Search

For two unital Kirchberg algebras with finitely generated K-groups, we introduce a property, called reciprocality, which is proved to be closely related to the homotopy theory of Kirchberg algebras. We show the Spanier–Whitehead duality for bundles of separable nuclear UCT C*-algebras with finitely generated K-groups and conclude that two reciprocal Kirchberg algebras share the same structure of their bundles. [ABSTRACT FROM AUTHOR]

We investigate ⁎-homomorphisms with nuclear dimension equal to zero. In the framework of classification of ⁎-homo-morphisms, we characterise such maps as those that can be approximately factorised through an AF-algebra. Along the way, we obtain various obstructions for the total invariant of zero-dimensional morphisms and show that in the presence of real rank zero, nuclear dimension zero can be completely determined at the level of the total invariant. We end by characterising when unital embeddings of Z have nuclear dimension equal to zero. [ABSTRACT FROM AUTHOR]

We develop a new method for studying the Baum–Connes conjecture, which we call the direct splitting method. We introduce what we call property (γ) for G -equivariant Kasparov cycles. We show that the existence of a G -equivariant Kasparov cycle with property (γ) implies the split-injectivity of the assembly map μ A G for any separable G - C ⁎ -algebra A. We also show that if such a cycle exists, the assembly map μ A G is an isomorphism if and only if the cycle acts as the identity on the right-hand side group K ⁎ (A ⋊ r G) of the Baum–Connes conjecture. In a separate paper, with J. Brodzki, E. Guentner and N. Higson, we use this method to give a finite-dimensional proof of the Baum–Connes conjecture for groups which act properly and co-compactly on a finite-dimensional CAT(0)-cubical space. [ABSTRACT FROM AUTHOR]

Alain Connes and Nigel Higson pointed out in the 1990s that the Connes-Kasparov "conjecture" for the K-theory of reduced group C ⁎ -algebras seemed, in the case of reductive Lie groups, to be a cohomological echo of a conjecture of George Mackey concerning the rigidity of representation theory along the deformation from a real reductive group to its Cartan motion group. For complex semisimple groups, Nigel Higson established in 2008 that Mackey's analogy is a real phenomenon, and does lead to a simple proof of the Connes-Kasparov isomorphism. We here turn to more general reductive groups and use our recent work on Mackey's proposal, together with Higson's work, to obtain a new proof of the Connes-Kasparov isomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we apply quantitative operator K -theory to develop an algorithm for computing K -theory for the class of filtered C ⁎ -algebras with asymptotic finite nuclear decomposition. As a consequence, we prove the Künneth formula for C ⁎ -algebras in this class. Our main technical tool is a quantitative Mayer–Vietoris sequence for K -theory of filtered C ⁎ -algebras. [ABSTRACT FROM AUTHOR]

We introduce a notion of (approximately) coarsely proper algebras for coarse embeddings of metric spaces to formulate and prove a version of twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebras. This may be regarded as a coarse geometric version of the generalized Green-Julg Theorem in the Baum-Connes conjecture for countable discrete groups. It also provides a conceptual framework for the Dirac-dual-Dirac method to the coarse Novikov conjecture for coarse embeddings into several different spaces, including Hilbert spaces, simply connected complete Riemannian manifolds with non-positive sectional curvature, Banach spaces with property (H) and Hilbert-Hadamard spaces. Moreover, for a group extension 1 → N → G → Q → 1 , we show that if N is coarsely embeddable into Hilbert space and Q is coarsely embeddable into an admissible Hilbert-Hadamard space, then the coarse Novikov conjecture holds for G. [ABSTRACT FROM AUTHOR]

We introduce a bivariant version of the Cuntz semigroup as equivalence classes of order zero maps generalizing the ordinary Cuntz semigroup. The theory has many features formally analogous to KK-theory including a composition product. We establish basic properties, like additivity, stability and continuity, and study categorical aspects in the setting of local C⁎-algebras. We determine the bivariant Cuntz semigroup for numerous examples such as when the second algebra is a Kirchberg algebra, and Cuntz homology for compact Hausdorff spaces which provides a complete invariant. Moreover, we establish identities when tensoring with strongly self-absorbing C⁎-algebras. Finally, we show how to use the bivariant Cuntz semigroup of the present work to classify unital and stably finite C⁎-algebras. [ABSTRACT FROM AUTHOR]

