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Quantum Euclidean spaces with noncommutative derivatives J. Noncommut. Geom. (IF 0.9) Pub Date : 2022-05-12 Li Gao, Marius Junge, Edward McDonald
Quantum Euclidean spaces, as Moyal deformations of Euclidean spaces, are the model examples of noncompact noncommutative manifold. In this paper, we study the quantum Euclidean space equipped with partial derivatives satisfying canonical commutation relation (CCR). This gives an example of semifinite spectral triple with nonflat geometric structure. We develop an abstract symbol calculus for the pseudo-differential
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Immersions and the unbounded Kasparov product: embedding spheres into Euclidean space J. Noncommut. Geom. (IF 0.9) Pub Date : 2022-04-26 Walter D. van Suijlekom,Luuk S. Verhoeven
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Measured quantum groupoids on a finite basis and equivariant Kasparov theory J. Noncommut. Geom. (IF 0.9) Pub Date : 2022-04-04 Jonathan Crespo
In this article, we generalize to the case of measured quantum groupoids on a finite basis some important results concerning the equivariant Kasparov theory for actions of locally compact quantum groups, see Baaj and Skandalis (1989, 1993). To every pair $(A,B)$ of $\mathrm{C}^*$-algebras continuously acted upon by a regular measured quantum groupoid on a finite basis $\mathcal{G}$, we associate a
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Leibniz bialgebras, relative Rota–Baxter operators, and the classical Leibniz Yang–Baxter equation J. Noncommut. Geom. (IF 0.9) Pub Date : 2022-01-31 Rong Tang,Yunhe Sheng
In this paper, first we introduce the notion of a Leibniz bialgebra and show that matched pairs of Leibniz algebras, Manin triples of Leibniz algebras and Leibniz bialgebras are equivalent. Then we introduce the notion of a (relative) Rota-Baxter operator on a Leibniz algebra and construct the graded Lie algebra that characterizes relative Rota-Baxter operators as Maurer-Cartan elements. By these structures
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Finitely summable $\gamma$-elements for word-hyperbolic groups J. Noncommut. Geom. (IF 0.9) Pub Date : 2022-01-28 Jean-Marie Cabrera, Michael Puschnigg
We present two explicit combinatorial constructions of finitely summable reduced “Gamma”-elements $\gamma_r\in KK(C^*_r(\Gamma),\mathbb{C})$ for any word-hyperbolic group $(\Gamma,S)$ and obtain summability bounds for them in terms of the cardinality of the generating set $S\subset\Gamma$ and the hyperbolicity constant of the associated Cayley graph.
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Geometric K-homology and the Freed–Hopkins–Teleman theorem J. Noncommut. Geom. (IF 0.9) Pub Date : 2022-01-18 Yiannis Loizides
We construct a map at the level of cycles from the equivariant twisted K-homology of a compact, connected, simply connected Lie group $G$ to the Verlinde ring, which is inverse to the Freed–Hopkins–Teleman isomorphism. As an application, we prove that two of the proposed definitions of the quantization of a Hamiltonian loop group space—one via twisted K-homology of $G$ and the other via index theory
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Cyclic Gerstenhaber–Schack cohomology J. Noncommut. Geom. (IF 0.9) Pub Date : 2022-01-17 Domenico Fiorenza, Niels Kowalzig
We show that the diagonal complex computing the Gerstenhaber–Schack cohomology of a bialgebra (that is, the cohomology theory governing bialgebra deformations) can be given the structure of an operad with multiplication if the bialgebra is a (not necessarily finite dimensional) Hopf algebra with invertible antipode; if the antipode is involutive, the operad is even cyclic. Therefore, the Gerstenhaber–Schack
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The relative Mishchenko–Fomenko higher index and almost flat bundles II: Almost flat index pairing J. Noncommut. Geom. (IF 0.9) Pub Date : 2022-01-10 Yosuke Kubota
This is the second part of a series of papers which bridges the Chang–Weinberger–Yu relative higher index and geometry of almost flat Hermitian vector bundles on manifolds with boundary. In this paper, we apply the description of the relative higher index given in part I to establish the relative version of the Hanke–Schick theorem, which relates the relative higher index with the index pairing of
