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We propose a new notion of unbounded $K\!K$-cycle, mildly generalising unbounded Kasparov modules, for which the direct sum is well-defined. To a pair $(A,B)$ of $\sigma$-unital $C^{*}$-algebras, we can then associate a semigroup $\overline{U\!K\!K}(A,B)$ of homotopy equivalence classes of unbounded cycles, and we prove that this semigroup is in fact an abelian group. In case $A$ is separable, our group $\overline{U\!K\!K}(A,B)$ is isomorphic to Kasparov's $K\!K$-theory group $K\!K(A,B)$ via the bounded transform. We also discuss various notions of degenerate cycles, and we prove that the homotopy relation on unbounded cycles coincides with the relation generated by operator-homotopies and addition of degenerate cycles., Comment: 33 pages

Let $\Gamma$ be a torsion-free arithmetic group acting on its associated global symmetric space $X$. Assume that $X$ is of non-compact type and let $\Gamma$ act on the geodesic boundary $\partial X$ of $X$. Via general constructions in KK-theory, we endow the K-groups of the arithmetic manifold $X/\Gamma$, of the reduced group C*-algebra of $\Gamma$ and of the boundary crossed product algebra associated to the action of $\Gamma$ on $\partial X$, with Hecke operators. The K-theory and K-homology groups of these C*-algebras are related by a Gysin six-term exact sequence. In the case when $\Gamma$ is a group of real hyperbolic isometries, we show that this Gysin sequence is Hecke equivariant. Finally, in the case when $\Gamma$ is a subgroup of a Bianchi group, we construct explicit Hecke-equivariant maps between the integral cohomology of $\Gamma$ and each of these K-groups. Our methods apply to torsion-free finite index subgroups of $PSL(2,\mathbb{Z})$ as well. These results are achieved in the context of unbounded Fredholm modules, shedding light on noncommutative geometric aspects of the purely infinite boundary crossed product algebra., Comment: 50 pages

In this paper we introduce the curvature of densely defined universal connections on Hilbert $C^{*}$-modules relative to a spectral triple (or unbounded Kasparov module), obtaining a well-defined curvature operator. Fixing the spectral triple, we find that modulo junk forms, the curvature only depends on the represented form of the universal connection. We refine our definition of curvature to factorisations of unbounded Kasparov modules. Our refined definition recovers all the curvature data of a Riemannian submersion of compact manifolds, viewed as a $KK$-factorisation., 47 pages

Let $B$ be a $C^{*}$-algebra, $X$ a Hilbert $C^{*}$-module over $B$ and $M,N\subset X$ a pair of complemented submodules. We prove the $C^{*}$-module version of von Neumann's alternating projections theorem: the sequence $(P_{N}P_{M})^{n}$ is Cauchy in the $*$-strong module topology if and only if $M\cap N$ is the complement of $\overline{M^{\perp}+N^{\perp}}$. In this case, the $*$-strong limit of $(P_{M}P_{N})^{n}$ is the orthogonal projection onto $M\cap N$. We use this result and the local-global principle to show that the cosine of the Friedrichs angle $c(M,N)$ between any pair of complemented submodules $M,N\subset X$ is well-defined and that $c(M,N), 19 pages. We added Lemma 3.10, sharpened Proposition 3.12, and discuss vector bundles in Remark 3.14

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 feedback

We consider the algebra $\dot\Sigma(\mathcal L)$ generated by the inner-limit derivations over the ${\rm GICAR}$ algebra of a fermion gas populating an aperiodic Delone set $\mathcal L$. Under standard physical assumptions such as finite interaction range, Galilean invariance and continuity with respect to the aperiodic lattice, we demonstrate that the image of $\dot \Sigma(\mathcal L)$ through the Fock representation can be completed to a groupoid-solvable pro-$C^\ast$-algebra. Our result is the first step towards unlocking the $K$-theoretic tools available for separable $C^\ast$-algebra for applications in the context of interacting fermions.

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 \'{e}tale 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 Schr\"{o}dinger 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., Comment: v2: name changed and other minor changes. To appear in J. Fourier Anal. Appl. 28 pages

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 pages

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"

We examine the noncommutative index theory associated to the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological phases of aperiodic lattices and materials, and applies to invariants from tilings as well. Our discussion concerns semifinite index pairings, factorisation properties of Kasparov modules and the construction of unbounded Fredholm modules for lattices with finite local complexity., 52 pages, Section 1.6 added and other minor improvements. To appear in Annales Henri Poincar\'{e}

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., Final version. Minor editorial changes

Let G be a semisimple Lie group and H a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of convolution operators on the Hilbert space L^2(G/H) associated to test functions. In this paper we present a cohomological interpretation of the trace formula involving the K-theory of the maximal group C*-algebras of G and H. As an application, we exploit the role of group C*-algebras as recipients of higher indices of elliptic differential operators and we obtain the index theoretic version of the Selberg trace formula developed by Barbasch and Moscovici from ours., Comment: Extended the introduction and added a couple of extra remarks in Section 4

Let $\Gamma$ be a lattice in a locally compact group $G$. In earlier work, we used $KK$-theory to equip the $K$-groups of any $\Gamma$-$C^{*}$-algebra on which the commensurator of $\Gamma$ acts with Hecke operators. When $\Gamma$ is arithmetic, this gives Hecke operators on the $K$-theory of certain $C^{*}$-algebras that are naturally associated with $\Gamma$. In this paper, we first study the topological $K$-theory of the arithmetic manifold associated to $\Gamma$. We prove that the Chern character commutes with Hecke operators. Afterwards, we show that the Shimura product of double cosets naturally corresponds to the Kasparov product and thus that the $KK$-groups associated to an arithmetic group $\Gamma$ become true Hecke modules. We conclude by discussing Hecke equivariant maps in $KK$-theory in great generality and apply this to the Borel-Serre compactification as well as various noncommutative compactifications associated with $\Gamma$. Along the way we discuss the relation between the $K$-theory and the integral cohomology of low-dimensional manifolds as Hecke modules., Comment: 22 pages. Revised version, to appear in Annals of K-theory

