scholarly journals CONTINUOUS TRACE C*-ALGEBRAS, GAUGE GROUPS AND RATIONALIZATION

2009 ◽  
Vol 01 (03) ◽  
pp. 261-288 ◽  
Author(s):  
JOHN R. KLEIN ◽  
CLAUDE L. SCHOCHET ◽  
SAMUEL B. SMITH
Keyword(s):  
Gauge Group ◽  
Metric Space ◽  
Principal Bundle ◽  
Rational Homotopy ◽  
Compact Metric Space ◽  
Gauge Groups ◽  
C Algebra ◽  
C Algebras ◽  
Rational Cohomology ◽  
Rational Homotopy Type

Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let Aζdenote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UAζ, the group of unitaries of Aζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.

scholarly journals Gauge groups and bialgebroids

2021 ◽  
Vol 111 (6) ◽  
Author(s):  
Xiao Han ◽  
Giovanni Landi
Keyword(s):  
Gauge Group ◽  
Galois Extension ◽  
Principal Bundle ◽  
Module Structure ◽  
Crossed Module ◽  
Gauge Groups ◽  
Quantum Sphere ◽  
Hopf Algebroid ◽  
Commutative Algebras

AbstractWe study the Ehresmann–Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the gauge groupoid of a classical principal bundle. We show that the gauge group of the noncommutative bundle is isomorphic to the group of bisections of the bialgebroid, and we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include: Galois objects of Taft algebras, a monopole bundle over a quantum sphere and a not faithfully flat Hopf–Galois extension of commutative algebras. For each of the latter two examples, there is in fact a suitable invertible antipode for the bialgebroid making it a Hopf algebroid.

scholarly journals A zoo of growth functions of mapping class sets

2018 ◽  
Vol 12 (03) ◽  
pp. 841-855 ◽  
Author(s):  
Fedor Manin
Keyword(s):  
Mapping Class ◽  
Rational Homotopy ◽  
Growth Functions ◽  
Lipschitz Maps ◽  
Rational Cohomology ◽  
Rational Homotopy Type ◽  
Number Formula ◽  
Mapping Classes ◽  
Simply Connected ◽  
The Universe

Suppose [Formula: see text] and [Formula: see text] are finite complexes, with [Formula: see text] simply connected. Gromov conjectured that the number of mapping classes in [Formula: see text] which can be realized by [Formula: see text]-Lipschitz maps grows asymptotically as [Formula: see text], where [Formula: see text] is an integer determined by the rational homotopy type of [Formula: see text] and the rational cohomology of [Formula: see text]. This conjecture was disproved in a recent paper of the author and Weinberger; we gave an example where the “predicted” growth is [Formula: see text] but the true growth is [Formula: see text]. Here we show, via a different mechanism, that the universe of possible such growth functions is quite large. In particular, for every rational number [Formula: see text], there is a pair [Formula: see text] for which the growth of [Formula: see text] is essentially [Formula: see text].

On p-local homotopy types of gauge groups

2014 ◽  
Vol 144 (1) ◽  
pp. 149-160 ◽  
Author(s):  
Daisuke Kishimoto ◽  
Akira Kono ◽  
Mitsunobu Tsutaya
Keyword(s):  
Gauge Group ◽  
Homotopy Type ◽  
Principal Bundle ◽  
Dimensional Sphere ◽  
Gauge Groups ◽  
Homotopy Types

The aim of this paper is to show that the p-local homotopy type of the gauge group of a principal bundle over an even-dimensional sphere is completely determined by the divisibility of the classifying map by p. In particular, for gauge groups of principal SU(n)-bundles over S2d for 2 ≤ d ≤ p − 1 and n ≤ 2p − 1, we give a concrete classification of their p-local homotopy types.

LIMITS OF HOMOMORPHISMS WITH FINITE-DIMENSIONAL RANGE

2005 ◽  
Vol 16 (07) ◽  
pp. 807-821 ◽  
Author(s):  
SHANWEN HU ◽  
HUAXIN LIN ◽  
YIFENG XUE
Keyword(s):  
Metric Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
C Algebra ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Unital Simple ◽  
Dimensional Range

Let X be a compact metric space and A be a unital simple C*-algebra with TR (A)=0. Suppose that ϕ : C(X) → A is a unital monomorphism. We study the problem when ϕ can be approximated by homomorphisms with finite-dimensional range. We give a K-theoretical necessary and sufficient condition for ϕ being approximated by homomorphisms with finite-dimensional range.

scholarly journals The structure of the C*-algebra of a locally injective surjection

2011 ◽  
Vol 32 (4) ◽  
pp. 1226-1248 ◽  
Author(s):  
TOKE MEIER CARLSEN ◽  
KLAUS THOMSEN
Keyword(s):  
Metric Space ◽  
Matrix Algebra ◽  
Finite Covering ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
Classification Result ◽  
Full Matrix ◽  
C Algebra ◽  
Simple Quotient ◽  
The Ideal

AbstractIn this paper we investigate the ideal structure of the C*-algebra of a locally injective surjection introduced by the second-named author. Our main result is that a simple quotient of the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is either a full matrix algebra, a crossed product of a minimal homeomorphism of a compact metric space of finite covering dimension, or it is purely infinite and hence covered by the classification result of Kirchberg and Phillips. It follows in particular that if the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is simple, then it is automatically purely infinite, unless the map in question is a homeomorphism. A corollary of this result is that if the C*-algebra of a one-sided subshift is simple, then it is also purely infinite.

