Vol. 306, No. 1, 2020

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Invariant Banach limits and applications to noncommutative geometry

Evgenii Semenov, Fedor Sukochev, Alexandr Usachev and Dmitriy Zanin

Vol. 306 (2020), No. 1, 357–373
Abstract

A linear functional $B$ on the space of bounded sequences ${l}_{\infty }$ is called a Banach limit if it is positive, normalised and invariant under the shift operator. There are Banach limits which possess additional invariance properties. We prove that every Banach limit invariant under the Cesaro operator is also invariant under all dilation operators. We also prove the “continuous version” of this result and apply it to the theory of singular traces.

Keywords
Banach limit, space of bounded sequences, Cesaro operator, dilation operator
Mathematical Subject Classification 2010
Primary: 46B45, 58B34
Secondary: 47B37