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Let X be a compact Hausdorff space. Then the radius of comparison rc ( C (X)) is related to the covering dimension dim (X) by rc ( C (X)) \geq [ dim (X) - 7 ] / 2. Except for the additive constant, this improves a result of Elliott and Niu, who proved that if X is metrizable then rc (C (X)) \geq [ dim_{\mathbb{Q}} (X) - 4 ] / 2. There are compact metric spaces X for which the estimate of Elliott and Niu gives no information, but for which rc ( C (X)) is infinite or has arbitrarily large finite values., Comment: AMSLaTeX; 8 pages

Let $1 \to N \to G \to G/N \to 1$ be a short exact sequence of countable discrete groups and let $B$ be any $G$-$C^*$-algebra. In this paper, we show that the strong Novikov conjecture with coefficients in $B$ holds for such a group $G$ when the normal subgroup $N$ and the quotient group $G/N$ are coarsely embeddable into Hilbert spaces. As a result, the group $G$ satisfies the Novikov conjecture under the same hypothesis on $N$ and $G/N$., Comment: 40 pages

For an \'etale correspondence $\Omega \colon G \to H$ of \'etale groupoids, we construct an induction functor $\mathrm{Ind}_\Omega \colon \mathrm{KK}^H \to \mathrm{KK}^G$ between equivariant Kasparov categories. We introduce the crossed product of an $H$-equivariant correspondence by $\Omega$, and use this to build a natural transformation $\alpha_\Omega \colon K_*( G \ltimes \mathrm{Ind}_\Omega -) \Rightarrow K_*(H \ltimes -)$. When $\Omega$ is proper these constructions naturally sit above an induced map in K-theory $K_*(C^*(G)) \to K_*(C^*(H))$., Comment: Version published in the Journal of Functional Analysis

For each odd integer $n \geq 3$, we construct a rank-3 graph $\Lambda_n$ with involution $\gamma_n$ whose real C*-algebra $C^*_\mathbb{R}(\Lambda_n, \gamma_n)$ is stably isomorphic to the exotic Cuntz algebra $\mathcal E_n^\mathbb{R}$. This construction is optimal, as we prove that a rank-2 graph with involution $(\Lambda,\gamma)$ can never satisfy $C^*_\mathbb{R}(\Lambda, \gamma)\sim_{ME} \mathcal E_n^\mathbb{R}$, and the first author reached the same conclusion in previous work. Our construction relies on a rank-1 graph with involution $(\Lambda, \gamma)$ whose real C*-algebra $C^*_\mathbb{R}(\Lambda, \gamma)$ is stably isomorphic to the suspension $ S \mathbb{R}$. In the Appendix, we show that the i-fold suspension $S^i \mathbb{R}$ is stably isomorphic to a graph algebra iff $-2 \leq i \leq 1$.

We determine the group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra $O_{n+1}$ for finite n in terms of K-theory. We show that there is an example of a space for which the homotopy set is a non-commutative group, and hence the classifying space of the automorphism group of the Cuntz algebra for finite n is not an H-space. We also make an improvement of Dadarlat's classification of continuous fields of the Cuntz algebras in terms of vector bundles., Comment: 14 pages, Our paper is motivated by arXiv:1004.1722

The Cuntz-Toeplitz algebra $E_{n+1}$ for $n\geq1$ is the universal C*-algebra generated by $n+1$ isometries with mutually orthogonal ranges. In this paper, we investigate the automorphism groups of the Cuntz-Toeplitz algebras and determine their homotopy groups., Comment: 20 pages

In a previous paper we have introduced the gauge-equivariant K-theory group of a bundle endowed with a continuous action of a bundle of compact Lie groups. These groups are the natural range for the analytic index of a family of gauge-invariant elliptic operators (i.e. a family of elliptic operators invariant with respect to the action of a bundle of compact groups). In this paper, we continue our study of gauge-equivariant K-theory. In particular, we introduce and study products, which helps us establish the Thom isomorphism in gauge-equivariant K-theory. Then we construct push-forward maps and define the topological index of a gauge-invariant family., Comment: 29 pages, LaTeX2e, amsart, xy

