In 1965, Thompson defined the groups F, T , and V [8]. Thompson’s group V is the group of right-continuous bijections v of [0, 1] that map dyadic rational numbers to dyadic rational numbers, that are differentiable except at finitely many dyadic rational numbers, and such that, on each interval on which v is differentiable, v is affine with derivative a power of 2. The group F is the subgroup o...

We study the projective linear group PGL2(A) associated with an arbitrary algebra A, and its subgroups from the point of view of their action on the space of involutions in A. This action formally resembles Möbius transformations known from complex geometry. By specifying A to be an algebra of bounded operators in a Hilbert space H, we rediscover the Möbius group μev(M) defined by Connes and st...

We study the projective linear group PGL 2 (A) associated with an arbitrary algebra A and its subgroups from the point of view of their action on the space of involutions in A. This action formally resembles M obius transformations known from complex geometry. By specifying A to be an algebra of bounded operators in a Hilbert space H, we rediscover the Mobius group ev (M) de ned by Connes and...

Let (A,H, D) be a spectral triple, namely: A is a C∗-algebra, H is a Hilbert space on which A acts and D is a selfadjoint operator with compact resolvent such that the set of elements of A having a bounded commutator with D is dense. A spectral metric space, the noncommutative analog of a complete metric space, is a spectral triple (A,H, D) with additional properties which guaranty that the Con...

We investigate the higher-dimensional amenability of tensor products A ⊗̂ B of Banach algebras A and B. We prove that the weak bidimension dbw of the tensor product A⊗̂B of Banach algebras A and B with bounded approximate identities satisfies dbwA ⊗̂ B = dbwA+ dbwB. We show that it cannot be extended to arbitrary Banach algebras. For example, for a biflat Banach algebra A which has a left or right...

Motivated by the search for new examples of “noncommutative manifolds”, we study the noncommutative geometry of the group C*-algebra of the three dimensional discrete Heisenberg group. We present a unified treatment of the K-homology, cyclic cohomology and derivations of this algebra. 1. The discrete Heisenberg group Recently, there has been considerable interest in the notion of a “noncommutat...

We show that Tsirelson’s problem concerning the set of quantum correlations and Connes’ embedding problem on finite approximations in von Neumann algebras (known to be equivalent to Kirchberg’s QWEP conjecture) are essentially equivalent. Specifically, Tsirelson’s problem asks whether the set of bipartite quantum correlations generated between tensor product separated systems is the same as the...

We give a superconnection proof of Connes’ index theorem for proper cocompact actions of étale groupoids. This includes Connes’ general foliation index theorem for foliations with Hausdor¤ holonomy groupoid.