Abstract We revisit Špakula's uniform K -homology, construct the external product for it and use this to deduce homotopy invariance of uniform K -homology. We define uniform K -theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry. We further construct a cap product with uniform K -homology and prove Poincaré duality between uniform K -theory and uniform K -homology on spin c manifolds of bounded geometry. [ABSTRACT FROM AUTHOR]

We develop a generalization of quantitative K -theory, which we call controlled K -theory. It is powerful enough to study the K -theory of crossed product of C ⁎ -algebras by action of étale groupoids and discrete quantum groups. In this article, we will use it to study groupoids crossed products. We define controlled assembly maps, which factorize the Baum–Connes assembly maps, and define the controlled Baum–Connes conjecture. We relate the controlled conjecture for groupoids to the classical conjecture, and to the coarse Baum–Connes conjecture. This allows to give applications to Coarse Geometry. In particular, we can prove that the maximal version of the controlled coarse Baum–Connes conjecture is satisfied for a coarse space which admits a fibred coarse embedding, which is a stronger version of a result of M. Finn-Sell. [ABSTRACT FROM AUTHOR]

Quantitative (or controlled) K -theory for C ⁎ -algebras was used by Guoliang Yu in his work on the Novikov conjecture, and later developed more formally by Yu together with Hervé Oyono-Oyono. In this paper, we extend their work by developing a framework of quantitative K -theory for the class of algebras of bounded linear operators on subquotients (i.e., subspaces of quotients) of L p spaces. We also prove the existence of a controlled Mayer–Vietoris sequence in this framework. [ABSTRACT FROM AUTHOR]

Let ι : F 0 → F 1 be a suitably oriented inclusion of foliations over a manifold M , then we extend the construction of the lower shriek maps given by Hilsum and Skandalis to adiabatic deformation groupoid C*-algebras: we construct an asymptotic morphism (ι a d [ 0 , 1)) ! ∈ E n (C ⁎ (G a d [ 0 , 1)) , C ⁎ (G a d [ 0 , 1))) , where G and H are the monodromy groupoids associated with F 0 and F 1 respectively. Furthermore, we prove an interior Kasparov product formula for foliated ϱ -classes associated with longitudinal metrics of positive scalar curvature in the case of Riemannian foliated bundles. [ABSTRACT FROM AUTHOR]

In this paper, we show that for unital, separable C*-algebras of stable rank one and real rank zero, the unitary Cuntz semigroup functor and the functor K ⁎ are natural equivalent. Then we introduce a refinement of the unitary Cuntz semigroup, say the total Cuntz semigroup, which is a new invariant for separable C ⁎ -algebras of stable rank one, is a well-defined continuous functor from the category of C ⁎ -algebras of stable rank one to the category Cu _. We prove that this new functor and the functor K _ are naturally equivalent for unital, separable, K-pure C ⁎ -algebras. Therefore, the total Cuntz semigroup is a complete invariant for a large class of C ⁎ -algebras of real rank zero. [ABSTRACT FROM AUTHOR]

Connectivity is a homotopy invariant property of separable C ⁎ -algebras which has three notable consequences: absence of nontrivial projections, quasidiagonality and a more geometric realization of KK-theory for nuclear C ⁎ -algebras using asymptotic morphisms. The purpose of this paper is to further explore the class of connective C ⁎ -algebras. We give new characterizations of connectivity for exact and for nuclear separable C ⁎ -algebras and show that an extension of connective separable nuclear C ⁎ -algebras is connective. We establish connectivity or lack of connectivity for C ⁎ -algebras associated to certain classes of groups: virtually abelian groups, linear connected nilpotent Lie groups and linear connected semisimple Lie groups. [ABSTRACT FROM AUTHOR]

