Search

In this paper, for 1 < p < ∞, the authors show that the coarse ℓp-Novikov conjecture holds for metric spaces with bounded geometry which are coarsely embeddable into a Banach space with Kasparov-Yu's Property (H). [ABSTRACT FROM AUTHOR]

In this paper, the authors consider the range of a certain class of ASH algebras in [An, Q., Elliott, G. A., Li, Z. and Liu, Z., The classification of certain ASH C*-algebras of real rank zero, J. Topol. Anal., 14(1), 2022, 183–202], which is under the scheme of the Elliott program in the setting of real rank zero C*-algebras. As a reduction theorem, they prove that all these ASH algebras are still the AD algebras studied in [Dadarlat, M. and Loring, T. A., Classifying C*-algebras via ordered, mod-p K-theory, Math. Ann., 305, 1996, 601–616]. [ABSTRACT FROM AUTHOR]

In this paper, we give a classification of real rank zero C*-algebras that can be expressed as inductive limits of a sequence of a subclass of Elliott-Thomsen algebras C . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the propagation of topological currents along magnetic interfaces (also known as magnetic walls) of a two-dimensional material. We consider tight-binding magnetic models associated to generic magnetic multi-interfaces and describe the K -theoretical setting in which a bulk-interface duality can be derived. Then, the (trivial) case of a localized magnetic field and the (non trivial) case of the Iwatsuka magnetic field are considered in full detail. This is a pedagogical preparatory work that aims to anticipate the study of more complicated multi-interface magnetic systems. [ABSTRACT FROM AUTHOR]

The main result in this paper is a fixed point formula for equivariant indices of elliptic differential operators, for proper actions by connected semisimple Lie groups on possibly noncompact manifolds, with compact quotients. For compact groups and manifolds, this reduces to the Atiyah-Segal-Singer fixed point formula. Other special cases include an index theorem by Connes and Moscovici for homogeneous spaces, and an earlier index theorem by the second author, both in cases where the group acting is connected and semisimple. As an application of this fixed point formula, we give a new proof of Harish-Chandra's character formula for discrete series representations. [ABSTRACT FROM AUTHOR]

Elliott dimension drop interval algebra is an important class among all C*-algebras in the classification theory. Especially, they are building stones of A H D algebra and the latter contains all AH algebras with the ideal property of no dimension growth. In this paper, the authors will show two decomposition theorems related to the Elliott dimension drop interval algebra. Their results are key steps in classifying all AH algebras with the ideal property of no dimension growth. [ABSTRACT FROM AUTHOR]

A crossed product functor is said to be injective if it takes injective morphisms to injective morphisms. In this paper we show that every locally compact group $G$ admits a maximal injective crossed product $A\mapsto A\rtimes _{\text{inj}}G$. Moreover, we give an explicit construction of this functor that depends only on the maximal crossed product and the existence of $G$ -injective $C^{\ast }$ -algebras; this is a sort of 'dual' result to the construction of the minimal exact crossed product functor, the latter having been studied for its relationship to the Baum–Connes conjecture. It turns out that $\rtimes _{\text{inj}}$ has interesting connections to exactness, the local lifting property, amenable traces, and the weak expectation property. [ABSTRACT FROM AUTHOR]

Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Using her construction, we show that the associated stable $C^{\ast }$ -algebra is the stationary inductive limit of a $C^{\ast }$ -stable Fell algebra that has a compact spectrum and trivial Dixmier–Douady invariant. This result applies in particular to Williams solenoids along with other examples. Beyond the structural implications of this inductive limit, one can use this result to, in principle, compute the $K$ -theory of the stable $C^{\ast }$ -algebra. A specific one-dimensional Smale space (the $aab/ab$ -solenoid) is considered as an illustrative running example throughout. [ABSTRACT FROM AUTHOR]

Persistence approximation property was introduced by Hervé Oyono-Oyono and Guoliang Yu. This property provides a geometric obstruction to Baum-Connes conjecture. In this paper, the authors mainly discuss the persistence approximation property for maximal Roe algebras. They show that persistence approximation property of maximal Roe algebras follows from maximal coarse Baum-Connes conjecture. In particular, let X be a discrete metric space with bounded geometry, assume that X admits a fibred coarse embedding into Hilbert space and X is coarsely uniformly contractible, then C*max(X) has persistence approximation property. The authors also give an application of the quantitative K-theory to the maximal coarse Baum-Connes conjecture. [ABSTRACT FROM AUTHOR]

