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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]

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]

We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of $C^{\ast }$ -algebras, including: $D$ -absorbing $C^{\ast }$ -algebras, where $D$ is a strongly self-absorbing $C^{\ast }$ -algebra; stable $C^{\ast }$ -algebras; $C^{\ast }$ -algebras with finite nuclear dimension (or decomposition rank); $C^{\ast }$ -algebras with finite stable rank (or real rank); and $C^{\ast }$ -algebras whose $K$ -theory is either trivial, rational, or $n$ -divisible for $n\in \mathbb{N}$. The combination of nuclearity and the universal coefficient theorem (UCT) is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous $C(X)$ -algebra with fibers that are stably isomorphic to the underlying algebra. The space $X$ is computed in some cases of interest, and we use its description to construct a $\mathbb{Z}_{2}$ -action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree. [ABSTRACT FROM AUTHOR]

Since their inception in the 1930s by von Neumann, operator algebras have been used to shed light on many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two has been sought since their emergence in the late 1960s. We connect these seemingly separate types of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^{*}$ -algebras with additional $C^{*}$ -algebraic structure. Our approach naturally applies to algebras arising from $C^{*}$ -correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs. [ABSTRACT FROM AUTHOR]

We construct a Baum–Connes assembly map localised at the unit element of a discrete group $\Gamma$. This morphism, called $\mu _\tau$ , is defined in $KK$ -theory with coefficients in $\mathbb {R}$ by means of the action of the idempotent $[\tau ]\in KK_{\mathbin {{\mathbb {R}}}}^\Gamma (\mathbb {C},\mathbb {C})$ canonically associated to the group trace of $\Gamma$. We show that the corresponding $\tau$ -Baum–Connes conjecture is weaker than the classical version, but still implies the strong Novikov conjecture. The right-hand side of $\mu _\tau$ is functorial with respect to the group $\Gamma$. [ABSTRACT FROM AUTHOR]

We consider Lie groupoids of the form G (M , M 1) : = M 0 × M 0 ⊔ H × M 1 × M 1 ⇉ M , where M 0 = M \ M 1 and the isotropy subgroup H is an exponential Lie group of dimension equal to the codimension of the manifold M 1 in M. The existence of such Lie groupoids follows from integration of almost injective Lie algebroids by Claire Debord. They correspond to (the s-connected version of) the problem of the existence of a holonomy groupoid associated to the singular foliation whose leaves are the connnected components of M 1 and the connected components of M 0 . We study the Lie groupoid structure of these groupoids, and verify that they are amenable and Fredholm in the sense recently introduced by Carvalho, Nistor and Qiao. We compute explicitly the K-groups of these groupoid’s C ∗ -algebras, we obtain K 0 (C ∗ (G (M , M 1))) ≅ Z , K 1 (C ∗ (G (M , M 1))) ≅ Z for M 1 of odd codimension, and K 0 (C ∗ (G (M , M 1))) ≅ Z ⊕ Z , K 1 (C ∗ (G (M , M 1))) ≅ { 0 } for M 1 of even codimension. When M and M 1 are compact we obtain, as an application of our previous K-theory computations, that in the odd codimensional case every elliptic operator (in the groupoid pseudodifferential calculus) can be perturbed (up to stabilization by an identity operator) with a regularizing operator, such that the perturbed operator is Fredholm; and in the even case, given an elliptic operator there is a topological obstruction to satisfy the previous Fredholm perturbation property given by the Atiyah-Singer topological index of the restriction operator to M 1 . [ABSTRACT FROM AUTHOR]

We show that the following properties of C* -algebras in a class Ω are inherited by simple unital C*-algebras in class TAΩ: (1) m-comparison of positive elements, (2) strong tracial m-comparison of positive elements, and (3) tracial nuclear dimension at most m. As an application, every unital simple C*-algebra with tracial topological rank at most k has tracial nuclear dimension at most k. Also as an application, let A be an infinite-dimensional simple unital exact C*-algebra such that A has one of the above-listed properties. Suppose that α: G → Aut(A) is an action of a finite group G on A which has the tracial Rokhlin property. Then the crossed product C*-algebra C* (G, A, α) also has the property under consider also. [ABSTRACT FROM AUTHOR]

We compute the homology groups of transformation groupoids associated with odometers and show that certain $(\mathbb{Z}\rtimes \mathbb{Z}_{2})$ -odometers give rise to counterexamples to the HK conjecture, which relates the homology of an essentially principal, minimal, ample groupoid $G$ with the K-theory of $C_{r}^{\ast }(G)$. We also show that transformation groupoids of odometers satisfy the AH conjecture. [ABSTRACT FROM AUTHOR]