We use K -theory to prove an isomorphism theorem for a large class of generalised Bunce–Deddens algebras constructed by Kribs and Solel from a directed graph E and a sequence ω of positive integers. In particular, we compute the torsion-free component of the K 0 -group for a class of generalised Bunce–Deddens algebras to show that supernatural numbers are a complete invariant for this class. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes p n , p n + 1 satisfying p n + 1 − p n ≥ c log ⁡ p n log 2 ⁡ p n log 4 ⁡ p n log 3 ⁡ p n , with c being a fixed positive constant, for which the interval ( p n , p n + 1 ) contains the k -th power of a prime number. [ABSTRACT FROM AUTHOR]

In this paper, we study the continuous Rokhlin property of C ⁎ -dynamical systems using techniques of equivariant KK-theory and quantum group theory. In particular, we determine the KK-equivalence class and give a classification of Kirchberg G -algebras when the G is a compact Lie group with the Hodgkin condition. [ABSTRACT FROM AUTHOR]

We consider the general properties of bounded approximate units in non-self-adjoint operator algebras. Such algebras arise naturally from the differential structure of spectral triples and unbounded Kasparov modules. Our results allow us to present a unified approach to characterising completeness of spectral metric spaces, existence of connections on modules, self-adjointness and regularity of induced operators on tensor product C ⁎ -modules and the lifting of Kasparov products to the unbounded category. In particular, we prove novel existence results for quasicentral approximate units in non-self-adjoint operator algebras, allowing us to strengthen Kasparov's technical theorem and extend it to this realm. Finally, 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. [ABSTRACT FROM AUTHOR]

Counterexamples to classification of purely infinite, nuclear, separable C ⁎ -algebras (in the ideal-related bootstrap class) and with primitive ideal space X using ideal-related K-theory occur for infinitely many finite primitive ideal spaces X , the smallest of which being six spaces with four points. All constructed counterexamples for spaces with at least five points are based on the counterexamples constructed for these six four-point spaces. With real rank zero added to the assumptions imposed on the C ⁎ -algebras, ideal-related K-theory is known to be strongly complete for four of these six spaces. In this article, we close the two remaining cases: the pseudo-circle and the diamond space. We show that ideal-related K-theory is strongly complete for real rank zero, purely infinite, nuclear, separable C ⁎ -algebras that have the pseudo-circle as primitive ideal space. In the opposite direction, we construct a Cuntz–Krieger algebra with the diamond space as its primitive ideal space for which an automorphism on ideal-related K-theory does not lift. [ABSTRACT FROM AUTHOR]

We characterise quasidiagonality of the C ⁎ -algebra of a cofinal k -graph in terms of an algebraic condition involving the coordinate matrices of the graph. This result covers all simple k -graph C ⁎ -algebras. In the special case of cofinal 2-graphs we further prove that AF-embeddability, quasidiagonality and stable finiteness of the 2-graph algebra are all equivalent. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Toeplitz algebra associated to actions of Ore semigroups on C ⁎ -algebras. In particular, we consider injective and surjective actions of such semigroups. We use the theory of groupoid dynamical systems to represent the Toeplitz algebra as a groupoid crossed product. We also discuss the K-theory of the Toeplitz algebra in some examples. For instance, we show that for the semigroup of positive matrices, the K-theory of the associated Toeplitz algebra vanishes. [ABSTRACT FROM AUTHOR]

It was recently shown that each C*-algebra generated by a faithful irreducible representation of a finitely generated, torsion free nilpotent group is classified by its ordered K-theory. For the three step nilpotent group U T ( 4 , Z ) we calculate the ordered K-theory of each C*-algebra generated by a faithful irreducible representation of U T ( 4 , Z ) and see that they are all simple A T algebras. We also point out that there are many simple non - A T algebras generated by irreducible representations of nilpotent groups. [ABSTRACT FROM AUTHOR]