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The Novikov conjecture and extensions of coarsely embeddable groups J. Noncommut. Geom. (IF 0.9) Pub Date : 2022-01-10 Jintao Deng
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
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Hopf–Galois structures on ambiskew polynomial rings J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-12-22 Julien Bichon, Agustín García Iglesias
We provide necessary and sufficient conditions to extend the Hopf–Galois algebra structure on an algebra $R$ to a generalized ambiskew ring based on $R$, in a way such that the added variables for the extension are skew-primitive in an appropriate sense. We show that the associated Hopf algebra is again a generalized ambiskew ring, based on a suitable Hopf algebra $\underline{H}(R)$. Several examples
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Quantizations of local surfaces and rebel instantons J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-12-23 Severin Barmeier, Elizabeth Gasparim
We construct explicit deformation quantizations of the noncompact complex surfaces $Z_k:=\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^1}(-k))$ and describe their effect on moduli spaces of vector bundles and instanton moduli spaces. We introduce the concept of rebel instantons, as being those which react badly to some quantizations, misbehaving by shooting off extra families of noncommutative instantons
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Kernels for noncommutative projective schemes J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-11-23 Matthew Ballard, Blake Farman
We give a noncommutative geometric description of the internal Hom dg-category in the homotopy category of dg-categories between two noncommutative projective schemes in the style of Artin–Zhang. As an immediate application, we give a noncommutative projective derived Morita statement along the lines of Rickard and Orlov.
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Strict quantization of coadjoint orbits J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-12-15 Philipp Schmitt
For every semisimple coadjoint orbit $\hat{\mathcal{O}}$ of a complex connected semisimple Lie group $\hat{G}$, we obtain a family of $\hat{G}$-invariant products $\hat{*}_\hbar$ on the space of holomorphic functions on $\hat{\mathcal{O}}$. For every semisimple coadjoint orbit $\mathcal{O}$ of a real connected semisimple Lie group $G$, we obtain a family of $G$-invariant products $*_\hbar$ on a space
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Quotients of singular foliations and Lie 2-group actions J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-12-10 Alfonso Garmendia, Marco Zambon
Androulidakis–Skandalis (2009) showed that every singular foliation has an associated topological groupoid, called holonomy groupoid. In this note, we exhibit some functorial properties of this assignment: if a foliated manifold $(M,\mathcal{F}_M)$ is the quotient of a foliated manifold $(P,\mathcal{F}_P)$ along a surjective submersion with connected fibers, then the same is true for the corresponding
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$L^p$ coarse Baum–Connes conjecture and $K$-theory for $L^p$ Roe algebras J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-12-10 Jianguo Zhang, Dapeng Zhou
In this paper, we verify the $L^p$ coarse Baum–Connes conjecture for spaces with finite asymptotic dimension for $p\in[1,\infty)$. We also show that the $K$-theory of $L^p$ Roe algebras is independent of $p\in(1,\infty)$ for spaces with finite asymptotic dimension.
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The coarse geometric $\ell^p$-Novikov conjecture for subspaces of nonpositively curved manifolds J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-11-19 Lin Shan, Qin Wang
In this paper, we prove the coarse geometric $\ell^p$-Novikov conjecture for metric spaces with bounded geometry which admit a coarse embedding into a simply connected complete Riemannian manifold of nonpositive sectional curvature.
