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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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