In the first part of the paper, we develop a theory of crossed products of a C ⁎ -algebra A by an arbitrary (not necessarily extendible) endomorphism α : A → A . We consider relative crossed products C ⁎ ( A , α ; J ) where J is an ideal in A , and describe up to Morita–Rieffel equivalence all gauge-invariant ideals in C ⁎ ( A , α ; J ) and give six term exact sequences determining their K -theory. We also obtain certain criteria implying that all ideals in C ⁎ ( A , α ; J ) are gauge-invariant, and that C ⁎ ( A , α ; J ) is purely infinite. In the second part, we consider a situation where A is a C 0 ( X ) -algebra and α is such that α ( f a ) = Φ ( f ) α ( a ) , a ∈ A , f ∈ C 0 ( X ) where Φ is an endomorphism of C 0 ( X ) . Pictorially speaking, α is a mixture of a topological dynamical system ( X , φ ) dual to ( C 0 ( X ) , Φ ) and a continuous field of homomorphisms α x between the fibers A ( x ) , x ∈ X , of the corresponding C ⁎ -bundle. For systems described above, we establish efficient conditions for the uniqueness property, gauge-invariance of all ideals, and pure infiniteness of C ⁎ ( A , α ; J ) . We apply these results to the case when X = Prim ( A ) is a Hausdorff space. In particular, if the associated C ⁎ -bundle is trivial, we obtain formulas for K -groups of all ideals in C ⁎ ( A , α ; J ) . In this way, we constitute a large class of crossed products whose ideal structure and K -theory is completely described in terms of ( X , φ , { α x } x ∈ X ; Y ) where Y is a closed subset of X . [ABSTRACT FROM AUTHOR]

We study the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal Λ, and the K -theory of an associated twisted groupoid algebra. In particular, we construct a finitely generated projective module over this algebra, and any multiwindow Gabor frame for Λ can be used to construct an idempotent representing this module in K -theory. For lattice subsets in dimension two, this allows us to prove a twisted version of Bellissard's gap labeling theorem. [ABSTRACT FROM AUTHOR]

Suppose A is a C ⁎ -algebra and H is a C ⁎ -correspondence over A . If H is regular in the sense that the left action of A is faithful and is given by compact operators, then we compute the K -theory of O A ( H ) ⋊ T where the action is the usual gauge action. The case where A is an AF-algebra is carefully analyzed. In particular, if A is AF, we show O A ( H ) ⋊ T is AF. Combining this with Takai duality and an AF-embedding theorem of N. Brown, we show the conditions AF-embeddability, quasidiagonality, and stable finiteness are equivalent for O A ( H ) . If H is also assumed to be regular, these finiteness conditions can be characterized in terms of the ordered K -theory of A . [ABSTRACT FROM AUTHOR]

A separated graph is a pair ( E , C ) consisting of a directed graph E and a set C = ⨆ v ∈ E 0 C v , where each C v is a partition of the set of edges whose terminal vertex is v . Given a separated graph ( E , C ) , such that all the sets X ∈ C are finite, the K-theory of the graph C ⁎ -algebra C ⁎ ( E , C ) is known to be determined by the kernel and the cokernel of a certain map, denoted by 1 C − A ( E , C ) , from Z ( C ) to Z ( E 0 ) . In this paper, we compute the K-theory of the tame graph C ⁎ -algebra O ( E , C ) associated to ( E , C ) , which has been recently introduced by the authors. Letting π denote the natural surjective homomorphism from C ⁎ ( E , C ) onto O ( E , C ) , we show that K 1 ( π ) is a group isomorphism, and that K 0 ( π ) is a split monomorphism, whose cokernel is a torsion-free abelian group. We also prove that this cokernel is a free abelian group when the graph E is finite, and determine its generators in terms of a sequence of separated graphs { ( E n , C n ) } n = 1 ∞ naturally attached to ( E , C ) . On the way to showing our main results, we obtain an explicit description of a connecting map arising in a six-term exact sequence computing the K-theory of an amalgamated free product, and we also exhibit an explicit isomorphism between ker ⁡ ( 1 C − A ( E , C ) ) and K 1 ( C ⁎ ( E , C ) ) . [ABSTRACT FROM AUTHOR]

