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 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
Authors
Evgenii Semenov
Voronezh State University
Voronezh
Russia
Fedor Sukochev
School of Mathematics and Statistics
University of New South Wales
Sydney
Australia
Alexandr Usachev
Department of Mathematical Sciences
Chalmers University of Technology
University of Gothenburg
Gothenburg
Sweden
School of Mathematics and Statistics
Central South University
Hunan
China
Dmitriy Zanin
School of Mathematics and Statistics
University of New South Wales
Sydney
Australia