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.

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