Transversals, duality, and irrational rotation

Abstract

An early result of Noncommutative Geometry was Connes' observation in the 1980's that the Dirac-Dolbeault cycle for the 2-torus T 2 , which induces a Poincaré self-duality for T 2 , can be 'quantized' to give a spectral triple and a K-homology class in KK 0 (A θ ⊗ A θ , C) providing the co-unit for a Poincaré self-duality for the irrational rotation algebra A θ for any θ ∈ R ∖ Q. Connes' proof, however, relied on a K-theory computation and does not supply a representative cycle for the unit of this duality. Since such representatives are vital in applications of duality, we supply such a cycle in unbounded form in this article. Our approach is to construct, for any non-zero integer b, a finitely generated projective module L b over A θ ⊗ A θ by using a reduction-toa-transversal argument of Muhly, Renault, and Williams, applied to a pair of Kronecker foliations along lines of slope θ and θ + b, using the fact that these flows are transverse to each other. We then compute Connes' dual of [L b ] and prove that we obtain an invertible τ b ∈ KK 0 (A θ , A θ), represented by an equivariant bundle of Dirac-Schrödinger operators. An application of equivariant Bott Periodicity gives a form of higher index theorem describing functoriality of such 'b-twists' and this allows us to describe the unit of Connes' duality in terms of a combination of two constructions in KK-theory. This results in an explicit spectral representative of the unit-a kind of 'quantized Thom class' for the diagonal embedding of the noncommutative torus.

A geometric representative for the fundamental class in KK-duality of Smale spaces

“…However, Ruelle algebras are not associated to Fock spaces in general, and Smale spaces are typically fractal rather than smooth manifolds. A concrete representation of the K-theory duality class of an irrational rotation algebra is given by Duwenig and Emerson in [18]. Further, the work of Rennie, Robertson and Sims [45] on crossed products by Z with coefficient algebras satisfying KK-duality doesn't apply to our situation, since the coefficient algebras of R s (X, ϕ) and R u (X, ϕ) don't have KK-duals in general, see [27,Subsection 5.1.2].…”