We present a new approach to C ⁎ -algebras recently constructed in the context of modular index theory by Carey, Phillips, Putnam and Rennie. It turns out that their constructions can be identified with full corners of ideals in semigroup C ⁎ -algebras. This new point of view leads to a systematic analysis of these algebras and their canonical maximal abelian subalgebras. [ABSTRACT FROM AUTHOR]

In this paper, we prove the coarse Novikov conjecture for metric spaces with bounded geometry which are coarsely embeddable into Banach spaces with a geometric condition, called Property (H), introduced by G. Kasparov and G. Yu. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show how zeta functions and excision in cyclic cohomology may be combined to obtain index theorems. In the first part, we obtain an index formula for “abstract elliptic pseudodifferential operators” associated to spectral triples, in the spirit of the one of Connes and Moscovici. This formula is notably well adapted when the zeta function has multiple poles. The second part is devoted to give a concrete realization of this formula by deriving an index theorem on the simple, but significant example of Heisenberg elliptic operators on a trivial foliation, which are in general not elliptic but hypoelliptic. The formula obtained is an extension of an index formula due to Fedosov. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the KK-theory of (generalized) dimension drop interval algebras (with possibly different dimension drops at the endpoints), with special emphasis on the problem which KK-class is representable by a ⁎-homomorphism between two such C*-algebras (allowing the tensor product with a matrix algebra for the codomain algebra). This lifting problem makes sense in its own right in KK-theory, and also has application to the classification of C*-algebras which are inductive limits of these building blocks. It turns out that when the dimension drops at the two endpoints are different, there exist KK-elements which preserve the order structure defined by M. Dadarlat and T.A. Loring in [4] on the mod p K-theory, but cannot be lifted to a ⁎-homomorphism. This is different from the equal dimension drop case, as shown by S. Eilers in [6] . [ABSTRACT FROM AUTHOR]

Abstract: We study the interplay of -dynamics and K-theory. Notions of chain recurrence for transformations groups and MF actions for non-commutative -dynamical systems are translated into K-theoretical language, where purely algebraic conditions are shown to be necessary and sufficient for a reduced crossed product to admit norm microstates. We are particularly interested in actions of free groups on AF algebras, in which case we prove that a K-theoretic coboundary condition determines whether or not the reduced crossed product is a matricial field (MF) algebra. One upshot is the equivalence of stable finiteness and being MF for these reduced crossed product algebras. [Copyright &y& Elsevier]

Abstract: We consider the properties of weak cancellation, -surjectivity, good index theory, and -injectivity, for the class of extremally rich -algebras, and for the smaller class of isometrically rich -algebras. We establish all four properties for isometrically rich -algebras and for extremally rich -algebras that are either purely infinite or of real rank zero, -injectivity in the real rank zero case following from a prior result of H. Lin. We also show that weak cancellation implies the other properties for extremally rich -algebras and that the class of extremally rich -algebras with weak cancellation is closed under extensions. Moreover, we consider analogous properties which replace the group with the extremal K-set as well as two versions of -surjectivity. [Copyright &y& Elsevier]

Abstract: We augment Restorff's classification of purely infinite Cuntz–Krieger algebras by describing the range of his invariant on purely infinite Cuntz–Krieger algebras. We also describe its range on purely infinite graph -algebras with finitely many ideals, and provide ‘unital’ range results for purely infinite Cuntz–Krieger algebras and unital purely infinite graph -algebras. [Copyright &y& Elsevier]

Abstract: Using an equivariant version of Connesʼ Thom isomorphism, we prove that equivariant K-theory is invariant under strict deformation quantization for a compact Lie group action. [Copyright &y& Elsevier]

Abstract: The Serre–Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. In this context it states that the category of Hilbert bundles of bounded geometry is equivalent to the category of operator ⁎-modules over the operator ⁎-algebra of continuously differentiable functions which vanish at infinity. Operator ⁎-modules are generalizations of Hilbert -modules where the category of -algebras has been replaced by a more flexible category of involutive algebras of bounded operators: The operator ⁎-algebras. Operator ⁎-modules play an important role in the study of the unbounded Kasparov product. [Copyright &y& Elsevier]

