On the bundle convergence of double orthogonal series in noncommutative $L_2$-spaces

Ferenc Móricz; Barthélemy Le Gac

ArticleOriginal scientific text

Title

On the bundle convergence of double orthogonal series in noncommutative $L_{2}$ -spaces

Authors ¹, ²

Affiliations

Bolyai Institute University of Szeged Aradi, vertanuk tere 1, 6720 Szeged, Hungary
Centre de Mathématiques et d'Informatique, Université de Provence, 39 Rue Joliot-Curie, 13453 Marseille, Cedex 13, France

Abstract

The notion of bundle convergence in von Neumann algebras and their

L_{2}

-spaces for single (ordinary) sequences was introduced by Hensz, Jajte, and Paszkiewicz in 1996. Bundle convergence is stronger than almost sure convergence in von Neumann algebras. Our main result is the extension of the two-parameter Rademacher-Men'shov theorem from the classical commutative case to the noncommutative case. To our best knowledge, this is the first attempt to adopt the notion of bundle convergence to multiple series. Our method of proof is different from the classical one, because of the lack of the triangle inequality in a noncommutative von Neumann algebra. In this context, bundle convergence resembles the regular convergence introduced by Hardy in the classical case. The noncommutative counterpart of convergence in Pringsheim's sense remains to be found.

Keywords

ENG

von Neumann algebra, faithful and normal state, completion, Gelfand-Naimark-Segal representation theorem, bundle convergence, almost sure convergence, regular convergence, orthogonal sequence of vectors in

L_{2}

, Rademacher-Men'shov theorem, convergence in Pringsheim's sense

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Title

On the bundle convergence of double orthogonal series in noncommutative L2-spaces

Affiliations

Abstract

Keywords

Bibliography

On the bundle convergence of double orthogonal series in noncommutative $L_{2}$ -spaces