In the setting of non-type II 1 representations, we propose a definition of deformed Fredholm module [ D T | D T | - 1 , · ] T for a modular spectral triple T , where D T is the deformed Dirac operator. D T is assumed to be invertible for the sake of simplicity, and its domain is an “essential” operator system E T . According to such a definition, we obtain [ D T | D T | - 1 , · ] T = | D T | - 1 d T (·) + d T (·) | D T | - 1 , where d T is the deformed derivation associated to D T . Since the “quantum differential” 1 / | D T | appears in a symmetric position, such a definition of Fredholm module differs from the usual one even in the undeformed case, that is in the tracial case. Therefore, it seems to be more suitable for the investigation of noncommutative manifolds in which the nontrivial modular structure might play a crucial role. We show that all models in Fidaleo and Suriano (J Funct Anal 275:1484–1531, 2018) of non-type II 1 representations of noncommutative 2-tori indeed provide modular spectral triples, and in addition deformed Fredholm modules according to the definition proposed in the present paper. Since the detailed knowledge of the spectrum of the Dirac operator plays a fundamental role in spectral geometry, we provide a characterisation of eigenvalues and eigenvectors of the deformed Dirac operator D T in terms of the periodic solutions of a particular class of eigenvalue Hill equations. [ABSTRACT FROM AUTHOR]

An n-parameter Fibonacci AF-algebra A is any stationary AF C ∗ -algebra with incidence matrix K of the form: K = m 1 m 2 ⋯ m n - 1 m n 1 0 ⋯ 0 0 0 1 ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ 1 0 , where n ≥ 2 and the m i are nonnegative integers with m n ≠ 0 . We write A = A K . The main result of the present paper is that A K is isomorphic to its transpose AF-algebra, A K t . [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of generalized inductive limit of C*-algebras with coherent maps being completely positive contractions of order zero: the nuclear dimension of generalized inductive limit of C*-algebras with finite nuclear dimension is finite; the generalized inductive limits of C*-algebras with the α-comparison property also have the α-comparison property. [ABSTRACT FROM AUTHOR]

In this paper, we compute $$K$$-groups $$\{K_{n}(C^{*}(x))\}_{n=0}^{\infty }$$ of the $$C^{*}$$-subalgebra $$C^{*}(x)$$ of $$B(H),$$ generated by a single operator $$x,$$ where $$H$$ is a separable infinite dimensional Hilbert space, and $$B(H)$$ is the operator algebra consisting of all (bounded linear) operators on $$H.$$ These computations not only provide nice examples in $$K$$-theory, but also characterize-and-classify projections in a $$C^{*}$$-algebra generated by a single operator. The main result of this paper shows that: the $$K$$-groups of $$C^{*}(x)$$ are completely characterized by those of $$C^{*}(q),$$ where $$q$$ is the positive-operator part of $$x$$ in the polar decomposition of $$x.$$ [ABSTRACT FROM AUTHOR]

Abstract: The goal of this paper is to study when uniform Roe algebras have certain C ∗‐algebraic properties in terms of the underlying space: in particular, we study properties like having stable rank one or real rank zero that are thought of as low dimensional, and connect these to low dimensionality of the underlying space in the sense of the asymptotic dimension of Gromov. Some of these results (for example, on stable rank one, cancellation, strong quasidiagonality, and finite decomposition rank) give definitive characterizations, while others (on real rank zero) are only partial and leave a lot open. We also establish results about K‐theory, showing that all K 0‐classes come from the inclusion of the canonical Cartan in low‐dimensional cases, but not in general; in particular, our K‐theoretic results answer a question of Elliott and Sierakowski about vanishing of K 0 groups for uniform Roe algebras of non‐amenable groups. Along the way, we extend some results about paradoxicality, proper infiniteness of projections in uniform Roe algebras, and supramenability from groups to general metric spaces. These are ingredients needed for our K‐theoretic computations, but we also use them to give new characterizations of supramenability for metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, the authors construct a Φ-group for n submodules, which generalizes the classical K-theory and gives more information than the classical ones. This theory is related to the classification theory for indecomposable systems of n subspaces. [ABSTRACT FROM AUTHOR]

In this paper we study the $$K$$-theory of free quantum groups in the sense of Wang and Van Daele, more precisely, of free products of free unitary and free orthogonal quantum groups. We show that these quantum groups are $$K$$-amenable and establish an analogue of the Pimsner-Voiculescu exact sequence. As a consequence, we obtain in particular an explicit computation of the $$K$$-theory of free quantum groups. Our approach relies on a generalization of methods from the Baum-Connes conjecture to the framework of discrete quantum groups. This is based on the categorical reformulation of the Baum-Connes conjecture developed by Meyer and Nest. As a main result we show that free quantum groups have a $$\gamma $$-element and that $$\gamma = 1$$. As an important ingredient in the proof we adapt the Dirac-dual Dirac method for groups acting on trees to the quantum case. We use this to extend some permanence properties of the Baum-Connes conjecture to our setting. [ABSTRACT FROM AUTHOR]

