Symmetric functionals on simply generated symmetric spaces

TL;DR

This paper introduces a method to construct symmetric functionals on symmetric spaces over semifinite von Neumann algebras, linking them to shift-invariant functionals on bounded sequences, and extends the Connes trace formula.

Contribution

It establishes a bijection between symmetric functionals on symmetric spaces and shift-invariant functionals, and extends the Connes trace formula to a broader class of operators.

Findings

Bijection between symmetric functionals and shift-invariant functionals.

Extension of the Connes trace formula.

Not all Dixmier traces are included in the bijection.

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.