- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 12:57:35
38
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JNCG&vol=8&update_since=2024-03-28
Journal of Noncommutative Geometry
J. Noncommut. Geom.
JNCG
1661-6952
1661-6960
Global analysis, analysis on manifolds
General
10.4171/JNCG
http://www.ems-ph.org/doi/10.4171/JNCG
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
8
2014
1
The resolvent cocycle in twisted cyclic cohomology and a local index formula for the Podleś sphere
Adam
Rennie
University of Wollongong, WOLLONGONG, AUSTRALIA
Roger
Senior
Australian National University, CANBERRA, AUSTRALIA
Spectral triple, cyclic cohomology, Kasparov theory, q-deformations, Podleś sphere
We continue the investigation of twisted homology theories in the context of dimension drop phenomena. This work unifies previous equivariant index calculations in twisted cyclic cohomology. We do this by proving the existence of the resolvent cocycle, a finitely summable analogue of the JLO cocycle, under weaker smoothness hypotheses and in the more general setting of ‘modular’ spectral triples. As an application we show that using our twisted resolvent cocycle, we can obtain a local index formula for the Podleś sphere. The resulting twisted cyclic cocycle has non-vanishing Hochschild class which is in dimension 2.
$K$-theory
Functional analysis
1
43
10.4171/JNCG/147
http://www.ems-ph.org/doi/10.4171/JNCG/147
On the Hochschild and cyclic (co)homology of rapid decay group algebras
Ronghui
Ji
Indiana University Purdue University Indianapolis, INDIANAPOLIS, UNITED STATES
Crichton
Ogle
Ohio State University, COLUMBUS, UNITED STATES
Bobby
Ramsey
University of Hawai‘i at Mānoa, HONOLULU, UNITED STATES
$\mathcal{B}$-bounded cohomology, isocohomological, weighted complex, Generalized Bass Conjecture
We show that the technical condition of solvable conjugacy bound, introduced in [JOR1], can be removed without affecting the main results of that paper. The result is a Burghelea-type description of the summands $\mathrm{HH}_*^t({\mathcal{H}_{\mathcal{B},L}(G)})_{\langle x\rangle}$ and $\mathrm{HC}_*^t({\mathcal{H}_{\mathcal{B},L}(G)})_{\langle x\rangle}$ for any bounding class $\mathcal{B}$, discrete group with word-length $(G,L)$ and conjugacy class $\langle x\rangle\in \langle G\rangle$. We use this description to prove the conjecture $\mathcal{B}$-SrBC of [JOR1] for a class of groups that goes well beyond the cases considered in that paper. In particular, we show that the conjecture $\ell^1$-SrBC (the Strong Bass Conjecture for the topological K-theory of $\ell^1(G)$) is true for all semihyperbolic groups which satisfy SrBC, a statement consistent with the rationalized Bost conjecture for such groups.
Global analysis, analysis on manifolds
General
45
59
10.4171/JNCG/148
http://www.ems-ph.org/doi/10.4171/JNCG/148
Line bundles and the Thom construction in noncommutative geometry
Edwin
Beggs
University of Wales Swansea, SWANSEA, UNITED KINGDOM
Tomasz
Brzeziński
University of Wales Swansea, SWANSEA, UNITED KINGDOM
Morita context, C*-algebra, bimodules, line bundles, Thom construction, Hopf–Galois extension, Chern class
The idea of a line bundle in classical geometry is transferred to noncommutative geometry by the idea of a Morita context. From this we construct $\mathbb{Z}$- and $\mathbb{N}$-graded algebras, the $\mathbb{Z}$-graded algebra being a Hopf–Galois extension. A non-degenerate Hermitian metric gives a star structure on this algebra, and an additional star operation on the line bundle gives a star operation on the $\mathbb{N}$-graded algebra. In this case, we carry out the associated circle bundle and Thom constructions. Starting with a C*-algebra as base, and with some positivity assumptions, the associated circle and Thom algebras are also C*-algebras. We conclude by examining covariant derivatives and Chern classes on line bundles after the method of Kobayashi and Nomizu.
Functional analysis
Associative rings and algebras
Global analysis, analysis on manifolds
61
105
10.4171/JNCG/149
http://www.ems-ph.org/doi/10.4171/JNCG/149
Quantum groups of GL(2) representation type
Colin
Mrozinski
Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE
Quantum groups, Hopf algebra, monoidal category, cogroupoid
We classify the cosemisimple Hopf algebras whose corepresentation semi-ring is isomorphic to that of GL(2). This leads us to define a new family of Hopf algebras which generalize the quantum similitude group of a non-degenerate bilinear form. A detailed study of these Hopf algebras gives us an isomorphic classification and the description of their corepresentation categories.
