A remark about supramenability and the Macaev norm

TL;DR

This paper establishes a link between a specific condition involving the Macaev norm and supramenability in finitely generated groups, introducing new insights into operator perturbation and group properties.

Contribution

It introduces a novel condition related to the Macaev norm that characterizes supramenability in finitely generated groups and explores its implications for operator theory.

Findings

Condition with Macaev norm implies supramenability in groups

Existence of quasicentral approximate units is not preserved under tensor products

The condition relates to Yamasaki parabolicity and operator perturbation

Abstract

We show that a finitely generated group G which satisfies a certain condition with respect to the Macaev norm is supramenable. The condition is equivalent to the existence of quasicentral approximate unit with respect to the Macaev norm relative to the left regular representation of the group and has been studied by the author in connection with perturbation questions for Hilbert space operators. The condition can be also viewed as an analogue with respect to the Macaev norm of Yamasaki parabolicity. We also show that existence of quasicentral approximate units relative to the Macaev norm for n-tuples of operators is not preserved when taking tensor products.