\), where \(S_{\psi, p}:= \{(x_k)\subseteq {\mathcal M}: \sum^\infty_{k= 1} x_k= \xi\) in \(H\) and \(\sum^\infty_{k= 1} x_k p\) converges in norm in \({\mathcal M}\}\). The following result transfers the classical Tandori theorem to the non-commutative setting.
Theorem: Let \((\xi_n)^\infty_{n= 1}\) be a sequence of pairwise orthogonal elements in \(H\) and \[ \sum^\infty_{k= 0} \Biggl( \sum_{n\in I_k} |\xi_n|^2\log^2(n+ 1)\Biggr)^{1/2}< \infty, \] where \(I_k= \{2^{2^k}+1,\dots, 2^{2^{k+ 1}}\}\). Then, for each permutation \(\pi\) of the set \(\mathbb{N}\) of positive integers, the series \(\sum^\infty_{k= 0} \xi_{\pi(k)}\) is almost surely convergent.

MSC:
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
60F15 Strong limit theorems
42C15 General harmonic expansions, frames
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