In the 1980's Pierre Julg and Alain Valette, and also Tadeusz Pytlik and Ryszard Szwarc, constructed and studied a certain Fredholm operator associated to a simplicial tree. The operator can be defined in at least two ways: from a combinatorial flow on the tree, similar to the flows in Forman's discrete Morse theory, or from the theory of unitary operator-valued coccyges. There are applications of the theory surrounding the operator to C*-algebra K-theory, to the theory of completely bounded representations of groups that act on trees, and to the Selberg principle in the representation theory of p-adic groups. The main aim of this paper is to extend the constructions of Julg and Valette, and Pytlik and Szwarc, to CAT(0) cubical spaces. A secondary aim is to illustrate the utility of the extended construction by developing an application to operator K-theory and giving a new proof of K-amenability for groups that act properly on bounded-geometry CAT(0)-cubical spaces.

We initiate the study of real $C^*$-algebras associated to higher-rank graphs $\Lambda$, with a focus on their $K$-theory. Following Kasparov and Evans, we identify a spectral sequence which computes the $\mathcal{CR}$ $K$-theory of $C^*_{\mathbb R} (\Lambda, \gamma)$ for any involution $\gamma$ on $\Lambda$, and show that the $E^2$ page of this spectral sequence can be straightforwardly computed from the combinatorial data of the $k$-graph $\Lambda$ and the involution $\gamma$. We provide a complete description of $K^{CR}(C^*_{\mathbb R}(\Lambda, \gamma))$ for several examples of higher-rank graphs $\Lambda$ with involution.

For a continuous field of the Cuntz algebra over a finite CW complex, we introduce a topological invariant, which is an element in Dadarlat-Pennig's generalized cohomology group, and prove that the invariant is trivial if and only if the field comes from a vector bundle via Pimsner's construction., Comment: 23 pages

The Baum-Connes map for finitely generated free abelian groups is a K-theoretic analogue of the Fourier-Mukai transform from algebraic geometry. We describe this K-theoretic transform in the language of topological correspondences, and compute its action on K-theory (of tori) described geometrically in terms of Baum-Douglas cocycles, showing that the Fourier-Mukai transform maps the class of a subtorus to the class of a suitably defined dual torus. We deduce the Fourier-Mukai inversion formula. We use these results to give a purely geometric description of the Baum-Connes assembly map for free abelian groups., Comment: Re-written and re-titled from the first version

We consider representations of the Cuntz algebras ${\mathcal O}_N$ as constructed by Bratteli-Jorgensen and use these to define a faithful action of the analytic loop group $L_{\text{an}}U(n)$ on ${\mathcal O}_N$ for $N \geq n$. This extends to a faithful action on the infinite Cuntz algebra ${\mathcal O}_{\infty}$, and we show that this action can be used to explicitly construct ${\mathcal O}_\infty \otimes \mathcal K$-bundles (where $ \mathcal K$ is the algebra of compact operators) over spaces, which are geometric twists in higher twisted $K$-theory., Comment: Fatal error in the definition of the action. The authors thank R. Conti for pointing this out

The stabilization theorem for $A$-Hilbert modules was established by G. G. Kasparov. The equivariant version, in which a locally compact group $H$ acts properly on a locally compact space $Y$, was proved by N. C. Phillips. This equivariant theorem involves the Hilbert $(H,C_{0}(Y))$-module $C_{0}(Y,L^{2}(H)^{\infty})$. It can naturally be interpreted in terms of a stabilization theorem for proper groupoids, and the paper establishes this theorem within the general proper groupoid context. The theorem has applications in equivariant KK-theory and groupoid index theory., Comment: 16 pages

In this paper we study continuous bundles of C*-algebras which are non-commutative analogues of principal torus bundles. We show that all such bundles, although in general being very far away from being locally trivial bundles, are at least locally trivial with respect to a suitable bundle version of bivariant K-theory (denoted RKK-theory) due to Kasparov. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to equivariant Morita equivalence. We then study these bundles as topological fibrations (forgetting the group action) and give necessary and sufficient conditions for any non-commutative principal torus bundle being RKK-equivalent to a commutative one. As an application of our methods we shall also give a K-theoretic characterization of those principal torus-bundles with H-flux, as studied by Mathai and Rosenberg which possess "classical" T-duals., Comment: 33 pages, to appear in the Proceedings of the London Mathematical Society

This paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with the Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on the Cuntz algebra. We introduce a modified $K_1$-group of the Cuntz algebra so as to pair with this twisted cocycle. As a corollary we obtain a noncommutative geometry interpretation for Araki's notion of relative entropy in this example. We also note the connection of this example to the theory of noncommutative manifolds., Comment: 27 pages, minor corrections

In this paper we begin a study of the space of unbounded self-adjoint Fredholm operators as a classifying space for K^{1}(X), with the goal of incorporating the information in the eigenspaces and eigenvalues of the operators. In particular, the role that the multiplicity of eigenvalues plays is developed here., Comment: Typos fixed and application improved

We give a survey on recent development of the Novikov conjecture and its applications to topological rigidity and non-rigidity. ., Comment: 16 pages. Dedicated to Sergei Novikov on the occasion of his 80th birthday. To appear in Russian Math Survey, 2019. arXiv admin note: text overlap with arXiv:1811.02086

We give a proof of the Bott periodicity theorem for topological K-theory of C*-algebras based on Loring's treatment of Voiculescu's almost commuting matrices and Atiyah's rotation trick. We also explain how this relates to the Dirac operator on the circle; this uses Yu's localization algebra and an associated explicit formula for the pairing between the first K-homology and first K-theory groups of a (separable) C*-algebra., Comment: 15 pages

The purpose of the paper is twofold: First, known results of the noncommutative spin geometry of the standard Podles sphere are extended by discussing Poincare duality and orientability. In the discussion of orientability, Hochschild homology is replaced by a twisted version which avoids the dimension drop. The twisted Hochschild cycle representing an orientation is related to the volume form of the distinguished covariant differential calculus. Integration over the volume form defines a twisted cyclic 2-cocycle which computes the q-winding numbers of quantum line bundles. Second, a "twisted" Chern character from equivariant K0-theory to even twisted cyclic homology is introduced which gives rise to a Chern-Connes pairing between equivariant K0-theory and twisted cyclic cohomology. The Chern-Connes pairing between the equivariant K0-group of the standard Podles sphere and the generators of twisted cyclic cohomology relative to the modular automorphism and its inverse are computed. This includes the pairings with the twisted cyclic 2-cocycle associated to the volume form, and the one corresponding to the "no-dimension drop" case. From explicit index computations, it follows that the pairings with these cocycles give the q-indices of the known equivariant 0-summable Dirac operator on the standard Podles sphere., Comment: 33 pages, title changed, Section 3.1 completely rewritten, Chern-Connes pairing in the "no-dimension drop" case included

The bulk spectrum of a possible Chern insulator on a quasicrystalline lattice is examined. The effect of being a 2D insulator seems to override any fractal properties in the spectrum. We compute that the spectrum is either two continuous bands, or that any gaps other than the main gap are small. After making estimates on the spectrum, we deduce a finite system size, above which the K-theory must coincide with the K-theory of the infinite system. Knowledge of the spectrum and $K$-theory of the infinite-area system will control the spectrum and K-theory of sufficiently large finite systems. The relation between finite volume $K$-theory and infinite volume Chern numbers is only proven to begin, for the model under investigation here, for systems on Hilbert space of dimension around 17 million. The real-space method based on the Clifford spectrum allows for computing Chern numbers for systems on Hilbert space of dimension around 2.7 million. New techniques in numerical K-theory are used to equate the K-theory of systems of different sizes., Comment: Added the missing acknowledgement section