Abstract: Assume that we are given a coaction δ of a locally compact group G on a -algebra A and a -valued Borel 2-cocycle ω on G. Motivated by the approach of Kasprzak to Rieffelʼs deformation we define a deformation of A. Among other properties of we show that is canonically isomorphic to . This, together with a slight extension of a result of Echterhoff et al., implies that for groups satisfying the Baum–Connes conjecture the K-theory of remains invariant under homotopies of ω. [Copyright &y& Elsevier]

Abstract: Let X be a unit interval or a unit circle and let B be a -unital, purely infinite, simple -algebra such that its multiplier algebra has real rank zero. Then we determine necessary and sufficient conditions for a projection in the corona algebra of to be liftable to a projection in the multiplier algebra. This generalizes a result proved by L. Brown and the author, Brown and Lee (2012) [3]. The main technical tools are divided into two parts. The first part is borrowed from the authorʼs result, Lee (2011) [14, Theorem 3.3]. The second part is a proposition showing that we can produce a sub-projection, with an arbitrary rank which is prescribed as K-theoretical data, of a projection or a co-projection in the multiplier algebra of under a suitable “infinite rank and co-rank” condition. [Copyright &y& Elsevier]

Abstract: We construct HNN extensions of discrete quantum groups, we study their representation theory and we show that an HNN extension of amenable discrete quantum groups is K-amenable. [Copyright &y& Elsevier]

Abstract: Using an appropriate notion of locally convex Kasparov modules, we show how to induce isomorphisms under a large class of functors on the category of locally convex algebras; examples are obtained from spectral triples. Our considerations are based on the action of algebraic K-theory on these functors, and involve compatibility properties of the induction process with this action, and with Kasparov-type products. This is based on an appropriate interpretation of the Connes–Skandalis connection formalism. As an application, we prove Bott periodicity and a Thom isomorphism for algebras of Schwartz functions. As a special case, this applies to the theories kk for locally convex algebras considered by Cuntz. [Copyright &y& Elsevier]

Abstract: We define E-theory for separable -algebras over second countable topological spaces and establish its basic properties. This includes an approximation theorem that relates the E-theory over a general space to the E-theories over finite approximations to this space. We obtain effective criteria for determining the invertibility of E-theory elements over possibly infinite-dimensional spaces. Furthermore, we prove a Universal Multicoefficient Theorem for -algebras over totally disconnected metrisable compact spaces. [Copyright &y& Elsevier]

Abstract: Dye proved that the discrete unitary group in a factor determines the algebraic type of the factor. We show that if the unitary groups of two simple unital AH-algebras of slow dimension growth and of real rank zero are isomorphic as abstract groups, then their -ordered groups are isomorphic. Also, using Gong and Dadarlatʼs classification theorem, we prove that such -algebras are isomorphic if and only if their unitary groups are isomorphic as topological groups. For simple, unital purely infinite -algebras, we show that two unital Kirchberg algebras are ⁎-isomorphic if and only if their unitary groups are isomorphic as abstract groups. [Copyright &y& Elsevier]

Abstract: The construction of the -algebra associated to a directed graph E is extended to incorporate a family C consisting of partitions of the sets of edges emanating from the vertices of E. These -algebras are analyzed in terms of their ideal theory and K-theory, mainly in the case of partitions by finite sets. The groups and are completely described via a map built from an adjacency matrix associated to . One application determines the K-theory of the -algebras , confirming a conjecture of McClanahan. A reduced -algebra is also introduced and studied. A key tool in its construction is the existence of canonical faithful conditional expectations from the -algebra of any row-finite graph to the -subalgebra generated by its vertices. Differences between and , such as simplicity versus non-simplicity, are exhibited in various examples, related to some algebras studied by McClanahan. [Copyright &y& Elsevier]

