A Inequality for Non-Microstates Free Entropy Dimension for Crossed Products by Finite Abelian Groups

TL;DR

This paper establishes an inequality relating non-microstates free entropy dimensions in subfactor pairs involving finite abelian groups, with applications to crossed product algebras and bounds on their free entropy dimensions.

Contribution

It introduces a new inequality for free entropy dimension in the context of crossed products by finite abelian groups, extending understanding of their structure.

Findings

Proves an inequality similar to Schreier's formula for free entropy dimensions.

Provides bounds on free entropy dimension for certain crossed product algebras.

Applies results to algebras involving infinite direct sums of cyclic groups.

Abstract

For certain generating sets of the subfactor pair $M\subset M\rtimes G$ where $G$ is a finite abelian group we prove an approximate inequality between their non-microstates free entropy dimension, resembling the Shreier formula for ranks of finite index subgroups of free groups. As an application, we give bounds on free entropy dimension of generating sets of crossed products of the form $M\rtimes(\mathbb{Z}/2\mathbb{Z})^{\oplus\infty}$ for a large class of algebras $M$.