We prove that the Novikov conjecture holds for any discrete group admitting an isometric and metrically proper action on an admissible Hilbert-Hadamard space. Admissible Hilbert-Hadamard spaces are a class of (possibly infinite-dimensional) non-positively curved metric spaces that contain dense sequences of closed convex subsets isometric to Riemannian manifolds. Examples of admissible Hilbert-Hadamard spaces include Hilbert spaces, certain simply connected and non-positively curved Riemannian-Hilbertian manifolds and infinite\-/dimensional symmetric spaces. Thus our main theorem can be considered as an infinite-dimensional analogue of Kasparov's theorem on the Novikov conjecture for groups acting properly and isometrically on complete, simply connected and non-positively curved manifolds. As a consequence, we show that the Novikov conjecture holds for geometrically discrete subgroups of the group of volume preserving diffeomorphisms of a closed smooth manifold. This result is inspired by Connes' theorem that the Novikov conjecture holds for higher signatures associated to the Gelfand-Fuchs classes of groups of diffeormorphisms., Comment: 55 pages; contains minor corrections and modifications from the last version; to be published in GAFA

In this paper we construct a bivariant Chern character for the equivariant KK-theory of a totally disconnected group with values in bivariant equivariant cohomology in the sense of Baum and Schneider. We prove in particular that the complexified left hand side of the Baum-Connes conjecture for a totally disconnected group is isomorphic to cosheaf homology. Moreover, it is shown that our transformation extends the Chern character defined by Baum and Schneider for profinite groups., Comment: 27 pages

Let $A$ be a C*-algebra, $J \subset A$ a C*-subalgebra, and let $B$ be a stable C*-algebra. Under modest assumptions we organize invertible C*-extensions of $A$ by $B$ that are trivial when restricted onto $J$ to become a group $Ext_J^{-1}(A,B)$, which can be computed by a six-term exact sequence which generalizes the excision six-term exact sequence in the first variable of $KK$-theory. Subsequently we investigate the relative K-homology which arises from the group of relative extensions by specializing to abelian C*-algebras. It turns out that this relative K-homology carries substantial information also in the operator theoretic setting from which the BDF theory was developed and we conclude the paper by extracting some of this information on approximation of normal operators., Comment: 24 pages

It is shown that the class of Fredholm operators over an arbitrary unital $C^{*}$--algebra, which may not admit adjoint ones, can be extended in such a way that this class of compact operators, used in the definition of the class of Fredholm operators, contains compact operators both with and without existence of adjoint ones. The main property of this new class is that a Fredholm operator which may not admit an adjoint one has a decomposition into a direct sum of an isomorphism and a finitely generated operator. In the space of compact operators in the Hilbert space a new IM-topology is defined. In the case when the $C^{*}$--algebra is a commutative algebra of continuous functions on a compact space the IM-topology fully describe the set of compact operators over the $C^{*}$--algebra without assumption of existence bounded adjoint operators over the algebra. In the revised version of the paper the proof of the theorem 8 has been added.

In the paper we describe the C*-algebras of noncommutative spherical tight frames over some C*-algebras and then apply to study the noncommutative version of the universal classifying space., Comment: LaTeX2e, no figure

In this paper we study the C*-algebras associated to continuous fields over locally compact metrisable zero dimensional spaces whose fibers are Kirchberg C*-algebras satisfying the UCT. We show that these algebras are inductive limits of finite direct sums of Kirchberg algebras and they are classified up to isomorphism by topological invariants., Comment: 17 pages, some corrections, additional remarks

Many index theorems (both classical and in noncommutative geometry) can be interpreted in terms of a Lie groupoid acting properly on a manifold and leaving an elliptic family of pseudodifferential operators invariant. Alain Connes in his book raised the question of an index theorem in this general context. In this paper, an analytic index for many such situations is constructed. The approach is inspired by the classical families theorem of Atiyah and Singer, and the proof generalizes, to the case of proper Lie groupoid actions, some of the results proved for proper locally compact group actions by N. C. Phillips.