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Koszul duality for compactly generated derived categories of second kind J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-12-07 Ai Guan, Andrey Lazarev
For any dg algebra $A$ we construct a closed model category structure on dg $A$-modules such that the corresponding homotopy category is compactly generated by dg $A$-modules that are finitely generated and free over $A$ (disregarding the differential). We prove that this closed model category is Quillen equivalent to the category of comodules over a certain, possibly nonconilpotent dg coalgebra, a
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Lie structure on the Hochschild cohomology of a family of subalgebras of the Weyl algebra J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-12-07 Samuel A. Lopes, Andrea Solotar
For each nonzero $h\in\mathbb{F}[x]$, where $\mathbb{F}$ is a field, let $\mathsf{A}_h$ be the unital associative algebra generated by elements $x,y$, satisfying the relation $yx-xy=h$. This gives a parametric family of subalgebras of the Weyl algebra $\mathsf{A}_1$, containing many well-known algebras which have previously been studied independently. In this paper, we give a full description of the
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Induced map on $K$-theory for certain $\Gamma$-equivariant maps between Hilbert spaces J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-11-22 Tsuyoshi Kato
Higson–Kasparov–Trout introduced an infinite-dimensional Clifford algebra of a Hilbert space, and verified Bott periodicity on $K$-theory. To develop an algebraic topology of maps between Hilbert spaces, in this paper we introduce an induced Hilbert Clifford algebra and construct an induced map between $K$-theory of the Higson–Kasparov–Trout Clifford algebra and the induced Clifford algebra. We also
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Doob equivalence and non-commutative peaking for Markov chains J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-11-09 Xinxin Chen, Adam Dor-On, Langwen Hui, Christopher Linden, Yifan Zhang
In this paper, we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize the coincidence of conditional probabilities in terms of (generalized) Doob transforms, which then leads to a stronger classification result for the associated operator algebras in terms of spectral
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Quadratic Lie conformal superalgebras related to Novikov superalgebras J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-12-10 Pavel S. Kolesnikov, Roman A. Kozlov, Aleksander S. Panasenko
We study quadratic Lie conformal superalgebras associated with Novikov superalgebras. For every Novikov superalgebra $(V,\circ)$, we construct an enveloping differential Poisson superalgebra $U(V)$ with a derivation $d$ such that $u\circ v=ud(v)$ and $\{u,v\}=u\circ v-(-1)^{|u||v|}v\circ u$ for $u,v\in V$. The latter means that the commutator Gelfand–Dorfman superalgebra of $V$ is special. Next, we
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Cyclic cocycles in the spectral action J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-12-21 Teun D. H. van Nuland,Walter D. van Suijlekom
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A proof of a conjecture of Shklyarov J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-12-21 Michael K. Brown,Mark E. Walker
We prove a conjecture of Shklyarov concerning the relationship between K. Saito's higher residue pairing and a certain pairing on the periodic cyclic homology of matrix factorization categories. Along the way, we give new proofs of a result of Shklyarov and Polishchuk-Vaintrob's Hirzebruch-Riemann-Roch formula for matrix factorizations.
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$G$-homotopy invariance of the analytic signature of proper co-compact $G$-manifolds and equivariant Novikov conjecture J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-10-07 Yoshiyasu Fukumoto
The main result of this paper is the $G$-homotopy invariance of the $G$-index of the signature operator of proper co-compact $G$-manifolds. If proper co-compact $G$-manifolds $X$ and $Y$ are $G$-homotopy equivalent, then we prove that the images of their signature operators by the $G$-index map are the same in the $K$-theory of the $C^*$-algebra of the group $G$. Neither discreteness of the locally
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The adiabatic groupoid and the Higson–Roe exact sequence J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-10-05 Vito Felice Zenobi
Let $\widetilde{X}$ be a smooth Riemannian manifold equipped with a proper, free, isometric, and cocompact action of a discrete group $\Gamma$. In this paper, we prove that the analytic surgery exact sequence of Higson–Roe for $\widetilde{X}$ is isomorphic to the exact sequence associated to the adiabatic deformation of the Lie groupoid $\widetilde{X}\times_\Gamma\widetilde{X}$. We then generalize
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Ideals of étale groupoid algebras and Exel’s Effros–Hahn conjecture J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-10-05 Benjamin Steinberg
We extend to arbitrary commutative base rings a recent result of Demeneghi that every ideal of an ample groupoid algebra over a field is an intersection of kernels of induced representations from isotropy groups, with a much shorter proof, by using the author’s Disintegration Theorem for groupoid representations. We also prove that every primitive ideal is the kernel of an induced representation from
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$C^*$ exponential length of commutators unitaries in AH-algebras J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-11-15 Chun Guang Li, Liangqing Li, Iván Velázquez Ruiz
For each unital $C^*$-algebra $A$, we denote $\operatorname{cel}_{\operatorname{CU}}(A)=\sup\{\operatorname{cel}(u):u\in\operatorname{CU}(A)\}$, where $\operatorname{cel}(u)$ is the exponential length of $u$ and $\operatorname{CU}(A)$ is the closure of the commutator subgroup of $U_0(A)$. In this paper, we prove that $\operatorname{cel}_{\operatorname{CU}}(A)\geq2\pi$ provided that $A$ is an AH-algebra
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Poisson cohomology, Koszul duality, and Batalin–Vilkovisky algebras J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-10-15 Xiaojun Chen, Youming Chen, Farkhod Eshmatov, Song Yang
We study the noncommutative Poincaré duality between the Poisson homology and cohomology of unimodular Poisson algebras, and show that Kontsevich’s deformation quantization as well as Koszul duality preserve the corresponding Poincaré duality. As a corollary, the Batalin– Vilkovisky algebra structures that naturally arise in these cases are all isomorphic.
