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

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.