We introduce a notion of rank completion for bi-modules over a finite tracial von Neumann algebra. We show that the functor of rank completion is exact and that the category of complete modules is abelian with enough projective objects. This leads to interesting computations in the L 2-homology for tracial algebras. As an application, we also give a new proof of a Theorem of Gaboriau on invaria...

We introduce and study the framework of compact metric structures and their associated notions of isomorphisms such as homeomorphic and bi-Lipschitz isomorphism. This is subsequently applied to model various classification problems in analysis such as isomorphism of C∗-algebras and affine homeomorphism of Choquet simplices, where among other things we provide a simple proof of the completeness ...

We investigate the bi-Hamiltonian structures associated with constrained dispersionless modified KP hierarchies which are constructed from truncations of the Lax operator of the dispersionless modified KP hierarchy. After transforming their second Hamiltonian structures to those of Gelfand-Dickey type, we obtain the Poisson algebras of the coefficient functions of the truncated Lax operators. T...

We solve the Jacobi identity of metric 3-Lie algebras for which an associated Lie algebra either does not admit bi-invariant 4-forms or it is the sum of up to three abelian directions and a semi-simple Lie algebra. In all cases, we find that the structure constants of the metric 3-Lie algebras are sums of orthogonal simple 4-forms verifying a conjecture in math/0211170. In particular, there is ...

The classical Cuntz semigroup has an important role in the study of C*-algebras, being one of the main invariants used to classify recalcitrant C*-algebras up to isomorphism. We consider C*-algebras that have Hopf algebra structure, and find additional structure in their Cuntz semigroups. We show that in many cases, isomorphisms of Cuntz semigroups that respect this additional structure can be ...

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.