An analogue of the thom isomorphism for crossed products of a C1 algebra by an action of R

In this paper we show the existence and uniqueness of a natural isomorphism ø^j_α of K_j(A) with K_j+1(A ?_α$\text{R}$), j ? $\text{Z}$/2 where (A, $\text{R}$, α) is a $\text{C1}$ dynamical $\text{R}$-system, K is the functor of topological K theory and A ?_α$\text{R}$ is the crossed product of A by the action of $\text{R}$. The Pimsner-Voiculescu exact sequence is obtained as a corollary. We show that given an α-invariant trace τ on A, with dual trace $\text{}\text{?}$gt, one has $\text{}\text{?}\text{gtø}{}^1{}_{\mathrm{\alpha }}\text{[u] = (}\text{1}\text{2}\text{iπ) τ(δ(u)u1)}$ for any unitary u in the domain of the derivation δ of A associated to the action α. Finally, we show that the crossed product of C(S³) (continuous functions on the 3 sphere) by a minimal diffeomorphism is a simple $\text{C1}$ algebra with no nontrivial idempotent.     

Mesure de Hausdorff des complementaires desU ε de zygmund

A. Zygmund originally introduced the notion of sets of unicityU _ε, and showed that they differ from classicalU-sets in that they can have positive measure. He then asked if they could be of full measure. J. P. Kahame and Y. Katznelson proved recently that there wereU _ε of full measure. The object of this paper is to show that, in terms of Hausdorff measure, one cannot go beyond that result, for a general sequence ε. In the case of a given sequence ε, and a given Hausdorff determining functionh, it gives a simple test for determining the existence ofU _ε with a complement of zero Hausdorff measure. In this paper the proof of the main known results concerning the measure ofU _ε sets is also reproduced.      

On hyperfinite factors of type III0 and Krieger's factors

A factor M is the cross product of an abelian von Neumann algebra by a single automorphism iff there exists an increasing sequence of normal conditional expectations of M onto finite-dimensional subalgebras N_k with (U N_k)^? = M. Assuming the uniqueness of the hyperfinite factor of type II_∞, we prove then that any hyperfinite factor of type III₀ is the cross product of an abelian von Neumann algebra by a single automorphism.     

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝ ⁿ . They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features. The new examples include the instanton algebra and the NC-4-spheres S ⁴ _θ. We construct the noncommutative algebras ?=C ^∞ (S ⁴ _θ) of functions on NC-spheres as solutions to the vanishing, ch _j (e) = 0, j < 2, of the Chern character in the cyclic homology of ? of an idempotent e∈M ₄ (?), e ²=e, e=e ^*. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmannian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere S ³ _θ distinct from quantum group deformations SU _q (2) of SU (2). We then construct the noncommutative geometry of S _θ ⁴ as given by a spectral triple ?, ℋ, D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric g _μν on S ⁴ whose volume form is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation,

Dynamical entropy ofC* algebras and von Neumann algebras

The definition of the dynamical entropy is extended for automorphism groups ofC* algebras. As an example, the dynamical entropy of the shift of a lattice algebra is studied, and it is shown that in some cases it coincides with the entropy density.     

Quantum gravity boundary terms from the spectral action of noncommutative space

We study the boundary terms of the spectral action of the noncommutative space, defined by the spectral triple dictated by the physical spectrum of the standard model, unifying gravity with all other fundamental interactions. We prove that the spectral action predicts uniquely the gravitational boundary term required for consistency of quantum gravity with the correct sign and coefficient. This is a remarkable result given the lack of freedom in the spectral action to tune this term.     

Trace formula in noncommutative geometry and the zeros of the Riemann zeta function

We give a spectral interpretation of the critical zeros of the Riemann zeta function as an absorption spectrum, while eventual noncritical zeros appear as resonances. We give a geometric interpretation of the explicit formulas of number theory as a trace formula on the noncommutative space of Adele classes. This reduces the Riemann hypothesis to the validity of the trace formula and eliminates the parameter of our previous approach.