Group theory and generalizations
Associative rings and algebras
107
140
10.4171/JNCG/150
http://www.ems-ph.org/doi/10.4171/JNCG/150
Polynomial realizations of some combinatorial Hopf algebras
Loïc
Foissy
Centre Universitaire de la Mi-Voix, CALAIS CEDEX, FRANCE
Jean-Christophe
Novelli
Université Paris-Est Marne-la-Vallée, MARNE-LA-VALLÉE CEDEX 2, FRANCE
Jean-Yves
Thibon
Université Paris-Est Marne-la-Vallée, MARNE-LA-VALLÉE CEDEX 2, FRANCE
Hopf algebras of decorated rooted trees, free quasi-symmetric functions, parking functions
We construct explicit polynomial realizations of some combinatorial Hopf algebras based on various kinds of trees or forests, and some more general classes of graphs, ranging from the Connes–Kreimer algebra to an algebra of labelled forests isomorphic to the Hopf algebra of parking functions and to a new noncommutative algebra based on endofunctions admitting many interesting subalgebras and quotients.
Combinatorics
Associative rings and algebras
141
162
10.4171/JNCG/151
http://www.ems-ph.org/doi/10.4171/JNCG/151
Group quasi-representations and almost flat bundles
Marius
Dadarlat
Purdue University, WEST LAFAYETTE, UNITED STATES
K-theory, discrete groups, deformations, almost flat bundles
We study the existence of quasi-representations of discrete groups $G$ into unitary groups $U(n)$ that induce prescribed partial maps $K_0(C^*(G))\to \mathbb{Z}$ on the K-theory of the group C*-algebra of $G$. We give conditions for a discrete group $G$ under which the K-theory group of the classifying space $BG$ consists entirely of almost flat classes.
Functional analysis
$K$-theory
163
178
10.4171/JNCG/152
http://www.ems-ph.org/doi/10.4171/JNCG/152
Projective Dirac operators, twisted K-theory, and local index formula
Dapeng
Zhang
California Institute of Technology, PASADENA, UNITED STATES
Twisted K-theory, spectral triple, Chern character
We construct a canonical noncommutative spectral triple for every oriented closed Riemannian manifold, which represents the fundamental class in the twisted K-homology of the manifold. This so-called “projective spectral triple” is Morita equivalent to the well-known commutative spin spectral triple provided that the manifold is spin-c. We give an explicit local formula for the twisted Chern character for K-theories twisted with torsion classes, and with this formula we show that the twisted Chern character of the projective spectral triple is identical to the Poincaré dual of the $\hat{A}$-genus of the manifold.
$K$-theory
Global analysis, analysis on manifolds
179
215
10.4171/JNCG/153
http://www.ems-ph.org/doi/10.4171/JNCG/153
The groupoid C*-algebra of a rational map
Klaus
Thomsen
Aarhus University, AARHUS C, DENMARK
C*-algebra, groupoid, holomorphic dynamics, rational map
This paper contains a quite detailed description of the C*-algebra arising from the transformation groupoid of a rational map of degree at least two on the Riemann sphere. The algebra is decomposed stepwise via extensions of familiar C*-algebras whose nature depend on the structure of the Julia set and the stable regions in the Fatou set, as well as on the behaviour of the critical points.
Global analysis, analysis on manifolds
Real functions
Ordinary differential equations
Functional analysis
217
264
10.4171/JNCG/154
http://www.ems-ph.org/doi/10.4171/JNCG/154
Noncommutative residue of projections in Boutet de Monvel’s calculus
Anders
Gaarde
, COPENHAGEN S, DENMARK
Noncommutative residues, K-theory, Boutet de Monvel’s algebra, pseudodifferential operators, pseudodifferential projections
Employing results by Melo, Nest, Schick and Schrohe on the K-theory of Boutet de Monvel’s calculus of boundary value problems, we show that the noncommutative residue introduced by Fedosov, Golse, Leichtnam and Schrohe vanishes on projections in the calculus. This partially answers a question raised in a recent collaboration with Grubb, namely whether the residue is zero on sectorial projections for boundary value problems: This is confirmed to be true when the sectorial projection is in the calculus.
Global analysis, analysis on manifolds
Partial differential equations
265
274
10.4171/JNCG/155
http://www.ems-ph.org/doi/10.4171/JNCG/155
Bost–Connes systems associated with function fields
Sergey
Neshveyev
University of Oslo, OSLO, NORWAY
Simen
Rustad
University of Oslo, OSLO, NORWAY
Bost–Connes systems, function fields, KMS-states, type III actions, Drinfeld modules
With a global function field $K$ with constant field $\mathbb{F}_q$, a finite set $S$ of primes in $K$ and an abelian extension $L$ of $K$, finite or infinite, we associate a C*-dynamical system. The systems, or at least their underlying groupoids, defined earlier by Jacob using the ideal action on Drinfeld modules and by Consani–Marcolli using commensurability of $K$-lattices are isomorphic to particular cases of our construction. We prove a phase transition theorem for our systems and show that the unique KMS$_\beta$-state for every $0
Functional analysis
Number theory
275
301
10.4171/JNCG/156
http://www.ems-ph.org/doi/10.4171/JNCG/156
2
On an extension of Knuth’s rotation correspondence to reduced planar trees
Kurusch
Ebrahimi-Fard
Universidad Autónoma de Madrid, MADRID, SPAIN
Dominique
Manchon
Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE
Trees, hypertrees, rotation correspondence, operads, Hopf algebras
We present a bijection from planar reduced trees to planar rooted hypertrees, which extends Knuth's rotation correspondence between planar binary trees and planar rooted trees. The operadic counterpart of the new bijection is explained. Related to this, the space of planar reduced forests is endowed with a combinatorial Hopf algebra structure. The corresponding structure on the space of planar rooted hyperforests is also described.