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On localized signature and higher rho invariant of fibered manifolds J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-10-27 Liu Hongzhi, Wang Jinmin
The higher index of the signature operator is a far-reaching generalization of the signature of a closed oriented manifold. When two closed oriented manifolds are homotopy equivalent, one can define a secondary invariant of the relative signature operator, called higher rho invariant. The higher rho invariant detects the topological nonrigidity of a manifold. In this paper, we prove product formulas
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Equivalences of (co)module algebra structures over Hopf algebras J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-10-19 Ana L. Agore, Alexey Gordienko, Joost Vercruysse
We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) an equivalence of gradings. We show that for each equivalence class of (co)module algebra structures on a given algebra $A$, there exists a unique universal Hopf algebra $H$ together with an $H$-(co)module structure on $A$ such that any other equivalent (co)module algebra
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Covering-monopole map and higher degree in non-commutative geometry J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-11-22 Tsuyoshi Kato
We analyze the monopole map over the universal covering space of a compact four-manifold. We induce the property of local properness of the covering-monopole map under the condition of closedness of the Atiyah–Hitchin–Singer (AHS) complex. In particular, we construct a higher degree of the covering-monopole map when the linearized equation is isomorphic. This induces a homomorphism between the $K$-groups
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An analog of the Krein–Milman theorem for certain non-compact convex sets J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-11-22 George A. Elliott, Zhiqiang Li, Xia Zhao
We make a contribution towards extending the remarkable Krein–Milman analog result of K. Thomsen and L. Li, in which a certain non-compact convex set is shown to be generated by its extreme points.
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Rapid decay and polynomial growth for bicrossed products J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-11-22 Pierre Fima, Hua Wang
We study the rapid decay property and polynomial growth for duals of bicrossed products coming from a matched pair of a discrete group and a compact group.
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Quasi-homogeneity of potentials J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-09-01 Zheng Hua, Guisong Zhou
In noncommutative differential calculus, Jacobi algebra (or potential algebra) plays the role of Milnor algebra in the commutative case. The study of Jacobi algebras is of broad interest to researchers in cluster algebra, representation theory and singularity theory. In this article, we study the quasi-homogeneity of a potential in a complete free algebra over an algebraic closed field of characteristic
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The unbounded Kasparov product by a differentiable module J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-09-01 Jens Kaad
In this paper we investigate the unbounded Kasparov product between a differentiable module and an unbounded cycle of a very general kind that includes all unbounded Kasparov modules and hence also all spectral triples. Our assumptions on the differentiable module are weak and we do in particular not require that it satisfies any kind of smooth projectivity conditions. The algebras that we work with
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A categorical characterization of quantum projective spaces J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-09-01 Izuru Mori, Kenta Ueyama
Let $R$ be a finite dimensional algebra of finite global dimension over a field $k$. In this paper, we will characterize a $k$-linear abelian category $\mathscr C$ such that $\mathscr C\cong \operatorname {tails} A$ for some graded right coherent AS-regular algebra $A$ over $R$. As an application, we will prove that if $\mathscr C$ is a smooth quadric surface in a quantum $\mathbb P^3$ in the sense
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Invariant Markov semigroups on quantum homogeneous spaces J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-09-28 Biswarup Das, Uwe Franz, Xumin Wang