Abstract: We construct a family of purely infinite -algebras, for that are classified by their K-groups. There is an action of the circle with a unique KMS state ψ on each . For , , with its usual action and KMS state. For , rational in lowest terms, () with UHF fixed point algebra of type . For any , for infinitely many λ with distinct KMS states and UHF fixed-point algebras. For any , . For λ irrational the fixed point algebras, are NOT AF and the are usually NOT Cuntz algebras. For λ transcendental, , so that is Cuntz'' [Cuntz (2008) ]. If λ and are both algebraic integers, the only which appear are those for which . For each λ, the representation of defined by the KMS state ψ generates a type factor. These algebras fit into the framework of modular index theory/twisted cyclic theory of Carey et al. (2010) , Carey et al. (2009) , Carey et al. (in press) . [Copyright &y& Elsevier]

Abstract: We prolong the list of -algebras which have the property that all extensions by a stable -algebra are semi-invertible. In particular, it is shown to include group -algebras, both reduced and full, of certain amalgamated free products of amenable groups, as well as all free products of nuclear -algebras with amalgamation over a common nuclear -subalgebra. [ABSTRACT FROM AUTHOR]

Abstract: We describe a method to compute the K-theory of the C⁎-algebra arising from the stable equivalence relation in the Smale space associated to a substitution tiling, and give detailed computations for one- and two-dimensional examples. We prove that for one-dimensional tilings the group is always torsion free and give an example of a two-dimensional tiling such that has torsion. [ABSTRACT FROM AUTHOR]

Abstract: We classify unital monomorphisms into certain simple -stable -algebras up to approximate unitary equivalence. The domain algebra C is allowed to be any unital separable commutative -algebra, or any unital simple separable nuclear -stable -algebra satisfying the UCT such that is of tracial rank zero for a UHF algebra B. The target algebra A is allowed to be any unital simple separable -stable -algebra such that has tracial rank zero for a UHF algebra B, or any unital simple separable exact -stable -algebra whose projections separate traces and whose extremal traces are finitely many. [Copyright &y& Elsevier]

Abstract: In this paper we generalize the notion of essential codimension of Brown, Douglas, and Fillmore using KK-theory and prove a result which asserts that there is a unitary of the form ‘identity+compact’ which gives the unitary equivalence of two projections if the ‘essential codimension’ of two projections vanishes for certain -algebras employing the proper asymptotic unitary equivalence of KK-theory found by M. Dadarlat and S. Eilers. We also apply our result to study the projections in the corona algebra of where X is , , , and . [Copyright &y& Elsevier]

Abstract: In this paper, we prove a higher Lefschetz formula for foliations in the presence of a closed Haefliger current. To this end, we associate with such a current an equivariant cyclic cohomology class of Connes''-algebra of the foliation, and compute its pairing with the localized equivariant K-theory in terms of local contributions near the fixed points. [Copyright &y& Elsevier]

Abstract: Let A be an AH algebra, that is, A is the inductive limit -algebra of with , where are compact metric spaces, and are positive integers, and are projections. Suppose that A has the ideal property: each closed two-sided ideal of A is generated by the projections inside the ideal, as a closed two-sided ideal. Suppose that . (This condition can be relaxed to a certain condition called very slow dimension growth.) In this article, we prove that if we further assume that is torsion free, then A is an approximate circle algebra (or an algebra), that is, A can be written as the inductive limit of where . One of the main technical results of this article, called the decomposition theorem, is proved for the general case, i.e., without the assumption that is torsion free. This decomposition theorem will play an essential role in the proof of a general reduction theorem, where the condition that is torsion free is dropped, in the subsequent paper Gong et al. (preprint) —of course, in that case, in addition to space , we will also need the spaces , , and , as in Gong (2002) . [Copyright &y& Elsevier]

Abstract: We extend the notion of Poincaré duality in KK-theory to the setting of quantum group actions. An important ingredient in our approach is the replacement of ordinary tensor products by braided tensor products. Along the way we discuss general properties of equivariant KK-theory for locally compact quantum groups, including the construction of exterior products. As an example, we prove that the standard Podleś sphere is equivariantly Poincaré dual to itself. [Copyright &y& Elsevier]