Combinatorics
Associative rings and algebras
303
320
10.4171/JNCG/157
http://www.ems-ph.org/doi/10.4171/JNCG/157
Dixmier traces generated by exponentiation invariant generalised limits
Fedor
Sukochev
University of New South Wales, SYDNEY, AUSTRALIA
Alexandr
Usachev
University of New South Wales, SYDNEY, AUSTRALIA
Dmitriy
Zanin
University of New South Wales, SYDNEY, AUSTRALIA
Dixmier traces, measurable elements, exponentiation invariant generalized limits, Lidskii formula
We define a new class of singular positive traces on the ideal $\mathcal{M}_{1,\infty}$ of $B(H)$ generated by exponentiation invariant generalized limits. We prove that this new class is strictly contained in the class of all Dixmier traces. We also prove a Lidskii-type formula for this class of traces.
Global analysis, analysis on manifolds
Functional analysis
321
336
10.4171/JNCG/158
http://www.ems-ph.org/doi/10.4171/JNCG/158
Thermodynamic semirings
Matilde
Marcolli
California Institute of Technology, PASADENA, UNITED STATES
Ryan
Thorngren
California Institute of Technology, PASADENA, UNITED STATES
Entropy (Shannon, Renyi, Tsallis, Kullback–Leibler divergence), semiring, Witt construction, multifractals, operads, binary guessing games, entropy algebras
The Witt construction describes a functor from the category of Rings to the category of characteristic 0 rings. It is uniquely determined by a few associativity constraints which do not depend on the types of the variables considered, in other words, by integer polynomials. This universality allowed Alain Connes and Caterina Consani to devise an analogue of the Witt ring for characteristic one, an attractive endeavour since we know very little about the arithmetic in this exotic characteristic and its corresponding field with one element. Interestingly, they found that in characteristic one, the Witt construction depends critically on the Shannon entropy. In the current work, we examine this surprising occurrence, defining a Witt operad for an arbitrary information measure and a corresponding algebra we call a thermodynamic semiring. This object exhibits algebraically many of the familiar properties of information measures, and we examine in particular the Tsallis and Renyi entropy functions and applications to non-extensive thermodynamics and multifractals. We find that the arithmetic of the thermodynamic semiring is exactly that of a certain guessing game played using the given information measure.
Information and communication, circuits
Commutative rings and algebras
Measure and integration
337
392
10.4171/JNCG/159
http://www.ems-ph.org/doi/10.4171/JNCG/159
L2-index formula for proper cocompact group actions
Hang
Wang
University of Adelaide, ADELAIDE, SA, AUSTRALIA
L2-index, K-theoretic index, G-trace, heat kernel
We study index theory of G-invariant elliptic pseudo-differential operators acting on a complete Riemannian manifold, where a unimodular, locally compact group G acts properly, cocompactly and isometrically. An L2-index formula is obtained using the heat kernel method.
$K$-theory
General
393
432
10.4171/JNCG/160
http://www.ems-ph.org/doi/10.4171/JNCG/160
Quantum gauge symmetries in noncommutative geometry
Jyotishman
Bhowmick
Indian Statistical Institute, KOLKATA, INDIA
Francesco
D'Andrea
Università degli Studi di Napoli “Federico II”, NAPOLI, ITALY
Biswarup
Das
University of Leeds, LEEDS, UNITED KINGDOM
Ludwik
Dąbrowski
SISSA, TRIESTE, ITALY
Quantum groups, noncommutative geometry, gauge symmetry, Standard Model
We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite-dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms in the framework of compact quantum group theory and spectral triples. The quantum analogue of these groups are defined as universal (initial) objects in some natural categories. After proving the existence of the universal objects, we discuss several examples that are of interest to physics, as they appear in the noncommutative geometry approach to particle physics: in particular, the C*-algebras $M_n(\mathbb{R})$, $M_n(\mathbb{C})$ and $M_n(\mathbb{H})$, describing the finite noncommutative space of the Einstein–Yang–Mills systems, and the algebras $\mathcal{A}_F=\mathbb{C}\oplus \mathbb{H} \oplus M_3(\mathbb{C})$ and $\mathcal{A}^{\mathrm{ev}}=\mathbb{H}\oplus \mathbb{H} \oplus M_4(\mathbb{C})$, that appear in Chamseddine–Connes derivation of the Standard Model of particle physics coupled to gravity. As a byproduct, we identify a “free” version of the symplectic group $\operatorname{Sp}(n)$ (quaternionic unitary group).