In this paper, we give a class of C*-algebras with non-stable K-group property which include the example non-simple tracial topological rank zero and stable rank two C*-algebra given by Lin and Osaka. [ABSTRACT FROM AUTHOR]

Let A be a unital separable nuclear C*-algebra which belongs to the bootstrap category N and B be a separable stable C*-algebra. In this paper, we consider the group Ext( A, B) consisting of the unitary equivalence classes of unital extensions τ: A→Q( B). The relation between Ext( A, B) and Ext( A, B) is established. Using this relation, we show the half-exactness of Ext(·, B) and the (UCT) for Ext( A, B). Furthermore, under certain conditions, we obtain the half-exactness and Bott periodicity of Ext( A, ·). [ABSTRACT FROM AUTHOR]

In this paper, we quantize universal gauge groups such as SU(∞), as well as their homogeneous spaces, in the σ- C*-algebra setting. More precisely, we propose concise definitions of σ- C*-quantum groups and σ- C*-quantum homogeneous spaces and explain these concepts here. At the same time, we put these definitions in the mathematical context of countably compactly generated spaces as well as C*-compact quantum groups and homogeneous spaces. We also study the representable K-theory of these spaces and compute these groups for the quantum homogeneous spaces associated to the quantum version of the universal gauge group SU(∞). [ABSTRACT FROM AUTHOR]

Let X be a noncompact discrete metric space with bounded geometry. Associated with X are two C*-algebras, the so-called uniform Roe algebra B*( X) and coarse Roe algebra C*( X), which arose from the index theory on noncompact complete Riemannian manifolds. In this paper, we describe the quasidiagonality of B*( X) and C*( X) in terms of coarse geometric invariants. Some necessary and sufficient conditions are given, which involve the Fredholm index and coarse connectedness of metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, we will discuss some properties of biprojection-commutative elements which are relevant to the classification of certain infinite C*-algebras, and define an important invariant s( A) of C*-algebra A as well as give some basic properties with regard to s( A). Moreover we prove that the invariant s( A) has continuity. [ABSTRACT FROM AUTHOR]

Let $$ \mathcal{H} $$ olenote a complex separable Hilbert space and $$ \mathcal{L}(\mathcal{H}) $$ denote the collection of bounded linear operators on $$ \mathcal{H} $$ . An operator T ∈ $$ \mathcal{L}(\mathcal{H}) $$ is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an “approximate inverse” of the Riesz decomposition theorem. In the paper, we give a more precise characterization of approximate invariants of strongly irreducible operators. The main result is: For any T ∈ $$ \mathcal{L}(\mathcal{H}) $$ with connected spectrum and ε > 0, there exists a strongly irreducible operator A, such that $$ \parallel A - T\parallel < \varepsilon , V(\mathcal{A}'(A)) \cong \mathbb{N}, K_0 (\mathcal{A}'(A)) \cong \mathbb{Z} $$ , and $$ \mathcal{A}'(A) $$ /rad $$ \mathcal{A}'(A) $$ is commutative, where $$ \mathcal{A}'(A) $$ denotes the commutant of A and rad $$ \mathcal{A}'(A) $$ denotes the Jacobson radical of $$ \mathcal{A}'(A) $$ . The research is inspired by the recent similarity classification technique of Cowen-Douglas operators of Jiang Chunlan. [ABSTRACT FROM AUTHOR]

In the present paper, a problem of Ioana Mihaila is negatively answered on the invertibility of composition operators on Riemann surfaces, and it is proved that the composition operator C p is Fredholm if and only if it is invertible if and only if p is invertible for some special cases. In addition, the Toeplitz operators on $$ \Lambda _{2,a}^1 (M) $$ for Riemann surface M are defined and some properties of these operators are discussed. [ABSTRACT FROM AUTHOR]

In the present paper, it is proved that the K 0-group of a Toeplitz algebra on any connected domain is always isomorphic to the K 0-group of the relative continuous function algebra. In addition, the cohomotopy groups of essential boundaries of some connected domains are computed, and the K 0-groups of the continuous function algebras on these domains are also computed. [ABSTRACT FROM AUTHOR]