Characterizations of topological dynamical systems whose transformation group C*-algebras are antiliminal and of type 1

1993 ◽  
Vol 13 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Nobuo Aoki ◽  
Jun Tomiyama
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Space ◽  
Transformation Group ◽  
Irreducible Representations ◽  
Topological Dynamical System ◽  
Compact Metric Space ◽  
C Algebras ◽  
Topological Dynamical Systems

AbstractFor a topological dynamical system Σ = (X, σ) where X is a compact metric space with a single homeomorphism σ, we determine the largest postliminal ideal of the transformation group C*-algebra A(Σ) as the intersection of all kernels of irreducible representations of A(Σ) induced from those recurrent points which are not periodic. The result implies characterizations of topological dynamical systems whose transformation group C*-algebras are anti-liminal and post-liminal, that is, of type 1.

scholarly journals Banach algebras, Samelson products, and the Wang differential

2014 ◽  
Vol 06 (02) ◽  
pp. 281-303
Author(s):  
Claude L. Schochet
Keyword(s):  
Banach Algebra ◽  
Principal Bundle ◽  
Banach Algebras ◽  
Homotopy Groups ◽  
Rational Homotopy ◽  
Connected Component ◽  
C Algebra ◽  
Close Analysis ◽  
K Theory ◽  
Compare And Contrast

Assume that given a principal G bundle ζ : P → Sk (with k ≥ 2) and a Banach algebra B upon which G acts continuously. Let [Formula: see text] denote the associated bundle and let [Formula: see text] denote the associated Banach algebra of sections. Then π* GL Aζ⊗B is determined by a mostly degenerate spectral sequence and by a Wang differential [Formula: see text] We show that if B is a C*-algebra then the differential is given explicitly in terms of an enhanced Samelson product with the clutching map of the principal bundle. Analogous results hold after localization and in the setting of topological K-theory. We illustrate our technique with a close analysis of the invariants associated to the C*-algebra of sections of the bundle [Formula: see text] constructed from the Hopf bundle ζ : S7 → S4 and by the conjugation action of S3 on M2 = M2(ℂ). We compare and contrast the information obtained from the homotopy groups π*( U ◦Aζ⊗M2), the rational homotopy groups π*( U ◦Aζ⊗M2) ⊗ ℚ and the topological K-theory groups K*(Aζ⊗M2), where U ◦B is the connected component of the unitary group of the C*-algebra B.

ALMOST MULTIPLICATIVE MORPHISMS AND K-THEORY

2000 ◽  
Vol 11 (08) ◽  
pp. 983-1000 ◽  
Author(s):  
GUIHUA GONG ◽  
HUAXIN LIN
Keyword(s):  
Metric Space ◽  
Classification Theory ◽  
Compact Metric Space ◽  
Dimensional Image ◽  
C Algebras ◽  
Finite Dimensional ◽  
Finite Set ◽  
K Theory

Let X be a compact metric space and A=C(X). Suppose that ℬ is a class of unital C*-algebras satisfying certain conditions, we prove the following: For any ∊>0, finite set F⊂A, there is an integer l such that if ϕ, ψ:A→B(B∈ℬ) are sufficiently multiplicative morphisms (e.g. when both ϕ and ψ are *-homomorphisms) which induce same K-theoretical maps, then there are a unitary u∈Ml+1(B) and a homomorphism σ:A→Ml(B) with finite dimensional image such that [Formula: see text] for all f∈F. In particular, the integer l does not depend on B, ϕ and ψ. This feature has important applications to the classification theory of nuclear C*-algebras.

MINIMAL DYNAMICS AND $\mathcal{Z}$-STABLE CLASSIFICATION

2011 ◽  
Vol 22 (01) ◽  
pp. 1-23 ◽  
Author(s):  
KAREN R. STRUNG ◽  
WILHELM WINTER
Keyword(s):  
Dynamical Systems ◽  
Compact Metric Space ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Minimal Dynamical Systems ◽  
Uniquely Ergodic ◽  
Stable Classification

Let X be an infinite compact metric space, α : X → X a minimal homeomorphism, u the unitary that implements α in the transformation group C*-algebra C(X) ⋊α ℤ, and [Formula: see text] a class of separable nuclear C*-algebras that contains all unital hereditary C*-subalgebras of C*-algebras in [Formula: see text]. Motivated by the success of tracial approximation by finite dimensional C*-algebras as an abstract characterization of classifiable C*-algebras and the idea that classification results for C*-algebras tensored with UHF algebras can be used to derive classification results up to tensoring with the Jiang-Su algebra [Formula: see text], we prove that (C(X) ⋊α ℤ) ⊗ Mq∞ is tracially approximately [Formula: see text] if there exists a y ∈ X such that the C*-subalgebra (C*(C(X), uC0(X\{y}))) ⊗ Mq∞ is tracially approximately [Formula: see text]. If the class [Formula: see text] consists of finite dimensional C*-algebras, this can be used to deduce classification up to tensoring with [Formula: see text] for C*-algebras associated to minimal dynamical systems where projections separate tracial states. This is done without making any assumptions on the real rank or stable rank of either C(X) ⋊α ℤ or C*(C(X), uC0(X\{y})), nor on the dimension of X. The result is a key step in the classification of C*-algebras associated to uniquely ergodic minimal dynamical systems by their ordered K-groups. It also sets the stage to provide further classification results for those C*-algebras of minimal dynamical systems where projections do not necessarily separate traces.

Sign in / Sign up

Export Citation Format

Share Document