Global analysis, analysis on manifolds
Group theory and generalizations
433
471
10.4171/JNCG/161
http://www.ems-ph.org/doi/10.4171/JNCG/161
Coupling of gravity to matter, spectral action and cosmic topology
Branimir
Ćaćić
Texas A&M University, COLLEGE STATION, UNITED STATES
Matilde
Marcolli
California Institute of Technology, PASADENA, UNITED STATES
Kevin
Teh
California Institute of Technology, PASADENA, UNITED STATES
Spectral action, cosmic topology, gravity coupled to matter, Poisson summation formula, heat kernel
We consider a model of modified gravity based on the spectral action functional, for a cosmic topology given by a spherical space form, and the associated slow-roll inflation scenario. We consider then the coupling of gravity to matter determined by an almost-commutative geometry over the spherical space form. We show that this produces a multiplicative shift of the amplitude of the power spectra for the density fluctuations and the gravitational waves, by a multiplicative factor equal to the total number of fermions in the matter sector of the model. We obtain the result by an explicit nonperturbative computation, based on the Poisson summation formula and the spectra of twisted Dirac operators on spherical space forms, as well as, for more general spacetime manifolds, using a heat kernel computation.
Global analysis, analysis on manifolds
General
473
504
10.4171/JNCG/162
http://www.ems-ph.org/doi/10.4171/JNCG/162
PBW-deformations and deformations à la Gerstenhaber of N-Koszul algebras
Estanislao
Herscovich
Universität Bielefeld, BIELEFELD, GERMANY
Andrea
Solotar
Universidad de Buenos Aires, BUENOS AIRES, ARGENTINA
Mariano
Suárez-Álvarez
Universidad de Buenos Aires, BUENOS AIRES, ARGENTINA
Deformation theory, Koszul algebras, Hochschild cohomology
In this article we establish an explicit link between the classical theory of deformations à la Gerstenhaber (and a fortiori with the Hochschild cohomology) and (weak) PBW-deformations of homogeneous algebras. Our point of view is of cohomological nature. As a consequence, we recover a theorem by R. Berger and V. Ginzburg, which gives a precise condition for a filtered algebra to satisfy the so-called PBW property, under certain assumptions.
Associative rings and algebras
General
505
539
10.4171/JNCG/163
http://www.ems-ph.org/doi/10.4171/JNCG/163
The space of Penrose tilings and the noncommutative curve with homogeneous coordinate ring k〈x,y〉/(y2)
S. Paul
Smith
University of Washington, SEATTLE, UNITED STATES
Homogeneous coordinate ring, noncommutative algebraic geometry, Penrose tiling, aperiodic tilings, AF-algebra, Grothendieck group
It is shown that the noncommutative algebraic curve with homogeneous coordinate ring $\mathbb{C}\langle x,y\rangle/(y^2)$ is a noncommutative algebraic-geometric analogue of the space of Penrose tilings of the plane. Individual tilings determine “points” on the noncommutative curve and the tilings coincide under isometry if and only if the skyscraper sheaves of the corresponding points are isomorphic. The category of quasi-coherent sheaves on the curve is equivalent to the category of modules over a von Neumann regular ring that is a direct limit of finite dimensional semisimple algebras. The norm closure of this von Neumann regular ring is the AF-algebra that Connes associates to the space of Penrose tilings. There is an algebraic analogue of the fact that every isometry-invariant subset of tilings is dense in the set of all Penrose tilings.
Algebraic geometry
Associative rings and algebras
541
586
10.4171/JNCG/164
http://www.ems-ph.org/doi/10.4171/JNCG/164
Twisted Calabi–Yau property of Ore extensions
Liyu
Liu
Fudan University, SHANGHAI, CHINA
Shengqiang
Wang
Fudan University, SHANGHAI, CHINA
Quanshui
Wu
Fudan University, SHANGHAI, CHINA
Ore extension, twisted Calabi–Yau algebra, Nakayama automorphism, Artin–Schelter regular algebra
Suppose that $E=A[x;\sigma,\delta]$ is an Ore extension with $\sigma$ an automorphism. It is proved that if $A$ is twisted Calabi–Yau of dimension $d$, then $E$ is twisted Calabi–Yau of dimension $d+1$. The relation between their Nakayama automorphisms is also studied. As an application, the Nakayama automorphisms of a class of 5-dimensional Artin–Schelter regular algebras are given explicitly.
Associative rings and algebras
General
587
609
10.4171/JNCG/165
http://www.ems-ph.org/doi/10.4171/JNCG/165
3
A Chern–Simons action for noncommutative spaces in general with the example SUq(2)
Oliver
Pfante
, LEIPZIG, GERMANY
Spectral triples, Chern–Simons action, quantum group SUq(2), path integral, gauge transformation
Witten constructed a topological quantum field theory with the Chern–Simons action as Lagrangian. We define a Chern–Simons action for 3-dimensional spectral triples. We prove gauge invariance of the Chern–Simons action, and we prove that it concurs with the classical one in the case the spectral triple comes from a 3-dimensional spin manifold. In contrast to the classical Chern–Simons action, or a noncommutative generalization of it introduced by A. H. Chamseddine, A. Connes, and M. Marcolli by use of cyclic cohomology, the formula of our definition contains a linear term which shifts the critical points of the action, i.e., the solutions of the corresponding variational problem. Additionally, we investigate and compute the action for a particular example: the quantum group SUq(2). Two different spectral triples were constructed for SUq(2). We investigate the Chern–Simons action defined in the present paper in both cases and conclude the non-topological nature of the action. Using the Chern–Simons action as Lagrangian we define and compute the path integral, at least conceptually.