Invariance properties of linear functionals and linear maps on algebras of functions on quantum homogeneous spaces are studied, in particular for the special case of expected co-ideal *-subalgebras. Several one-to-one correspondences between such invariant functionals are established. Adding a positivity condition, this yields one-to-one correspondences of invariant quantum Markov semigroups acting
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Structure and $K$-theory of $\ell^p$ uniform Roe algebras J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-09-28 Yeong Chyuan Chung, Kang Li
In this paper, we characterize when the $\ell^p$ uniform Roe algebra of a metric space with bounded geometry is (stably) finite and when it is properly infinite in standard form for $p\in [1,\infty)$. Moreover, we show that the $\ell^p$ uniform Roe algebra is a (non-sequential) spatial $L^p$ AF algebra in the sense of Phillips and Viola if and only if the underlying metric space has asymptotic dimension
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Odd characteristic classes in entire cyclic homology and equivariant loop space homology J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-10-13 Sergio L. Cacciatori, Batu Güneysu
Given a compact manifold $M$ and a smooth map $g\colon M\to U(l\times l;\mathbb{C})$ from $M$ to the Lie group of unitary $l\times l$ matrices with entries in $\mathbb{C}$, we construct a Chern character $\mathrm{Ch}^-(g)$ which lives in the odd part of the equivariant (entire) cyclic Chen-normalized cyclic complex $\mathscr{N}_{\epsilon}(\Omega_{\mathbb{T}}(M\times \mathbb{T}))$ of $M$, and which
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Polyvector fields and polydifferential operators associated with Lie pairs J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-10-13 Ruggero Bandiera, Mathieu Stiénon, Ping Xu
We prove that the spaces $\operatorname{tot} \big(\Gamma ({\Lambda^\bullet A^\vee}) \otimes_R{{\mathcal{T}_{poly}^{\bullet}}}\big)$ and $\operatorname{tot} \big(\Gamma ({\Lambda^\bullet A^\vee}) \otimes_R{{\mathcal{D}_{poly}^{\bullet}}}\big)$ associated with a Lie pair $(L,A)$ each carry an $L_\infty$ algebra structure canonical up to an $L_\infty$ isomorphism with the identity map as linear part.
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Quasifolds, diffeology and noncommutative geometry J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-08-30 Patrick Iglesias-Zemmour, Elisa Prato
After embedding the objects quasifolds into the category $\{$Diffeology$\}$, we associate a $\boldsymbol{C}^*$-algebra with every atlas of any quasifold, and show how different atlases give Morita equivalent algebras. This builds a new bridge between diffeology and noncommutative geometry – beginning with the today classical example of the irrational torus – which associates a Morita class of $\bo
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Cyclic $A_\infty$-algebras and double Poisson algebras J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-04-21 David Fernández, Estanislao Herscovich
In this article we prove that there exists an explicit bijection between nice $d$-pre-Calabi–Yau algebras and $d$-double Poisson differential graded algebras, where $d \in \mathbb{Z}$, extending a result proved by N. Iyudu and M. Kontsevich. We also show that this correspondence is functorial in a quite satisfactory way, giving rise to a (partial) functor from the category of $d$-double Poisson dg
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The universal Boolean inverse semigroup presented by the abstract Cuntz–Krieger relations J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-04-21 Mark V. Lawson, Alina Vdovina
This paper is a contribution to the theory of what might be termed 0-dimensional non-commutative spaces. We prove that associated with each inverse semigroup $S$ is a Boolean inverse semigroup presented by the abstract versions of the Cuntz–Krieger relations. We call this Boolean inverse semigroup the tight completion of $S$ and show that it arises from Exel's tight groupoid under non-commutative Stone
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Vector bundles over multipullback quantum complex projective spaces J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-04-21 Albert Jeu-Liang Sheu
We work on the classification of isomorphism classes of finitely generated projective modules over the C*-algebras $C(\mathbb{P}^{n}(\mathcal{T})) $ and $C(\mathbb{S}_{H}^{2n+1})$ of the quantum complex projective spaces $\mathbb{P}^{n}(\mathcal{T})$ and the quantum spheres $\mathbb{S}_{H}^{2n+1}$, and the quantum line bundles $L_{k}$ over $\mathbb{P}^{n}(\mathcal{T})$, studied by Hajac and collaborators
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The dual modular Gromov–Hausdorff propinquity and completeness J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-04-21 Frédéric Latrémolière
