Capacity and the quasicentral modulus

TL;DR

This paper explores the quasicentral modulus as a noncommutative analogue of Sobolev condenser capacity, relating it to capacities on Cayley graphs and discussing related capacities.

Contribution

It establishes the connection between the quasicentral modulus and nonlinear capacities on Cayley graphs, expanding the understanding of noncommutative capacities.

Findings

Quasicentral modulus coincides with nonlinear condenser capacity on Cayley graphs.

Discussion of capacities related to the quasicentral modulus.

Provides insights into noncommutative analogues of classical capacities.

Abstract

We point out that the quasicentral modulus is a noncommutative analogue of a nonlinear rearrangement invariant Sobolev condenser capacity. In the case of the shifts by the generators of a finitely generated group, the quasicentral modulus coincides with a corresponding nonlinear condenser capacity on the Cayley graph of the group. Some other capacities related to the quasicentral modulus are also discussed.