We study L-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes in [1], in the presence of a bi-finite correspondence and proof a proportionality formula.

We show that the noncommutative spheres of Connes and Landi are quantum homogeneous spaces for certain compact quantum groups. We give a general construction of homogeneous spaces which support noncommutative spin geometries.

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.

In this paper we give a uni ed description of the archimedean and the totally split degenerate bers of an arithmetic surface, using operator algebras and Connes' theory of spectral triples in noncommutative geometry.

We prove a version of the L2-index Theorem of Atiyah, which uses the universal center-valued trace instead of the standard trace. We construct for G-equivariant K-homology an equivariant Chern character, which is an isomorphism and lives over the ring Z ⊂ ΛG ⊂ Q obtained from the integers by inverting the orders of all finite subgroups of G. We use these two results to show that the Baum-Connes...

The Connes Embedding Problem (CEP) asks whether every separable II1 factor embeds into an ultrapower of the hyperfinite II1 factor. We show that the CEP is equivalent to the statement that every type II1 tracial von Neumann algebra has a computable universal theory.

We extend the BRS and anti-BRS symmetry to the two point space of Connes’ non-commutative model building scheme. The constraint relations are derived and the quantum Lagrangian constructed. We find that the quantum Lagrangian can be written as a functional of the curvature for symmetric gauges with the BRS, anti-BRS auxiliary field finding a geometrical interepretation as the extension of the H...

Let A and B be separable C-algebras, B stable. We show that the Connes-Higson construction gives rise to an isomorphism between the group of unitary equivalence classes of extensions of SA by B, modulo the extensions which are asymptotically split, and the homotopy classes of asymptotic homomorphisms from SA to B.

We construct an associative algebra with a decomposition into the direct sum of the underlying vector spaces of another associative algebra and its dual space such that both of them are subalgebras and the natural symmetric bilinear form is invariant or the natural antisymmetric bilinear form is a Connes 2-cocycle. The former is called a double construction of Frobenius algebra and the latter i...