We study irreducible Smale spaces with totally disconnected stable sets and their associated $K$-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one $K$-theoretic. Using Wieler’s theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyse an explicit groupoid Morita equivalence between the groupoids of Deaconu–Renault and Putnam–Spielberg, extending results of Thomsen. The Deaconu–Renault groupoid and the explicit Morita equivalence lead to a Cuntz–Pimsner model for the stable Ruelle algebra. The $K$-theoretic invariants of Cuntz–Pimsner algebras are then studied using the Cuntz–Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions, we characterize the Kubo–Martin–Schwinger (KMS) weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples.

We explore factorizations of noncommutative Riemannian spin geometries over commutative base manifolds in unbounded KK-theory. After setting up the general formalism of unbounded KK-theory and improving upon the construction of internal products, we arrive at a natural bundle-theoretic formulation of gauge theories arising from spectral triples. We find that the unitary group of a given noncommutative spectral triple arises as the group of endomorphisms of a certain Hilbert bundle; the inner fluctuations split in terms of connections on, and endomorphisms of, this Hilbert bundle. Moreover, we introduce an extended gauge group of unitary endomorphisms and a corresponding notion of gauge fields. We work out several examples in full detail, to wit Yang--Mills theory, the noncommutative torus and the $\theta$-deformed Hopf fibration over the two-sphere., Comment: 50 pages. Accepted version. Section 2 has been rewritten. Results in sections 3-6 are unchanged

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 pages

We give a construction of an odd spectral triple on the Cuntz algebra O N , whose K-homology class generates the odd K-homology group K1(O N ). Using a metric measure space structure on the Cuntz-Renault groupoid, we introduce a singular integral operator which is the formal analogue of the logarithm of the Laplacian on a Riemannian manifold. Assembling this operator with the infinitesimal generator of the gauge action on O N yields a θ-summable spectral triple whose phase is finitely summable. The relation to previous constructions of Fredholm modules and spectral triples on O N is discussed.

We examine the index data associated to twisted spectral triples and higher order spectral triples. In particular, we show that a Lipschitz regular twisted spectral triple can always be `logarithmically dampened' through functional calculus, to obtain an ordinary (i.e. untwisted) spectral triple. The same procedure turns higher order spectral triples into spectral triples. We provide examples of highly regular twisted spectral triples with nontrivial index data for which Moscovici's ansatz for a twisted local index formula is identically zero., 57 pages, full proof of meromorphic extensions added in this version

By considering the general properties of approximate units in differentiable algebras, we are able to present a unified approach to characterising completeness of spectral metric spaces, existence of connections on modules, and the lifting of Kasparov products to the unbounded category. In particular, by strengthening Kasparov's technical theorem, we show that given any two composable KK-classes, we can find unbounded representatives whose product can be constructed to yield an unbounded representative of the Kasparov product., Comment: 65 pages

This paper extends the notion of a spectral triple to a relative spectral triple, an unbounded analogue of a relative Fredholm module for an ideal $J\triangleleft A$. Examples include manifolds with boundary, manifolds with conical singularities, dimension drop algebras, $\theta$-deformations and Cuntz-Pimsner algebras of vector bundles. The bounded transform of a relative spectral triple is a relative Fredholm module, making the image of a relative spectral triple under the boundary mapping in $K$-homology easy to compute. We introduce an additional operator called a Clifford normal with which a relative spectral triple can be doubled into a spectral triple. The Clifford normal also provides a boundary Hilbert space, a representation of the quotient algebra, a boundary Dirac operator and an analogue of the Calderon projection. In the examples this data does assemble to give a boundary spectral triple, though we can not prove this in general. When we do obtain a boundary spectral triple, we provide sufficient conditions for the boundary triple to represent the $K$-homological boundary. Thus we abstract the proof of Baum-Douglas-Taylor's "boundary of Dirac is Dirac on the boundary" theorem into the realm of non-commutative geometry.

This article is about erroneous attempts to weaken the standard definition of unbounded Kasparov module (or spectral triple). We present counterexamples to claims in the literature that Fredholm modules can be obtained from these weaker variations of spectral triple. Our counterexamples are constructed using self-adjoint extensions of symmetric operators.

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 publication

This survey deals with the construction of a category of spectral triples that is compatible with the Kasparov product in $KK$-theory. These notes serve as an intuitive guide to these results, avoiding the necessary technical proofs. We will also add some background and a broader perspective on noncommutative geometry.

Let $c:\mathcal{G}\to\R$ be a cocycle on a locally compact Hausdorff groupoid $\mathcal{G}$ with Haar system. Under some mild conditions (satisfied by all integer valued cocycles on \'{e}tale groupoids), $c$ gives rise to an unbounded odd $\R$-equivariant bimodule $(\mathpzc{E},D)$ for the pair of $C^{*}$-algebras $(C^{*}(\mathcal{G}),C^{*}(\mathcal{H}))$. If the cocycle comes from a continuous quasi-invariant measure on the unit space $\mathcal{G}^{(0)}$, the corresponding element in $KK_{1}^{\R}(C^{*}(\mathcal{G}),C^{*}(\mathcal{H}))$ gives rise to an index map $K_{1}^{\R}(C^{*}(\mathcal{G}))\to \C$., Comment: Update: proof of exactness of integral cocycles corrected

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 6.1