Invariant Banach limits and applications to noncommutative geometry

Evgenii Semenov, Fedor Sukochev, Alexandr Usachev and Dmitriy Zanin

Vol. 306 (2020), No. 1, 357–373

DOI: 10.2140/pjm.2020.306.357

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

Milestones

Received: 22 March 2018

Revised: 8 June 2019

Accepted: 7 December 2019

Published: 14 June 2020