We introduce in this paper the dual modular propinquity, a complete metric, up to full modular quantum isometry, on the class of metrized quantum vector bundles, i.e. of Hilbert modules endowed with a type of densely defined norm, called a D-norm, which generalize the operator norm given by a connection on a Riemannian manifold. The dual modular propinquity is weaker than the modular propinquity yet
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Cup product on $A_\infty$-cohomology and deformations J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-02-10 Alexey A. Sharapov, Evgeny D. Skvortsov
We propose a simple method for constructing formal deformations of differential graded algebras in the category of minimal $A_\infty$-algebras. The basis for our approach is provided by the Gerstenhaber algebra structure on $A_\infty$-cohomology, which we define in terms of the brace operations. As an example, we construct a minimal $A_\infty$-algebra from the Weyl–Moyal $\ast$-product algebra of polynomial
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Coarse assembly maps J. Noncommut. Geom. (IF 0.9) Pub Date : 2020-11-30 Ulrich Bunke, Alexander Engel
For every strong coarse homology theory we construct a coarse assembly map as a natural transformation between coarse homology theories. We provide various conditions implying that this assembly map is an equivalence. These results generalize known results for the analytic coarse assembly map for $K$-homology to general coarse homology theories. Furthermore, we calculate the domain of the coarse assembly
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Noncommutative Borsuk–Ulam-type conjectures revisited J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-01-18 Ludwik Dąbrowski, Piotr M. Hajac, Sergey Neshveyev
Let $H$ be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra $A$. It was recently conjectured that there does not exist an equivariant *-homomorphism from $A$ (type-I case) or $H$ (type-II case) to the equivariant noncommutative join C*-algebra $A\circledast^\delta H$. When $A$ is the C*-algebra of functions on a sphere, and $H$ is the C*-algebra of functions
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A weak homotopy equivalence type result related to Kirchberg algebras J. Noncommut. Geom. (IF 0.9) Pub Date : 2020-11-30 Masaki Izumi, Hiroki Matui
We obtain a weak homotopy equivalence type result between two topological groups associated with a Kirchberg algebra: the unitary group of the continuous asymptotic centralizer and the loop group of the automorphism group of the stabilization. This result plays a crucial role in our subsequent work on the classification of poly-$\mathbb Z$ group actions on Kirchberg algebras. As a special case, we
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Periodicity and cyclic homology. Para-$S$-modules and perturbation lemmas J. Noncommut. Geom. (IF 0.9) Pub Date : 2020-11-30 Raphaël Ponge
In this paper, we introduce a paracyclic version of $S$-modules. These new objects are called para-$S$-modules. Paracyclic modules and parachain complexes give rise to para-$S$-modules much in the same way as cyclic modules and mixed complexes give rise to $S$-modules. More generally, para-$S$-modules provide us with a natural framework to get analogues for paracyclic modules and parachain complexes
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Duality of Gabor frames and Heisenberg modules J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-01-25 Mads S. Jakobsen, Franz Luef
Given a locally compact abelian group $G$ and a closed subgroup $\Lambda$ in $G\times\hat{G}$, Rieffel associated to $\Lambda$ a Hilbert $C^*$-module $\mathcal E$, known as a Heisenberg module. He proved that $\mathcal E$ is an equivalence bimodule between the twisted group $C^*$-algebra $C^*(\Lambda, \mathsf{c})$ and $C^*(\Lambda^\circ,\bar{\mathsf{c}})$, where $\Lambda^{\circ}$ denotes the adjoint
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The Higson–Roe sequence for étale groupoids. II. The universal sequence for equivariant families J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-02-08 Moulay-Tahar Benameur, Indrava Roy
This is the second part of our series about the Higson–Roe sequence for étale groupoids. We devote this part to the proof of the universal K-theory surgery exact sequence which extends the seminal results of N. Higson and J. Roe to the case of transformation groupoids. In the process, we prove the expected functoriality properties as well as the Paschke–Higson duality theorem.
