Hecke operators in KK-theory and the K-homology of Bianchi groups

Let Γ be a torsion-free arithmetic group acting on its associated global symmetric space X. Assume that X is of non-compact type and let Γ act on the geodesic boundary ∂X of X. Via general constructions in KK-theory, we endow the K-groups of the arithmetic manifold X/Γ, of the reduced group C∗-algebra C∗r(Γ) and of the boundary crossed product algebra C(∂X)⋊Γ with Hecke operators. In the case when Γ is a group of real hyperbolic isometries, the K-theory and K-homology groups of these C∗-algebras are related by a Gysin six-term exact sequence and we prove that this Gysin sequence is Hecke equivariant. Finally, when Γ is a Bianchi group, we assign explicit unbounded Fredholm modules (i.e. spectral triples) to (co)homology classes, inducing Hecke-equivariant isomorphisms between the integral cohomology of Γ and each of these K-groups. Our methods apply to case Γ⊂PSL(Z) as well. In particular we employ the unbounded Kasparov product to push the Dirac operator an embedded surface in the Borel–Serre compactification of H/Γ to a spectral triple on the purely infinite geodesic boundary crossed product algebra C(∂H)⋊Γ

Operator *-correspondences in analysis and geometry

An operator *-algebra is a non-selfadjoint operator algebra with completely isometric involution. We show that any operator *-algebra admits a faithful representation on a Hilbert space in such a way that the involution coincides with the operator adjoint up to conjugation by a symmetry. We introduce operator *-correspondences as a general class of inner product modules over operator *-algebras and prove a similar representation theorem for them. From this we derive the existence of linking operator *-algebras for operator *-correspondences. We illustrate the relevance of this class of inner product modules by providing numerous examples arising from noncommutative geometry.Comment: 31 pages. This work originated from the MFO workshop "Operator spaces and noncommutative geometry in interaction

Localised module frames and Wannier bases from groupoid morita equivalences

Following the operator algebraic approach to Gabor analysis, we construct frames of translates for the Hilbert space localisation of the Morita equivalence bimodule arising from a groupoid equivalence between Hausdorff groupoids, where one of the groupoids is etale and with a compact unit space. For finitely generated and projective submodules, we show these frames are orthonormal bases if and only if the module is free. We then apply this result to the study of localised Wannier bases of spectral subspaces of Schrodinger operators with atomic potentials supported on (aperiodic) Delone sets. The noncommutative Chern numbers provide a topological obstruction to fast-decaying Wannier bases and we show this result is stable under deformations of the underlying Delone set.Analysis and Stochastic

Describing distance: from the plane to spectral triples

Analysis and Stochastic

Gauge theory on noncommutative Riemannian principal bundles

We present a new, general approach to gauge theory on principal G-spectral triples, where G is a compact connected Lie group. We introduce a notion of vertical Riemannian geometry for G-C*-algebras and prove that the resulting noncommutative orbitwise family of Kostant's cubic Dirac operators defines a natural unbounded KKG-cycle in the case of a principal G-action. Then, we introduce a notion of principal G-spectral triple and prove, in particular, that any such spectral triple admits a canonical factorisation in unbounded KKG-theory with respect to such a cycle: up to a remainder, the total geometry is the twisting of the basic geometry by a noncommutative superconnection encoding the vertical geometry and underlying principal connection. Using these notions, we formulate an approach to gauge theory that explicitly generalises the classical case up to a groupoid cocycle and is compatible in general with this factorisation; in the unital case, it correctly yields a real affine space of noncommutative principal connections with affine gauge action. Our definitions cover all locally compact classical principal G-bundles and are compatible with theta-deformation; in particular, they cover the theta-deformed quaternionic Hopf fibration C-infinity(S-theta(7))hooked left arrow C-infinity(S-theta(4)) as a noncommutative principal SU(2)-bundle.Analysis and Stochastic

A K-theoretic Selberg trace formula

Number theory, Algebra and Geometr

Non-commutative fermion mass matrix and gravity

The first part is an introductory description of a small cross-section of the literature on algebraic methods in non-perturbative quantum gravity with a specific focus on viewing algebra as a laboratory in which to deepen understanding of the nature of geometry. This helps to set the context for the second part, in which we describe a new algebraic characterisation of the Dirac operator in non-commutative geometry and then use it in a calculation on the form of the fermion mass matrix. Assimilating and building on the various ideas described in the first part, the final part consists of an outline of a speculative perspective on (non-commutative) quantum spectral gravity. This is the second of a pair of papers so far on this project.Comment: To appear in Int. J. Mod. Phys. A Previous title: An outlook on quantum gravity from an algebraic perspective. 39 pages, 1 xy-pic figure, LaTex Reasons for new version: added references, change of title and some comments more up-to-dat