Spectral triples and finite summability on Cuntz-Krieger algebras

We produce a variety of odd bounded Fredholm modules and odd spectral triples on Cuntz-Krieger algebras by means of realizing these algebras as "the algebra of functions on a non-commutative space" coming from a sub shift of finite type. We show that any odd $K$-homology class can be represented by such an odd bounded Fredholm module or odd spectral triple. The odd bounded Fredholm modules that are constructed are finitely summable. The spectral triples are $\theta$-summable although their bounded transform, when constructed using the sign-function, will already on the level of analytic $K$-cycles be finitely summable bounded Fredholm modules. Using the unbounded Kasparov product, we exhibit a family of generalized spectral triples, possessing mildly unbounded commutators, whilst still giving well defined $K$-homology classes.Comment: 67 pages, minor changes in Section 5.1 and 6.

Sums of regular selfadjoint operators in Hilbert-C*-modules

We introduce a notion of weak anticommutativity for a pair (S,T) of self-adjoint regular operators in a Hilbert-C*-module E. We prove that the sum $S+T$ of such pairs is self-adjoint and regular on the intersection of their domains. A similar result then holds for the sum S^2+T^2 of the squares. We show that our definition is closely related to the Connes-Skandalis positivity criterion in $KK$-theory. As such we weaken a sufficient condition of Kucerovsky for representing the Kasparov product. Our proofs indicate that our conditions are close to optimal.Comment: Final version. Minor editorial change

Unbounded bivariant KK-theory and correspondences in noncommutative geometry

By introducing a notion of smooth connection for unbounded $KK$-cycles, we show that the Kasparov product of such cycles can be defined directly, by an algebraic formula. In order to achieve this it is necessary to develop a framework of smooth algebras and a notion of differentiable $C^{*}$-module. The theory of operator spaces provides the required tools. Finally, the above mentioned $KK$-cycles with connection can be viewed as the morphisms in a category whose objects are spectral triples.Comment: 67 pages. Final version. Accepted for publicatio

Shift tail equivalence and an unbounded representative of the Cuntz-Pimsner extension

We show how the fine structure in shift-tail equivalence, appearing in the noncommutative geometry of Cuntz-Krieger algebras developed by the first two authors, has an analogue in a wide range of other Cuntz-Pimsner algebras. To illustrate this structure, and where it appears, we produce an unbounded representative of the defining extension of the Cuntz-Pimsner algebra constructed from a finitely generated projective bi-Hilbertian module, extending work by the third author with Robertson and Sims. As an application, our construction yields new spectral triples for Cuntz- and Cuntz-Krieger algebras and for Cuntz-Pimsner algebras associated to vector bundles twisted by equicontinuous $*$-automorphisms.Comment: 30 page

Bivariant <i>K</i>-theory of groupoids and the noncommutative geometry of limit sets