In this paper we discuss how the question about the rationality of L^2-Betti numbers is related to the Isomorphism Conjecture in algebraic K-theory and why in this context noncommutative localization appears as an important tool., Comment: 28 pages

Alternative titles of this paper would have been `Index theory without index' or `The Baum-Connes conjecture without Baum.' In 1989, Rieffel introduced an analytic version of deformation quantization based on the use of continuous fields of C*-algebras. We review how a wide variety of examples of such quantizations can be understood on the basis of a single lemma involving amenable groupoids. These include Weyl-Moyal quantization on manifolds, C*-algebras of Lie groups and Lie groupoids, and the E-theoretic version of the Baum-Connes conjecture for smooth groupoids as described by Connes in his book Noncommutative Geometry. Concerning the latter, we use a different semidirect product construction from Connes. This enables one to formulate the Baum-Connes conjecture in terms of twisted Weyl-Moyal quantization. The underlying mechanical system is a noncommutative desingularization of a stratified Poisson space, and the Baum-Connes conjecture actually suggests a strategy for quantizing such singular spaces., Comment: 21 pages

This is an appendix to the paper {\bf Asymptotic K-theory for groups acting on $\tA_2$ buildings}, and contains the results of the computations performed by the authors., Comment: 7 pages

In this paper we investigate the extremal richness of the multiplier algebra $M(A)$ and the corona algebra $M(A)/A$, for a simple C*-algebra $A$ with real rank zero and stable rank one. We show that the space of extremal quasitraces and the scale of $A$ contain enough information to determine whether $M(A)/A$ is extremally rich. In detail, if the scale is finite, then $M(A)/A$ is extremally rich. In important cases, and if the scale is not finite, extremal richness is characterized by a restrictive condition: the existence of only one infinite extremal quasitrace which is isolated in a convex sense., Comment: 20 pages (revised 1999); to appear in Journal of Operator Theory

We compute K-theory for the reduced group C*-algebras of generalized Lamplighter groups., Comment: minor revisions; final version, accepted for publication in Proc. Amer. Math. Soc.; 6 pages

Algorithmic issues concerning Elliott local semigroups are seldom considered in the literature, although these combinatorial structures completely classify AF algebras. In general, the addition operation of an Elliott local semigroup is {\it partial}, but for every AF algebra $\mathfrak B$ whose Murray-von Neumann order of projections is a lattice, this operation is uniquely extendible to the addition of an involutive monoid $E(\mathfrak B)$. Let $\mathfrak M_1$ be the Farey AF algebra introduced by the present author in 1988 and rediscovered by F. Boca in 2008. The freeness properties of the involutive monoid $E(\mathfrak M_1)$ yield a natural word problem for every AF algebra $\mathfrak B$ with singly generated $E(\mathfrak B)$, because $\mathfrak B$ is automatically a quotient of $\mathfrak M_1$. Given two formulas $\phi$ and $\psi$ in the language of involutive monoids, the problem asks to decide whether $\phi$ and $\psi$ code the same equivalence of projections of $\mathfrak B$. This mimics the classical definition of the word problem of a group presented by generators and relations. We show that the word problem of $\mathfrak M_1$ is solvable in polynomial time, and so is the word problem of the Behnke-Leptin algebras $\mathcal A_{n,k}$, and of the Effros-Shen algebras $\mathfrak F_{\theta}$, for $\theta\in [0,1]\setminus \mathbb Q$ a real algebraic number, or $\theta = 1/e$. We construct a quotient of $\mathfrak M_1$ having a G\"odel incomplete word problem, and show that no primitive quotient of $\mathfrak M_1$ is G\"odel incomplete.

Let $A$, $B$ be separable C*-algebras, $B$ stable. Elements of the E-theory group $E(A,B)$ are represented by asymptotic homomorphisms from the second suspension of $A$ to $B$. Our aim is to represent these elements by (families of) maps from $A$ itself to $B$. We have to pay for that by allowing these maps to be even further from $*$-homomorphisms. We prove that $E(A,B)$ can be represented by pairs $(\varphi^+,\varphi^-)$ of maps from $A$ to $B$ that are not necessarily asymptotic homomorphisms, but have the same deficiency from being ones. Not surprisingly, such pairs of maps can be viewed as pairs of asymptotic homomorphisms from some C*-algebra $C$ that surjects onto $A$, and the two maps in a pair should agree on the kernel of this surjection. We give examples of full surjections $C\to A$, i.e. those, for which all classes in $E(A,B)$ can be obtained from pairs of asymptotic homomorphisms from $C$., Comment: 18 pages, some stupid errors fixed, the construction of homotopies is modified to become unoform