Quantum theory
Functional analysis
Global analysis, analysis on manifolds
611
654
10.4171/JNCG/166
http://www.ems-ph.org/doi/10.4171/JNCG/166
Crossed interval groups and operations on the Hochschild cohomology
Michael
Batanin
Macquarie University, SYDNEY, AUSTRALIA
Martin
Markl
Czech Academy of Sciences, PRAGUE 1, CZECH REPUBLIC
Crossed interval group, Hochschild cohomology, natural operation
We prove that the operad $\mathcal{B}$ of natural operations on the Hochschild cohomology has the homotopy type of the operad of singular chains on the little disks operad. To achieve this goal, we introduce crossed interval groups and show that $\mathcal{B}$ is a certain crossed interval extension of an operad $\mathcal{T}$ whose homotopy type is known. This completes the investigation of the algebraic structure on the Hochschild cochain complex that has lasted for several decades.
Algebraic topology
Category theory; homological algebra
655
693
10.4171/JNCG/167
http://www.ems-ph.org/doi/10.4171/JNCG/167
Index of elliptic operators for diffeomorphisms of manifolds
Anton
Savin
Peoples’ Friendship University of Russia, MOSCOW, RUSSIAN FEDERATION
Boris
Sternin
Peoples’ Friendship University of Russia, MOSCOW, RUSSIAN FEDERATION
Noncommutative elliptic theory, index, cyclic cohomology, crossed product, Haefliger cohomology, Todd class
We develop an elliptic theory for operators associated with a diffeomorphism of a closed smooth manifold. The aim of the present paper is to obtain an index formula for such operators in terms of topological invariants of the manifold and the symbol of the operator. The symbol in this situation is an element of a certain crossed product. We express the index as the pairing of the class in K-theory defined by the symbol and the Todd class in periodic cyclic cohomology of the crossed product.
Global analysis, analysis on manifolds
$K$-theory
695
734
10.4171/JNCG/168
http://www.ems-ph.org/doi/10.4171/JNCG/168
Asymptotic morphisms and superselection theory in the scaling limit
Roberto
Conti
Università di Roma La Sapienza, ROMA, ITALY
Gerardo
Morsella
Università di Roma Tor Vergata, ROMA, ITALY
Scaling algebras, asymptotic morphisms, superselection theory
Given a local Haag–Kastler net of von Neumann algebras and one of its scaling limit states, we introduce a variant of the notion of asymptotic morphism by Connes and Higson, and we show that the unitary equivalence classes of (localized) morphisms of the scaling limit theory of the original net are in bijection with classes of suitable pairs of such asymptotic morphisms. In the process, we also show that the quasi-local C*-algebras of two nets are isomorphic under very general hypotheses, and we construct an extension of the scaling algebra whose representation on the scaling limit Hilbert space contains the local von Neumann algebras. We also study the relation between our asymptotic morphisms and superselection sectors preserved in the scaling limit.
Quantum theory
Functional analysis
735
770
10.4171/JNCG/169
http://www.ems-ph.org/doi/10.4171/JNCG/169
Isomorphism invariants for multivariable C*-dynamics
Evgenios
Kakariadis
Newcastle University, NEWCASTLE UPON TYNE, UNITED KINGDOM
Elias
Katsoulis
East Carolina University, GREENVILLE, UNITED STATES
C*-algebra, dynamical system, piecewise conjugacy, Fell spectrum, outer conjugacy
To a given multivariable C*-dynamical system $(A, \alpha)$ consisting of *-automorphisms, we associate a family of operator algebras $\mathrm{alg}(A, \alpha)$, which includes as specific examples the tensor algebra and the semicrossed product. It is shown that if two such operator algebras $\mathrm{alg}(A, \alpha)$ and $\mathrm{alg}(B, \beta)$ are isometrically isomorphic, then the induced dynamical systems $(\hat{A}, \hat{\alpha})$ and $(\hat{B}, \hat{\beta})$ on the Fell spectra are piecewise conjugate in the sense of Davidson and Katsoulis. In the course of proving the above theorem we obtain several results of independent interest. If $\mathrm{alg}(A, \alpha)$ and $\mathrm{alg}(B, \beta)$ are isometrically isomorphic, then the associated correspondences $X_{(A, \alpha)}$ and $X_{(B, \beta)}$ are unitarily equivalent. In particular, the tensor algebras are isometrically isomorphic if and only if the associated correspondences are unitarily equivalent. Furthermore, isomorphism of semicrossed products implies isomorphism of the associated tensor algebras. In the case of multivariable systems acting on C*-algebras with trivial center, unitary equivalence of the associated correspondences reduces to outer conjugacy of the systems. This provides a complete invariant for isometric isomorphisms between semicrossed products as well.
