• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-06-30
Süleyman Kağan Samurkaş

We derive a lower and an upper bound for the rank of the finite part of operator $K$-theory groups of maximal and reduced $C^*$-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy classes of finite order elements in the group. The upper bound is based on the amount of torsion elements in the group. We use the lower bound to give lower bounds

更新日期：2020-07-30
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-28
Baojie Jiang; Hongzhi Liu

In [16], Weinberger, Xie, and Yu proved that higher rho invariant associated to homotopy equivalence defines a group homomorphism from the topological structure group to the analytic structure group, $K$-theory of certain geometric $C^*$-algebras, by piecewise-linear approach. In this paper, we adapt part of Weinberger, Xie, and Yu’s work, to give a differential geometry theoretic proof of the additivity

更新日期：2020-07-30
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-06-25
Tyrone Crisp

Let $F$ be a right Hilbert $C$*-module over a $C$*-algebra $B$, and suppose that $F$ is equipped with a left action, by compact operators, of a second $C$*-algebra $A$. Tensor product with $F$ gives a functor from Hilbert $C$*-modules over $A$ to Hilbert $C$*-modules over $B$. We prove that under certain conditions (which are always satisfied if, for instance, $A$ is nuclear), the image of this functor

更新日期：2020-07-30
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-06-26
Cris Negron; Yury Volkov; Sarah Witherspoon

We show that Hochschild cohomology of an algebra over a field is a space of infinity coderivations on an arbitrary projective bimodule resolution of the algebra. The Gerstenhaber bracket is the graded commutator of infinity coderivations. We thus generalize, to an arbitrary resolution, Stasheff’s realization of the Gerstenhaber bracket on Hochschild cohomology as the graded commutator of coderivations

更新日期：2020-07-30
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-02
Angshuman Bhattacharya; Michael Brannan; Alexandru Chirvasitu; Shuzhou Wang

We investigate connections between various rigidity and softness properties for discrete quantum groups. After introducing a notion of residual finiteness, we show that it implies the Kirchberg factorization property for the discrete quantum group in question. We also prove the analogue of Kirchberg’s theorem, to the effect that conversely, the factorization property and property (T) jointly imply

更新日期：2020-07-30
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-13
Jonathan P. Pridham

We develop Tannaka duality theory for dg categories. To any dg functor from a dg category $\mathcal A$ to finite-dimensional complexes, we associate a dg coalgebra $C$ via a Hochschild homology construction. When the dg functor is faithful, this gives a quasi-equivalence between the derived dg categories of $\mathcal A$-modules and of $C$-comodules. When $\mathcal A$ is Morita fibrant (i.e. an idempotent-complete

更新日期：2020-07-30
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-19
Julio Gutiérrez; Christian Valqui

We compute the isomorphism class in $\mathfrak{KK}^{\mathrm {alg}}$ of all noncommutative generalized Weyl algebras $A=\mathbb C[h](\sigma, P)$,where $\sigma(h)=qh+h_0$ is an automorphism of $\mathcal C[h]$, except when $q\neq 1$ is a root of unity. In particular, we compute the isomorphism class in $\mathfrak{KK}^{\mathrm {alg}}$ of the quantum Weyl algebra, the primitive factors $B_{\lambda}$ of

更新日期：2020-07-30
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-21
Ulrich Krähmer; Manuel Martins

In a recent article, the coordinate ring of the nodal cubic was given the structure of a quantum homogeneous space. Here the corresponding coalgebra Galois extension is expressed in terms of quantum groups at roots of unity, and is shown to be cleft. Furthermore, the minimal quotient extensions are determined.

更新日期：2020-07-30
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-13
Andrew J. Bruce

We introduce the notion of almost commutative $Q$-algebras and demonstrate how the derived bracket formalism of Kosmann–Schwarzbach generalises to this setting. In particular, we construct ‘almost commutative Lie algebroids’ following Vaıntrob’s $Q$-manifold understanding of classical Lie algebroids. We show that the basic tenets of the theory of Lie algebroids carry over verbatim to the almost commutative

更新日期：2020-07-30
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-29
Michael K. Brown; Mark E. Walker

Let $k$ be a field of characteristic 0 and $\mathcal{A}$ a curved $k$-algebra. We obtain a Chern–Weil-type formula for the Chern character of a perfect $\mathcal{A}$-module taking values in $HN^{II}_0(\mathcal{A})$, the negative cyclic homology of the second kind associated to $\mathcal{A}$, when $\mathcal{A}$ satisfies a certain smoothness condition.

