Fuglede-Kadison determinants of matrix-valued semicircular elements and capacity estimates

TL;DR

This paper computes the Fuglede-Kadison determinant for matrix-valued semicircular operators using capacity of covariance mappings and improves existing bounds to be dimension-independent.

Contribution

It introduces a method to calculate determinants of matrix-valued semicircular operators and refines capacity bounds to be dimension-independent.

Findings

Explicit formula for Fuglede-Kadison determinants in terms of capacity

Improved lower bound on capacity that is dimension-independent

Enhanced understanding of operator determinants in free probability

Abstract

We calculate the Fuglede-Kadison determinant of arbitrary matrix-valued semicircular operators in terms of the capacity of the corresponding covariance mapping. We also improve a lower bound by Garg, Gurvits, Oliveira, and Widgerson on this capacity, by making it dimension-independent.