Operator theory
General
771
787
10.4171/JNCG/170
http://www.ems-ph.org/doi/10.4171/JNCG/170
Leafwise homotopies and Hilbert-Poincaré complexes I. Regular HP-complexes and leafwise pull-back maps
Moulay Tahar
Benameur
Université Montpellier 2, MONTPELLIER CEDEX 5, FRANCE
Indrava
Roy
Università di Roma “La Sapienza”, ROMA, ITALY
Foliations, laminations, leafwise homotopy equivalence, K-theoretic signatures, Morita equivalence, K-theory of C*-algebras, maximal Baum--Connes conjecture, imprimitivity bimodules, rho-invariants
In the first part of our series of papers, we prove the leafwise homotopy invariance of K-theoretic signatures of foliations and laminations, using the formalism of Hilbert–Poincaré complexes as revisited by Higson and Roe. We use a generalization of their methods to give a homotopy equivalence of de Rham–Hilbert–Poincaré complexes associated with leafwise homotopy equivalence of foliations. In particular, we obtain an explicit path connecting the signature classes in K-theory, up to isomorphism induced by Morita equivalence. Applications of this path on the stability properties of rho-invariants à la Keswani will be carried out in the later parts of this series.
Global analysis, analysis on manifolds
General
789
836
10.4171/JNCG/171
http://www.ems-ph.org/doi/10.4171/JNCG/171
Equivariant Kasparov theory of finite groups via Mackey functors
Ivo
Dell'Ambrogio
Université de Lille 1, Villeneuve-d’Ascq Cedex, FRANCE
Equivariant KK-theory, universal coefficient theorem, Künneth formula, Mackey functors, Green functors
Let $G$ be any finite group. In this paper we systematically exploit general homological methods in order to reduce the computation of $G$-equivariant KK-theory to topological equivariant K-theory. The key observation is that the functor on $\mathsf{KK}^G$ that assigns to a $G$-C*-algebra $A$ the collection of its K-theory groups $\{ K^H_*(A) : H\leqslant G \}$ admits a lifting to the abelian category of $\mathbb{Z}/2$-graded Mackey modules over the representation Green functor for $G$; moreover, this lifting is the universal exact homological functor for the resulting relative homological algebra in $\mathsf{KK}^G$. It follows that there is a spectral sequence abutting to $\mathsf{KK}^G_*(A,B)$, whose second page displays Ext groups computed in the category of Mackey modules. Due to the nice properties of Mackey functors, we obtain a similar Künneth spectral sequence which computes the equivariant K-theory groups of a tensor product $A\otimes B$. Both spectral sequences behave nicely if $A$ belongs to the localizing subcategory of $\mathsf{KK}^G$ generated by the algebras $C(G/H)$ for all subgroups $H\leqslant G$.
Functional analysis
$K$-theory
837
871
10.4171/JNCG/172
http://www.ems-ph.org/doi/10.4171/JNCG/172
On the arithmetic of the BC-system
Alain
Connes
Le Bois-Marie, BURES-SUR-YVETTE, FRANCE
Caterina
Consani
The Johns Hopkins University, BALTIMORE, UNITED STATES
Witt rings, finite fields, BC-system
For each prime $p$ and each embedding $\sigma$ of the multiplicative group of an algebraic closure of $\mathbb{F}_p$ as complex roots of unity, we construct a $p$-adic indecomposable representation $\pi_\sigma$ of the integral BC-system as additive endomorphisms of the big Witt ring of $\bar{\mathbb{F}}_p$. The obtained representations are the $p$-adic analogues of the complex, extremal KMS$_\infty$ states of the BC-system. The role of the Riemann zeta function, as partition function of the BC-system over $\mathbb{C}$ is replaced, in the $p$-adic case, by the $p$-adic $L$-functions and the polylogarithms whose values at roots of unity encode the KMS states. We use Iwasawa theory to extend the KMS theory to a covering of the completion $\mathbb{C}_p$ of an algebraic closure of $\mathbb{Q}_p$. We show that our previous work on the hyperring structure of the adèle class space, combines with $p$-adic analysis to refine the space of valuations on the cyclotomic extension of $\mathbb{Q}$ as a noncommutative space intimately related to the integral BC-system and whose arithmetic geometry comes close to fulfill the expectations of the “arithmetic site”. Finally, we explain how the integral BC-system appears naturally also in de Smit and Lenstra construction of the standard model of $\bar{\mathbb{F}}_p$ which singles out the subsystem associated to the $\hat{\mathbb{Z}}$-extension of $\mathbb{Q}$.
Number theory
Functional analysis
Global analysis, analysis on manifolds
873
945
10.4171/JNCG/173
http://www.ems-ph.org/doi/10.4171/JNCG/173
4
Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets
Michel
Lapidus
University of California, RIVERSIDE, UNITED STATES
Jonathan
Sarhad
University of California, RIVERSIDE, UNITED STATES
Analysis on fractals, noncommutative fractal geometry, Laplacians and Dirac operators on fractals, spectral triples, spectral dimension, measurable Riemannian geometry, geodesics on fractals, geodesic and noncommutative metrics, fractals built on curves, Euclidean and harmonic Sierpinski gaskets, geometric analysis on fractals, fractal manifold
We construct Dirac operators and spectral triples for certain, not necessarily selfsimilar, fractal sets built on curves. Connes’ distance formula of noncommutative geometry provides a natural metric on the fractal. To motivate the construction, we address Kigami’s measurable Riemannian geometry, which is a metric realization of the Sierpinski gasket as a self-a ne space with continuously di erentiable geodesics. As a fractal analog of Connes’ theorem for a compact Riemmanian manifold, it is proved that the natural metric coincides with Kigami’s geodesic metric. This present work extends to the harmonic gasket and other fractals built on curves a significant part of the earlier results of E. Christensen, C. Ivan, and the first author obtained, in particular, for the Euclidean Sierpinski gasket. (As is now well known, the harmonic gasket, unlike the Euclidean gasket, is ideally suited to analysis on fractals. It can be viewed as the Euclidean gasket in harmonic coordinates.) Our current, broader framework allows for a variety of potential applications to geometric analysis on fractal manifolds.