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Classification, Koszulity and Artin–Schelter regularity of certain graded twisted tensor products J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-02-08 Andrew Conner, Peter Goetz
Let $\mathbb K$ be an algebraically closed field. We classify all of the quadratic twisted tensor products $A \otimes_{\tau} B$ in the cases where $(A, B) = (\mathbb K[x], \mathbb K[y])$ and $(A, B) = (\mathbb K[x, y], \mathbb K[z])$. We determine when a quadratic twisted tensor product of this form is Koszul, and when it is Artin–Schelter regular.
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Equivariant vector bundles over quantum spheres J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-02-08 Andrey Mudrov
We quantize $SO(2n+1)$-equivariant vector bundles over an even complex sphere $\mathbb{S}^{2n}$ as one-sided projective modules over its quantized coordinate ring. We realize them in two different ways: as linear maps between pseudo-parabolic modules and as induced modules of the orthogonal quantum group. Based on this alternative, we study representations of a quantum symmetric pair related to $\mathbb{S}^{2n}_q$
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K-theory and homotopies of twists on ample groupoids J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-02-08 Christian Bönicke
This paper investigates the K-theory of twisted groupoid C*-algebras. It is shown that a homotopy of twists on an ample groupoid satisfying the Baum–Connes conjecture with coefficients gives rise to an isomorphism between the K-theory groups of the respective twisted groupoid C*-algebras. The results are also interpreted in an inverse semigroup setting and applied to generalized Renault–Deaconu groupoids
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Algebraic bivariant $K$-theory and Leavitt path algebras. J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-02-02 Guillermo Cortiñas, Diego Montero
We investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras $L(E)$ and $L(F)$ of graphs $E$ and $F$ over a commutative ground ring $\ell$. We approach this by studying the structure of such algebras under bivariant algebraic $K$-theory $kk$, which is the universal homology theory with the properties above. We
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Homotopy Poisson algebras, Maurer–Cartan elements and Dirac structures of CLWX 2-algebroids J. Noncommut. Geom. (IF 0.9) Pub Date : 2021-01-21 Jiefeng Liu, Yunhe Sheng
In this paper, we construct a homotopy Poisson algebra of degree 3 associated to a split Lie 2-algebroid, by which we give a new approach to characterize a split Lie 2-bialgebroid. We develop the differential calculus associated to a split Lie 2-algebroid and establish the Manin triple theory for split Lie 2-algebroids. More precisely, we give the notion of a strict Dirac structure and define a Manin
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The index of $G$-transversally elliptic families. I J. Noncommut. Geom. (IF 0.9) Pub Date : 2020-11-03 Alexandre Baldare
We define and study the index map for families of $G$-transversally elliptic operators and introduce the multiplicity for a given irreducible representation as a virtual bundle over the base of the fibration. We then prove the usual axiomatic properties for the index map extending the Atiyah–Singer results [1]. Finally, we compute the Kasparov intersection product of our index class against the K-homology
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The index of $G$-transversally elliptic families. II J. Noncommut. Geom. (IF 0.9) Pub Date : 2020-11-03 Alexandre Baldare
We define the Chern character of the index class of a $G$-invariant family of $G$-transversally elliptic operators, see [6]. Next we study the Berline–Vergne formula for families in the elliptic and transversally elliptic case.
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The relative Mishchenko–Fomenko higher index and almost flat bundles. I. The relative Mishchenko–Fomenko index J. Noncommut. Geom. (IF 0.9) Pub Date : 2020-11-03 Yosuke Kubota
In this paper, the first of two, we introduce an alternative definition of the Chang–Weinberger–Yu relative higher index, which is thought of as a relative analogue of the Mishchenko–Fomenko index pairing. A main result of this paper is that our map coincides with the existing relative higher index maps. We make use of this fact for understanding the relative higher index. First, we relate the relative