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Journal of Operator Theory

Volume 65, Issue 1, Winter 2011  pp. 3-15.

Riesz summability of orthogonal series in noncommutative $L_2$-spaces

Authors:  Barthelemy Le Gac (1) and Ferenc Moricz (2)
Author institution: (1) 5 Passage de L'oratoire, 84000 Avignon, France
(2) University of Szeged, Bolyai Institute, Aradi\break v\'ertan\'uk tere 1, 6720 Szeged, Hungary

Summary:  A Riesz summability method is defined by means of a sequence $0=\lambda_0 < \lambda_1<\cdots <\lambda_n \to \infty$ of real numbers. The following theorem is known in commutative $L_2$-spaces: If a sequence $\{\xi_n : n=0,1,\ldots\}$ of pairwise orthogonal functions in some $L_2 = L_2 (X, \mathcal{F}, \mu)$ over a positive measure space is such that \begin{equation*} \sum_{n: \lambda_n\geqslant 4} (\log \log \lambda_n)^2 \|\xi_n\|^2<\infty, \end{equation*} then the series $\sum \xi_n$ is Riesz summable almost everywhere to its sum in the norm of $L_2$. In this paper, we extend this theorem to noncommutative $L_2(\mathfrak{A} , \phi)$ spaces, where $\mathfrak{A}$ is a von Neumann algebra, $\phi$ is a faithful, normal state acting on $\mathfrak{A}$, and bundle convergence plays the role of almost everywhere convergence. An interesting corollary of our Theorem \ref{thm1} reads as follows: For any sequence $\{A_n : n=0,1, \ldots\}$ of pairwise orthogonal operators in a von Neumann algebra $\mathfrak{A}$ with a faithful, normal state $\phi$ acting on $\mathfrak{A}$ for which $\sum\phi (|A_n|^2)< \infty$, there exists a Riesz method of summability such that the series $\sum \pi(A_n) \omega$ is summable in the sense of bundle convergence, where $\pi$ is a one-to-one $*$-homomorphism of $\mathfrak{A}$ into the algebra of all bounded linear operators on $L_2$ and $\omega$ is a cyclic, separating vector in $L_2$ according to the Gelfand--Naimark--Segal representation theorem.

Keywords:  von Neumann algebra $\mathfrak{A}$, faithful and normal state $\phi$, completion $L_2 = L_2(\mathfrak{A} , \phi)$, Gelfand--Naimark--Segal representation theorem, bundle convergence, orthogonal series, Riesz summability

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