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Fuglede-Kadison determinants for operators in the von Neumann algebra of an equivalence relation. (English) Zbl 1282.47061
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 [J. Feldman and 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.
Reviewer: Guoxing Ji (Xian)

##### MSC:
 47C15 Linear operators in $$C^*$$- or von Neumann algebras 47A35 Ergodic theory of linear operators 47B47 Commutators, derivations, elementary operators, etc.
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