Fuglede Kadison determinants for operators in the von Neumann algebra of an equivalence relation

TL;DR

This paper computes the Fuglede-Kadison determinant for certain operators in the von Neumann algebra associated with an ergodic equivalence relation, providing explicit formulas under specific conditions.

Contribution

It introduces formulas for the Fuglede-Kadison determinant of operators generated by partial isometries and multiplication operators in this setting, extending previous understanding.

Findings

Derived formulas for treeable equivalence relations.

Established determinant calculations under specific restrictions on functions and isomorphisms.

Enhanced the analytical tools for operators in von Neumann algebras of equivalence relations.

Abstract

We calculate the Fuglede-Kadison determinant for operators of the form $\sum_{i=1}^n M_{f_i}L_{g_i}$ where $L_{g_i}$ are unitaries or partial isometries coming from Borel (partial) isomorphisms $g_i$ on a probability space which generate an ergodic equivalence relation, and $M_{f_i}$ are multiplication operators. We obtain formulas for the cases when the relation is treeable or the $f_i$'s and $g_i$'s satisfy some restrictions.