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We prove that the approximation conjecture of Luck holds for all amenable groups in the complex group algebra case. This result was previously proved by Dodziuk, Linnell, Mathai, Schick and Yates under the assumption that the group is torsion-free elementary amenable.

In \cite{Elek} 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 $II_1$-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., Comment: submitted

We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing., Comment: 6 pages, to appear in Combinatorica

We introduce the notion of ends for algebras. The definition is analogous to the one in geometric group theory. We establish some relations to growth conditions and cyclic cohomology., Comment: 12 pages

We prove that Connes' Embedding Conjecture holds for the von Neumann algebras of sofic groups, that is sofic groups are hyperlinear. Hence we provide some new examples of hyperlinearity. We also show that the Determinant Conjecture holds for sofic groups as well. We introduce the notion of essentially free actions and amenable actions and study their properties., Comment: A rather annoying typo in the statement of Proposition 6.1 has been corrected

We introduce the notion of almost finite dimensionality of algebras and study its connection with the classical finiteness conditions.

We investigate the amenability of skew filed extensions of the complex numbers. We prove that all skew fields of finite Gelfand-Kirillov transcendence degree are amenable. However there are both amenable and non-amenable skew fields of infinite Gelfand-Kirillov transcendence degree. For example, the free field is non-amenable.

We introduce the notion of amenability for affine algebras. We characterize amenability by Folner-sequences, paradoxicality and the existence of finitely invariant dimension-measures. Then we extend the results of Rowen on ranks, from affine algebras of subexponential growth to amenable affine algebras.

In this note we prove that in the case of finitely generated amenable groups the classical zero divisor conjecture implies the analytic zero divisor conjecture of Linnell.

We investigate an analogue of the L^2-Betti numbers for amenable linear subshifts. The role of the von Neumann dimension shall be played by the topological entropy., Comment: 19 pages

The notion of topological entropy is originally defined for a single action. Later it was extended by Kieffer for arbitrary discrete amenable groups. Recently Friedland defined topological entropy for any discrete group actions amenable or not. Lind, Schmidt and Ward extended Yuzvinskii's addition formula from single action to abelian group actions. A result of Ward and Zhang suggests that such extension might be possible in the amenable case. In this paper we prove that the addition formula can not be extended for groups of finite type with nonzero Euler-characteristcs e.g. for free groups or surface groups., Comment: 7 pages LaTex

The existence of nonconstant harmonic Dirichlet functions on a Cayley graph of a discrete group is equivalent to the nonvanishing of the first L2-cohomology of the given group. It was first proven by Cheeger and Gromov that such functions do not exists on the Cayley-graph of an amenable group. The result was extended using Foster's averaging formula to transitive amenable graphs by Medolla and Soardi. In this paper we extend this result further for amenable graphs which are transitive in the category of quasi-isometries. Such graphs are usually not quasi-isometric to transitive graphs. The extension includes nets of amenable unimodular Lie-groups and geometrically amenable homogeneous spaces., Comment: 7 pages, LaTeX