Fuglede-Kadison determinants for operators in the von Neumann algebra of an equivalence relation (Q2862180)

Let \((X,\mathcal B,\mu)\) be a Borel standard probability space without atoms, \(\{A_i\}_{i\in I}\) and \(\{B_i\}_{i\in I}\) be two families of measurable subsets of \(X\), and \(\Lambda=\{g_i:A_i\to B_i\mid i\in I\}\) be a family of measure preserving bijections, where the index set \(I\) is at most countable. Let \(\mathcal R_{\Lambda}\) be the equivalence relation generated by the \(g_i\); i.e., \((x,y)\in\mathcal R_{\Lambda}\) if and only if \(x=y\) or there exists a map \(\omega=g_{i_1}^{\epsilon_1}g_{i_2}^{\epsilon_2}\dots g_{i_k}^{\epsilon_k}\) such that the domain of \(\omega\) contains \(x\) and \(\omega x=y\), where all exponents \(\epsilon_i=\pm 1\). By the Feldman-Moore construction [\textit{J. Feldman} and \textit{C. C. Moore}, Trans. Am. Math. Soc. 234, No. 2, 289--324 (1977; Zbl 0369.22009); ibid., 325--359 (1977; Zbl 0369.22010)], it is known that the von Neumann algebra \(\mathcal M(\mathcal R_{\Lambda})\) generated by \(L_{g}\)(\(g\in\Lambda)\)) and \(M_f\)(\(f\in L^{\infty}(X)\)) on \(L^2(\mathcal R_{\Lambda})\) is a \(II_1\)-factor if \(\mathcal R_{\Lambda}\) is ergodic and (SP1). The authors in this paper calculate the Fuglede-Kadison determinant for operators of the form \(\sum_{i=1}^n M_{f_i}L_{g_i}\) under some restrictions.