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Let Γ be a discrete group. Following Linnell and Schick one can define a continuous ring c(Γ) associated with Γ. They proved that if the Atiyah Conjecture holds for a torsion-free group Γ, then c(Γ) is a skew field. Also, if Γ has torsion and the Strong Atiyah Conjecture holds for Γ, then c(Γ) is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group Γ = ℤ2≀ℤ. It is known that ℂ(ℤ2≀ℤ) does not even have a classical ring of quotients. Our main result is that if H is amenable, then c(ℤ2≀ H) is isomorphic to a continuous ring constructed by John von Neumann in the 1930s. [ABSTRACT FROM AUTHOR]

The analytic von Neumann regular closure R(Γ) of a complex group algebra CG was introduced by Linnell and Schick. This ring is the smallest *-regular subring in the algebra of affiliated operators UΓ containing CΓ. We prove that all the algebraic von Neumann regular closures corresponding to sofic representations of an amenable group are isomorphic to RΓ. This result can be viewed as a structural generalization of Lück's approximation theorem. The main tool of the proof which might be of independent interest is that an amenable group algebra KΓ over any field K can be embedded to the rank completion of an ultramatricial algebra. [ABSTRACT FROM AUTHOR]

Abstract: We introduce the notion of sofic measurable equivalence relations. Using them we prove that Connes'' Embedding Conjecture as well as the Measurable Determinant Conjecture of Lück, Sauer and Wegner hold for treeable equivalence relations. [Copyright &y& Elsevier]

Abstract: Using the spectral theory of weakly convergent sequences of finite graphs, we prove the uniform existence of the integrated density of states for a large class of infinite graphs. [Copyright &y& Elsevier]

Abstract: In [G. Elek, On limits of finite graphs, Combinatorica, in press, URL: http://www.arxiv.org/pdf/math.CO/0505335] we proved that the limit of a weakly convergent sequence of finite graphs can be viewed as a graphing or a continuous field of infinite graphs. Thus one can associate a type -von Neumann algebra to such graph sequences. We show that in this case the integrated density of states exists, that is, the weak limit of the spectra of the graph Laplacians of the finite graphs is the KNS-spectral measure of the graph Laplacian of the limit graphing. Using this limit technique we prove a Cheeger type inequality for finite graphs. [Copyright &y& Elsevier]

Abstract: Let G be a finitely generated group and be normal subgroups such that . Let and be the images of A under the maps induced by the epimorphisms . According to the strong form of the Approximation Conjecture of Lück [W. Lück, -Invariants: Theory and Applications to Geometry and K-theory, Ergeb. Math. Grenzgeb. (3), vol. 44, Springer-Verlag, Berlin, 2002] where denotes the von Neumann dimension. In [J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates, Approximating -invariants and the Atiyah conjecture, Comm. Pure Appl. Math. 56 (7) (2003) 839–873] Dodziuk et al. proved the conjecture for torsion free elementary amenable groups. In this paper we extend their result for all amenable groups, using the quasi-tilings of Ornstein and Weiss [D.S. Ornstein, B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math. 48 (1987) 1–141]. [Copyright &y& Elsevier]

Abstract: We construct an analogue of von Neumann''s affiliated algebras for sofic group algebras over arbitrary fields. Consequently, we settle Kaplansky''s direct finiteness conjecture for sofic groups. [Copyright &y& Elsevier]