Symmetric functionals on simply generated symmetric spaces

2024/04/22 by Galina Levitina, Levitina, Galina, Alexandr Usachev +1
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2404.13870

openalex publication_date 2024/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the present paper we suggest a construction of symmetric functionals on a large class of symmetric spaces over a semifinite von Neumann algebra. This approach establishes a bijection between the symmetric functionals on symmetric spaces and shift-invariant functionals on the space of bounded sequences. It allows to obtain a bijection between the classes of all continuous symmetric functionals on different symmetric spaces. Notably, we show that this mapping is not bijective on the class of all Dixmier traces. As an application of our results we prove an extension of the Connes trace formula for a wide class of operators and symmetric functionals.

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