Measure and integration
Ordinary differential equations
Functional analysis
Differential geometry
947
985
10.4171/JNCG/174
http://www.ems-ph.org/doi/10.4171/JNCG/174
On noncommutative principal bundles with finite abelian structure group
Stefan
Wagner
University of Helsinki, HELSINKI, FINLAND
Noncommutative differential geometry, dynamical systems, (trivial) principal bundles with finite abelian structure group, (trivial) noncommutative principal bundles with finite abelian structure group, graded algebras, crossed-product algebras, factor systems
Let $\Lambda$ be a finite abelian group. A dynamical system with transformation group $\Lambda$ is a triple $(A,\Lambda,\alpha)$, consisting of a unital locally convex algebra $A$, the finite abelian group $\Lambda$ and a group homomorphism $\alpha:\Lambda\rightarrow\Aut(A)$, which induces an action of $\Lambda$ on $A$. In this paper we present a new, geometrically oriented approach to the noncommutative geometry of principal bundles with finite abelian structure group based on such dynamical systems.
Functional analysis
Nonassociative rings and algebras
Dynamical systems and ergodic theory
Algebraic topology
987
1022
10.4171/JNCG/175
http://www.ems-ph.org/doi/10.4171/JNCG/175
$\mathcal{Z}$ is universal
Bhishan
Jacelon
Universität Münster, MÜNSTER, GERMANY
Wilhelm
Winter
Westfälische Wilhelms-Universität Münster, MÜNSTER, GERMANY
Jiang–Su algebra, strongly self-absorbing C*-algebra, stably projectionless C*-algebra, order zero map, classification
We use order zero maps to express the Jiang–Su algebra $\mathcal{Z}$ as a universal C* -algebra on countably many generators and relations, and we show that a natural deformation of these relations yields the stably projectionless algebra $\mathcal{W}$ studied by Kishimoto, Kumjian and others. Our presentation is entirely explicit and involves only *-polynomial and order relations.
Functional analysis
General
1023
1042
10.4171/JNCG/176
http://www.ems-ph.org/doi/10.4171/JNCG/176
On weakly group-theoretical non-degenerate braided fusion categories
Sonia
Natale
Universidad Nacional de Cordoba, CORDOBA, ARGENTINA
Braided fusion category, braided $G$-crossed fusion category, Tannakian category, Witt class, solvability
We show that the Witt class of a weakly group-theoretical non-degenerate braided fusion category belongs to the subgroup generated by classes of non-degenerate pointed braided fusion categories and Ising braided categories. This applies in particular to solvable non-degenerate braided fusion categories. We also give some su cient conditions for a braided fusion category to be weakly group-theoretical or solvable in terms of the factorization of its Frobenius–Perron dimension and the Frobenius–Perron dimensions of its simple objects. As an application, we prove that every non-degenerate braided fusion category whose Frobenius–Perron dimension is a natural number less than 1800, or an odd natural number less than 33075, is weakly group-theoretical.
Category theory; homological algebra
General
1043
1060
10.4171/JNCG/177
http://www.ems-ph.org/doi/10.4171/JNCG/177
Kirchberg $X$-algebras with real rank zero and intermediate cancellation
Rasmus
Bentmann
Georg-August Universität Göttingen, GÖTTINGEN, GERMANY
Kirchberg algebras, K-theory
A universal coe cient theorem is proved for $C*$ -algebras over an arbitrary finite $T_0$-space $X$ which have vanishing boundary maps. Under bootstrap assumptions, this leads to a complete classification of unital/stable real-rank-zero Kirchberg $X$-algebras with intermediate cancellation. Range results are obtained for (unital) purely infinite graph $C*$ -algebras with intermediate cancellation and Cuntz–Krieger algebras with intermediate cancellation. Permanence results for extensions of these classes follow.
Category theory; homological algebra
General
$K$-theory
Functional analysis
1061
1081
10.4171/JNCG/178
http://www.ems-ph.org/doi/10.4171/JNCG/178
Combinatorial descent data for gerbes
Amnon
Yekutieli
Ben Gurion University, BEER SHEVA, ISRAEL
Cosimplicial crossed groupoids, descent, gerbes
We consider descent data in cosimplicial crossed groupoids. This is a combinatorial abstraction of the descent data for gerbes in algebraic geometry. The main result is this: a weak equivalence between cosimplicial crossed groupoids induces a bijection on gauge equivalence classes of descent data.