We discuss various aspects of noncommutative geometry of smooth subalgebras of Bunce-Deddens-Toeplitz algebras. [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]

We show that the following properties of the C*-algebras in a class P are inherited by simple unital C*-algebras in the class of asymptotically tracially in P : (1) n-comparison, (2) α-comparison (1 ≤ α < ∞). [ABSTRACT FROM AUTHOR]

We show that if a countable discrete group acts properly and isometrically on a spin manifold of bounded Riemannian geometry and uniformly positive scalar curvature, then, under a suitable condition on the group action, the maximal higher index of the Dirac operator vanishes in K-theory of the maximal equivariant Roe algebra. The group action is not assumed to be cocompact. A key step in the proof is to establish a functional calculus for the Dirac operator in the maximal equivariant uniform Roe algebra. This allows us to prove vanishing of the index of the Dirac operator in K-theory of this algebra, which in turn yields the result for the maximal higher index. [ABSTRACT FROM AUTHOR]

An error is identified and corrected in the construction of a non- Z -stable, stably projectionless, simple, nuclear C ⁎ -algebra carried out in a paper by the second author. [ABSTRACT FROM AUTHOR]

We give a proof of the index theorem of lattice Wilson–Dirac operators, which states that the index of a twisted Dirac operator on the standard torus is described in terms of the corresponding lattice Wilson–Dirac operator. Our proof is based on the higher index theory of almost flat vector bundles. [ABSTRACT FROM AUTHOR]

In this article, we present a trichotomy (a division into three classes) on Noetherian and Artinian C*-algebras and obtain some structural results about Noetherian (and/or Artinian) C*-algebras. We show that every Noetherian, purely infinite and σ-unital C*-algebra A is generated as a C*-ideal by a single projection. We show that if A is a purely infinite, nuclear, separable, Noetherian and Artinian C*-algebra, then A ≅ A ⊗ Z ≅ A ⊗ O ∞ . This is a partial extension of Kirchberg's O ∞ -absorption theorem and Kirchberg's exact embedding theorem. Finally, we show that each Noetherian AF-algebra has a full finite-dimensional C*-subalgebra. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we estimate the stable ranks of a Banach algebra in terms of the stable ranks of its quotient algebra and ideal under the assumption that the quotient map splits. As an application, several results about the Bass and the connected stable ranks of nest algebras are obtained. [Copyright &y& Elsevier]

Abstract: Penrose hyperbolic tilings are tilings of the hyperbolic plane which admit, up to affine transformations a finite number of prototiles. In this paper, we give a complete description of the -algebras and of the -theory for such tilings. Since the continuous hull of these tilings have no transversally invariant measure, these -algebras are traceless. Nevertheless, harmonic currents give rise to -cyclic cocycles and we discuss in this setting a higher-order version of the gap-labeling. [Copyright &y& Elsevier]

Abstract: In [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], we presented a -theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of the modular group of the Haar state. In contrast to the Cuntz algebras studied in [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], the computations are considerably more complex and interesting, because there are non-trivial ‘eta’ contributions to this index. [Copyright &y& Elsevier]

Abstract: The central result of this paper is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby generalising some fundamental results of Connes and Hochschild–Kostant–Rosenberg. The Connes–Chern character is also identified here with the twisted Chern character. [Copyright &y& Elsevier]

Abstract: In this paper we study the -algebras associated to continuous fields over locally compact metrizable zero-dimensional spaces whose fibers are Kirchberg -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. [Copyright &y& Elsevier]

Abstract: We study here the simple infinite-dimensional quotients of the group C *-algebras of two discrete6-dimensional nilpotent groups H 6,1 and H 6,2 as the higher-dimensional analogues of the irrational rotation algebras. P Milnes and S. Walters, jointly and individually, have studied the lower-dimensional cases in a series of papers, and also have started the study of some other 6-dimensional groups. For G = H 6,1 or H 6,2, we can determine the crossed product presentations for the simple quotients of C * (G), and matrix representations for those arising from non-faithful representations of the groups. The isomorphism classifications of these quotients are obtained using K-theoretic tools, namely, the K-groups and the range of trace on K 0. This marks the first use of K-theory in the classification of quotients for 6-dimensional groups. [Copyright &y& Elsevier]

In this paper we estimate the stable rank and connected stable rank of C&ast;-algebras under a technical assumption on their spectrums. Using this result we obtain some stable rank formulas for type I C&ast;-algebras. In particular, we obtain a product formula for the stable rank of type I C&ast;-algebras. [Copyright &y& Elsevier]