I show that quasidiagonality and AF embeddability are equivalent properties for traceless C ∗ -algebras and are characterised in terms of the primitive ideal space. For nuclear C ∗ -algebras the same characterisation determines when Connes and Higson's E-theory can be unsuspended. [ABSTRACT FROM AUTHOR]

We show that the following properties of the C*-algebras in a class Ω are inherited by simple unital C -algebras in the class TAΩ: (1) (m,n)-decomposable, (2) weakly (m,n)-divisible, (3) weak Riesz interpolation. As an application, let A be an infinite dimensional simple unital C*-algebra such that A has one of the above-listed properties. Suppose that α: G → Aut(A) is an action of a finite group G on A which has the tracial Rokhlin property. Then the crossed product C*-algebra C*(G, A, α) also has the property under consideration. [ABSTRACT FROM AUTHOR]

We show that Matui's HK conjecture holds for groupoids of unstable equivalence relations and their corresponding $C^{\ast }$ -algebras on one-dimensional solenoids. [ABSTRACT FROM AUTHOR]

The semigroups of unital extensions of separable C *-algebras come in two flavours: a strong and a weak version. By the unital Ext-groups, we mean the groups of invertible elements in these semigroups. We use the unital Ext-groups to obtain K -theoretic classification of both unital and non-unital extensions of C *-algebras, and in particular we obtain a complete K -theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras. [ABSTRACT FROM AUTHOR]

Let G be a unimodular type I second countable locally compact group and let Ĝ be its unitary dual. Motivated by a recent pseudo‐differential calculus, we develop a positive Berezin‐type quantization with operator‐valued symbols defined on Ĝ×G. [ABSTRACT FROM AUTHOR]

We introduce the concept of Roe C∗-algebra for a locally compact groupoid whose unit space is in general not compact, and that is equipped with an appropriate coarse structure and Haar system. Using Connes’ tangent groupoid method, we introduce an analytic index for an elliptic differential operator on a Lie groupoid equipped with additional metric structure, which takes values in the K-theory of the Roe C∗-algebra. We apply our theory to derive a Lichnerowicz type vanishing result for foliations on open manifolds. [ABSTRACT FROM AUTHOR]

Abstract: Shelstad's character identity is an equality between sums of characters of tempered representations in corresponding L‐packets of two real, semisimple, linear, algebraic groups that are inner forms to each other. We reconstruct this character identity in the case of the discrete series, using index theory of elliptic operators in the framework of K‐theory. Our geometric proof of the character identity is evidence that index theory can play a role in the classification of group representations via the Langlands program. [ABSTRACT FROM AUTHOR]

We prove the existence of a map of spectra τA:kA→ℓA between connective topological K-theory and connective algebraic L-theory of a complex C∗-algebra A which is natural in A and compatible with multiplicative structures. We determine its effect on homotopy groups and as a consequence obtain a natural equivalence KA12→≃LA12 of periodic K- and L-theory spectra after inverting 2. We show that this equivalence extends to K- and L-theory of real C∗-algebras. Using this we give a comparison between the real Baum-Connes conjecture and the L-theoretic Farrell-Jones conjecture. We conclude that these conjectures are equivalent after inverting 2 if and only if a certain completion conjecture in L-theory is true. [ABSTRACT FROM AUTHOR]

We show that the Cuntz splice induces stably isomorphic graph $$C^*$$ -algebras. This result is a key step towards the recent complete classification of unital graph $$C^*$$ -algebras both with respect to K-theoretical classification and with respect to geometrical classification in terms of an equivalence relation on the underlying graphs. [ABSTRACT FROM AUTHOR]

A Floquet systems is a periodically driven quantum system. It can be described by a Floquet operator. If this unitary operator has a gap in the spectrum, then one can define associated topological bulk invariants which can either only depend on the bands of the Floquet operator or also on the time as a variable. It is shown how a K-theoretic result combined with the bulk-boundary correspondence leads to edge invariants for the half-space Floquet operators. These results also apply to topological quantum walks. [ABSTRACT FROM AUTHOR]

We study criteria for a ring-or more generally, for a small category-to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to develop machinery for proving new ones. Among the universal coefficient theorems covered by our methods we find, besides all the classic examples, several exotic examples arising from the KK-theory of C*-algebras and also Neeman's Brown-Adams representability theorem for compactly generated categories. [ABSTRACT FROM AUTHOR]