Category theory; homological algebra
Group theory and generalizations
1083
1099
10.4171/JNCG/179
http://www.ems-ph.org/doi/10.4171/JNCG/179
On universal gradings, versal gradings and Schurian generated categories
Claude
Cibils
Université de Montpellier 2, MONTPELLIER CEDEX 5, FRANCE
María Julia
Redondo
Universidad Nacional del Sur, BAHIA BLANCA, ARGENTINA
Andrea
Solotar
Universidad de Buenos Aires, BUENOS AIRES, ARGENTINA
Grading, universal, versal, fundamental group, Schurian, Grothendieck, category
Categories over a field $k$ can be graded by di erent groups in a connected way; we consider morphisms between these gradings in order to define the fundamental grading group. We prove that this group is isomorphic to the fundamental group à la Grothendieck as considered in previous papers. In case the $k$-category is Schurian generated we prove that a universal grading exists. Examples of non-Schurian generated categories with universal grading, versal grading or none of them are considered.
Associative rings and algebras
General
Category theory; homological algebra
Algebraic topology
1101
1122
10.4171/JNCG/180
http://www.ems-ph.org/doi/10.4171/JNCG/180
Almost normal operators mod Hilbert–Schmidt and the $K$-theory of the algebras $E \Lambda (\Omega)$
Dan-Virgil
Voiculescu
University of California, BERKELEY, UNITED STATES
Trace-class self-commutator, K-theory, Dirichlet algebras, bidual Banach algebra
Is there a mod Hilbert–Schmidt analogue of the BDF-theorem, with the Pincus $g$-function playing the role of the index? We show that part of the question is about the $K$-theory of certain Banach algebras. These Banach algebras, related to Lipschitz functions and Dirichlet algebras have nice Banach-space duality properties. Moreover their corona algebras are $C*$ -algebras.
Functional analysis
General
Operator theory
1123
1145
10.4171/JNCG/181
http://www.ems-ph.org/doi/10.4171/JNCG/181
Noncommutative unfolding of hypersurface singularity
Vladimir
Hinich
University of Haifa, HAIFA, ISRAEL
Dan
Lemberg
University of Haifa, HAIFA, ISRAEL
Kontsevich formality theorem, dg algebras, operads, hypersurface singularity
A version of Kontsevich formality theorem is proven for smooth DG algebras. As an application of this, it is proven that any quasiclassical datum of noncommutative unfolding of an isolated surface singularity can be quantized.
Commutative rings and algebras
General
Algebraic geometry
Several complex variables and analytic spaces
1147
1169
10.4171/JNCG/182
http://www.ems-ph.org/doi/10.4171/JNCG/182
Lefschetz and Hirzebruch–Riemann–Roch formulas via noncommutative motives
Denis-Charles
Cisinski
Université Paul Sabatier, TOULOUSE CEDEX 9, FRANCE
Gonçalo
Tabuada
Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES
Lefschetz formula, Hirzebruch–Riemann–Roch formula, Euler characteristic, Fourier–Mukai functors, noncommutative motives
V. Lunts has recently established Lefschetz fixed point theorems for Fourier–Mukai functors and dg algebras. In the same vein, D. Shklyarov introduced the noncommutative analogue of the Hirzebruch–Riemman–Roch theorem. In this article, we see how these constructions and computations formally stem from their motivic counterparts.
Algebraic geometry
General
Category theory; homological algebra
$K$-theory
1171
1190
10.4171/JNCG/183
http://www.ems-ph.org/doi/10.4171/JNCG/183
Cyclic homology, tight crossed products, and small stabilizations
Guillermo
Cortiñas
Universidad de Buenos Aires, BUENOS AIRES, ARGENTINA
Operator ideal, Calkin’s theorem, crossed product, Karoubi’s cone, cyclic homology
In [1] we associated an algebra $\Gamma^\infty (\mathfrak A)$ to every bornological algebra $\mathfrak A$ and an ideal $I_{S(\mathfrak A)}\vartriangleleft \Gamma^\infty (\mathfrak A)$ to every symmetric ideal $S\vartriangleleft \ell^{infty}$. We showed that $I_{S(\mathfrak A)}$ has $K$-theoretical properties which are similar to those of the usual stabilization with respect to the ideal $J_S\vartriangleleft \mathcal B$ of the algebra $\mathcal B$ of bounded operators in Hilbert space which corresponds to $S$ under Calkin's correspondence. In the current article we compute the relative cyclic homology $HC_*(\Gamma^\infty (\mathfrak A):I_{S(\mathfrak A)})$. Using these calculations, and the results of loc. cit., we prove that if $\mathfrak A$ is a $C^*$-algebra and $c_0$ the symmetric ideal of sequences vanishing at infinity, then $K_*(I_{c_0(\mathfrak A)})$ is homotopy invariant, and that if $*\ge 0$, it contains $K^{\top}_*(\mathfrak A)$ as a direct summand. This is a weak analogue of the Suslin-Wodzicki theorem [20] that says that for the ideal $\mathcal K=J_{c_0}$ of compact operators and the $C^*$-algebra tensor product $\mathfrak A\overset{\sim}{\otimes}\mathcal K$, we have $K_*(\mathfrak A\overset{\sim}{\otimes} \mathcal K)=K^{\top}_*(\mathfrak A)$. Similarly, we prove that if $\mathfrak A$ is a unital Banach algebra and $\ell^{\infty-}=\bigcup_{q
$K$-theory
Operator theory
1191
1223
10.4171/JNCG/184
http://www.ems-ph.org/doi/10.4171/JNCG/184