We present a categorical setting for noncommutative geometry in the sense of Connes. This is done by introducing a notion of morphism for spectral triples. Spectral triples are the unbounded cycles for $K$-homology (\cite{Con}), and their bivariant generalization are the cycles for Kasparov's $KK$-theory (\cite{Kas}). The central feature of $KK$-theory is the Kasparov product $$KK_{i}(A,B)\otimes KK_{j}(B,C)\rightarrow KK_{i+j}(A,C).$$ Here $A,B$ and $C$ are $C^{*}$-algebras, and the product allows one to view $KK$ as a category. The unbounded picture of this theory was introduced by Baaj and Julg (\cite{BJ}). In this picture the external product KK_{i}(A,B)\otimes KK_{j}(A',B')\rightarrow KK_{i+j}(A\minotimes B,A'\minotimes B'), is given by an algebraic formula, as opposed to Kasparov's original approach, which is more analytic in nature, and highly technical. In order to describe the internal Kasparov product of unbounded $KK$-cycles, we introduce a notion of connection for unbounded cycles (\mathpzc{E},D). This is a universal connection \nabla:\mathpzc{E}\rightarrow\mathpzc{E}\tildeotimes_{B}\Omega^{1}(B), in the sense of Cuntz and Quillen (\cite{CQ}), such that $[\nabla,D]$ extends to a completely bounded operator. The topological tensor product used here is the Haagerup tensor product for operator spaces. Blecher (\cite{Blech}) showed this tensor product coincides with the $C^{*}$-module tensor product, in case both operator spaces are $C^{*}$-modules. His work plays a crucial role in our construction. The product of two cycles with connection is given by an algebraic formula and the product of connections can also be defined. Thus, cycles with connection form a category, and the bounded transform (\mathpzc{E},D,\nabla)\mapsto(\mathpzc{E},D(1+D^{2})^{-\frac{1}{2}}), defines a functor from this category to the category $KK$. We also describe a general construction for obtaining $KK$-cycles from real-valued groupoid cocycles. If $\mathcal{G}$ is a locally compact Hausdorff groupoid with Haar system and $c:\mathcal{G}\rightarrow\R$ a continuous closed cocycle, we show that pointwise multiplication by $c$ in the convolution algebra $C_{c}(\mathcal{G})$, extends to an unbounded regular operator on the completion of $C_{c}(\mathcal{G})$ as a $C^{*}$-module over $C^{*}(\mathcal{H})$, where $\mathcal{H}$ is the kernel of $c$. It gives a $KK$-cycle for $(C^{*}(\mathcal{G}),C^{*}(\mathcal{H}))$. In case the groupoid $\mathcal{H}$ is unimodular with respect to a quasi-invariant measure, or more general, if $C^{*}(\mathcal{H})$ carries a trace, this $KK$-cycle gives rise to an index map K_{1}(C^{*}(\mathcal{G}))\rightarrow \C. This result is general enough to be applied in a wide variety of examples. We use it to obtain the noncommutative torus as a smooth quotient (in the above categorical sense) of the irrational rotation action on the circle. In the last chapter we sketch the promising range of applications the above categorical setting and cocycle construction may have in the noncommutative geometry of limit sets

Classifying the Dynamics of Architected Materials by Groupoid Methods

We consider synthetic materials consisting of self-coupled identical resonators carrying classical internal degrees of freedom. The architecture of such material is specified by the positions and orientations of the resonators. Our goal is to calculate the smallest C*-algebra that covers the dynamical matrices associated to a fixed architecture and adjustable internal structures. We give the answer in terms of a groupoid C*-algebra that can be canonically associated to a uniformly discrete subset of the group of isometries of the Euclidean space. Our result implies that the isomorphism classes of these C*-algebras split these architected materials into classes containing materials that are identical from the dynamical point of view

The bordism group of unbounded KK-cycles

We consider Hilsum's notion of bordism as an equivalence relation on unbounded $KK$-cycles and study the equivalence classes. Upon fixing two $C^*$-algebras, and a $*$-subalgebra dense in the first $C^*$-algebra, a $\mathbb{Z}/2\mathbb{Z}$-graded abelian group is obtained; it maps to the Kasparov $KK$-group of the two $C^*$-algebras via the bounded transform. We study properties of this map both in general and in specific examples. In particular, it is an isomorphism if the first $C^*$-algebra is the complex numbers (i.e., for $K$-theory) and is a split surjection if the first $C^*$-algebra is the continuous functions on a compact manifold with boundary when one uses the Lipschitz functions as the dense $*$-subalgebra.Comment: 38 page

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 $G$-$C^\ast$-algebras and prove that the resulting noncommutative orbitwise family of Kostant's cubic Dirac operators defines a natural unbounded $KK^G$-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 $KK^G$-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 $C^\infty(S^7_\theta) \hookleftarrow C^\infty(S^4_\theta)$ as a noncommutative principal $\operatorname{SU}(2)$-bundle.Comment: Final version to appear in Commun. Math. Phys. encompassing various clarifications and corrections including thorough revisions of Prop. 2.35, Prop. 2.36, and Lemma 2.45 and a correction to Def. B.2. The authors thank the anonymous reviewers for their extraordinarily thoughtful, thorough, and useful feedbac