更新日期：2020-07-30
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-21

In this note, making use of the recent theory of noncommutative motives, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injectivity properties: in the case of a regular integral quasi-compact quasi-separated scheme, it is injective; in the case of an integral normal Noetherian scheme with a single isolated singularity, it distinguishes

更新日期：2020-07-30
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-29
Boris Shoikhet

Given two small dg categories $C,D$, defined over a field, we introduce their (non-symmetric) twisted tensor product $C\sotimes D$. We show that $-\sotimes D$ is left adjoint to the functor $\mathcal {Coh}(D,-)$, where $\mathcal {Coh}(D,E)$ is the dg category of dg functors $D\to E$ and their coherent natural transformations. This adjunction holds in the category of small dg categories (not in the

更新日期：2020-07-30
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-14
Moulay-Tahar Benameur; Indrava Roy

We introduce the dual Roe algebras for proper étale groupoid actions and deduce the expected Higson–Roe short exact sequence. When the action is co-compact, we show that the Roe $C^*$-ideal of locally compact operators is Morita equivalent to the reduced $C^*$-algebra of our groupoid, and we further identify the boundary map of the associated periodic six-term exact sequence with the Baum–Connes map

更新日期：2020-07-20
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-18
Špela Špenko; Michel Van den Bergh

Using the theory of dimer models Broomhead proved that every 3-dimensional Gorenstein affine toric variety Spec $R$ admits a toric non-commutative crepant resolution (NCCR). We give an alternative proof of this result by constructing a tilting bundle on a (stacky) crepant resolution of Spec $R$ using standard toric methods. Our proof does not use dimer models.

更新日期：2020-07-20
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-14
Bram Mesland; Mehmet Haluk Şengün

Let $\Gamma$ be a torsion-free arithmetic group acting on its associated global symmetric space $X$. Assume that $X$ is of non-compact type and let $\Gamma$ act on the geodesic boundary $\partial X$ of $X$. Via general constructions in $KK$-theory, we endow the $K$-groups of the arithmetic manifold $X / \Gamma$, of the reduced group $C^*$-algebra $C^*_r(\Gamma)$ and of the boundary crossed product

更新日期：2020-07-20
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-25
Sophie Chemla

M. Van den Bergh [20] defined the notion of a double Lie algebroid and showed that a double quasi-Poisson algebra gives rise to a double Lie algebroid.We give new examples of double Lie algebroids and develop a differential calculus in that context. We recover the non commutative Karoubi–de Rham complex [7, 9] and the double Poisson–Lichnerowicz cohomology [16] as particular cases of our construction

更新日期：2020-07-20
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-14
Shamindra Kumar Ghosh; Corey Jones; B. Madhav Reddy

We provide a description of the annular representation category of the free product of two rigid C*-tensor categories.

更新日期：2020-07-20
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-28
George A. Elliott; Guihua Gong; Huaxin Lin; Zhuang Niu

We study a class of stably projectionless simple C*-algebras which may be viewed as having generalized tracial rank one in analogy with the unital case. A number of structural questions concerning these simple C*-algebras are studied, pertinent to the classification of separable stably projectionless simple amenable Jiang–Su stable C*-algebras.

更新日期：2020-07-20
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-27
Daniel Kaplan

In this paper, we study a preprojective algebra for quivers decorated with $k$-algebras and bimodules, which generalizes work of Gabriel for ordinary quivers, work of Dlab and Ringel fork-species, and recent work of de Thanhoffer de Völcsey and Presotto, which has recently appeared from a different perspective in work of Külshammer. As for undecorated quivers, we show that its moduli space of representations

更新日期：2020-07-20
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-04-30
Paweł Kasprzak; Piotr M. Sołtan

The main result of the paper is the characterization of those locally compact quantum groups with projection, i.e. non-commutative analogs of semidirect products, which are extensions as defined by L. Vainerman and S. Vaes. It turns out that quantum groups with projection are usually not extensions.We discuss several examples including the quantum $\mathrm{U}_q(2)$. The major tool used to obtain these

更新日期：2020-04-30
• J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-02-11
Terry A. Loring; Hermann Schulz-Baldes

Even index pairings are integer-valued homotopy invariants combining an even Fredholm module with a $K_0$-class specified by a projection. Numerous classical examples are known from differential and non-commutative geometry and physics. Here it is shown how to construct a finite-dimensional self-adjoint and invertible matrix, called the spectral localizer, such that half its signature is equal to the

更新日期：2020-02-11
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