scholarly journals Groupoid algebras as Cuntz-Pimsner algebras

2017 ◽  
Vol 120 (1) ◽  
pp. 115 ◽  
Author(s):  
Adam Rennie ◽  
David Robertson ◽  
Aidan Sims
Keyword(s):  
Exact Sequence ◽  
Locally Compact ◽  
C Algebra ◽  
C Algebras ◽  
K Theory ◽  
Étale Groupoid

We show that if $G$ is a second countable locally compact Hausdorff étale groupoid carrying a suitable cocycle $c\colon G\to\mathbb{Z}$, then the reduced $C^*$-algebra of $G$ can be realised naturally as the Cuntz-Pimsner algebra of a correspondence over the reduced $C^*$-algebra of the kernel $G_0$ of $c$. If the full and reduced $C^*$-algebras of $G_0$ coincide, we deduce that the full and reduced $C^*$-algebras of $G$ coincide. We obtain a six-term exact sequence describing the $K$-theory of $C^*_r(G)$ in terms of that of $C^*_r(G_0)$.

scholarly journals C*-Algebras over Topological Spaces: Filtrated K-Theory

2012 ◽  
Vol 64 (2) ◽  
pp. 368-408 ◽  
Author(s):  
Ralf Meyer ◽  
Ryszard Nest
Keyword(s):  
Exact Sequence ◽  
Topological Space ◽  
Spectral Sequence ◽  
Topological Spaces ◽  
C Algebra ◽  
C Algebras ◽  
Finite Spaces ◽  
K Theory ◽  
Universal Coefficient Theorem ◽  
Kasparov Theory

AbstractWe define the filtrated K-theory of a C*-algebra over a finite topological spaceXand explain how to construct a spectral sequence that computes the bivariant Kasparov theory overXin terms of filtrated K-theory.For finite spaces with a totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated K-theory is not yet a complete invariant. We describe two C*-algebras over a spaceXwith four points that have isomorphic filtrated K-theory without being KK(X)-equivalent. For this spaceX, we enrich filtrated K-theory by another K-theory functor to a complete invariant up to KK(X)-equivalence that satisfies a Universal Coefficient Theorem.

scholarly journals Finitely generated nilpotent group C*-algebras have finite nuclear dimension

2018 ◽  
Vol 2018 (738) ◽  
pp. 281-298 ◽  
Author(s):  
Caleb Eckhardt ◽  
Paul McKenney
Keyword(s):  
Irreducible Representation ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Irreducible Representations ◽  
Finitely Generated ◽  
C Algebra ◽  
C Algebras ◽  
K Theory ◽  
Universal Coefficient Theorem ◽  
Torsion Free

Abstract We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra A generated by an irreducible representation of such a group has decomposition rank at most 3. If, in addition, A satisfies the universal coefficient theorem, another string of deep results shows it is classifiable by its ordered K-theory and is approximately subhomogeneous. We observe that all C*-algebras generated by faithful irreducible representations of finitely generated, torsion free nilpotent groups satisfy the universal coefficient theorem.

Non-Commutative Deformations of C(T2) and K-Theory

1997 ◽  
Vol 08 (05) ◽  
pp. 555-571
Author(s):  
Cristina Cerri
Keyword(s):  
Positive Element ◽  
Crossed Product ◽  
C Algebra ◽  
Basic Properties ◽  
C Algebras ◽  
The Family ◽  
K Theory

For each α ≥ 0, let Bα be the universal C*-algebra generated by unitary elements uα, vα and a self-adjoint hα such that ||hα|| ≤ α and [Formula: see text]. In this work we prove that the family {Bα}α ∈ [0,∞[ extend the family of soft torus with the same basic properties, i.e., the field of C*-algebras {Bα}α ∈ [0,α0] is continuous and each Bα is a crossed product of a C*-algebra homotopically equivalent to C(S1) by Z. We then show that the K-groups of Bα are isomorphic to Z ⊕ Z. Applying results from the theory of rotation algebras we prove that every positive element (n,m) in K0(Bα) satisfies |m|α ≤ 2πn. It follows that these C*-algebras are not all homotopically equivalent to each other, although they have the same K-groups.

scholarly journals Equivariant KK-theory for generalised actions and Thom isomorphism in groupoid twisted K-theory

2013 ◽  
Vol 13 (1) ◽  
pp. 83-113 ◽  
Author(s):  
El-Kaïoum M. Moutuou
Keyword(s):  
Vector Bundles ◽  
Isomorphism Theorem ◽  
Locally Compact ◽  
Bott Periodicity ◽  
C Algebras ◽  
Thom Isomorphism ◽  
K Theory ◽  
Thom Isomorphism Theorem

AbstractWe develop equivariant KK–theory for locally compact groupoid actions by Morita equivalences on real and complex graded C*-algebras. Functoriality with respect to generalised morphisms and Bott periodicity are discussed. We introduce Stiefel-Whitney classes for real or complex equivariant vector bundles over locally compact groupoids to establish the Thom isomorphism theorem in twisted groupoid K–theory.

QUANTIZATION AND SUPERSELECTION SECTORS I: TRANSFORMATION GROUP C*-ALGEBRAS

1990 ◽  
Vol 02 (01) ◽  
pp. 45-72 ◽  
Author(s):  
N.P. LANDSMAN
Keyword(s):  
Irreducible Representation ◽  
Large Class ◽  
Configuration Space ◽  
Topological Charge ◽  
Transformation Group ◽  
Irreducible Representations ◽  
Locally Compact ◽  
C Algebra ◽  
C Algebras ◽  
Charge Quantization

Quantization is defined as the act of assigning an appropriate C*-algebra [Formula: see text] to a given configuration space Q, along with a prescription mapping self-adjoint elements of [Formula: see text] into physically interpretable observables. This procedure is adopted to solve the problem of quantizing a particle moving on a homogeneous locally compact configuration space Q=G/H. Here [Formula: see text] is chosen to be the transformation group C*-algebra corresponding to the canonical action of G on Q. The structure of these algebras and their representations are examined in some detail. Inequivalent quantizations are identified with inequivalent irreducible representations of the C*-algebra corresponding to the system, hence with its superselection sectors. Introducing the concept of a pre-Hamiltonian, we construct a large class of G-invariant time-evolutions on these algebras, and find the Hamiltonians implementing these time-evolutions in each irreducible representation of [Formula: see text]. “Topological” terms in the Hamiltonian (or the corresponding action) turn out to be representation-dependent, and are automatically induced by the quantization procedure. Known “topological” charge quantization or periodicity conditions are then identically satisfied as a consequence of the representation theory of [Formula: see text].

HYPERREFLEXIVITY CONSTANTS OF THE BOUNDED -COCYCLE SPACES OF GROUP ALGEBRAS AND C*-ALGEBRAS

2019 ◽  
Vol 109 (1) ◽  
pp. 112-130
Author(s):  
EBRAHIM SAMEI ◽  
JAFAR SOLTANI FARSANI
Keyword(s):  
Polynomial Growth ◽  
Banach Algebras ◽  
Amenable Group ◽  
Open Subgroup ◽  
Group Algebras ◽  
Convolution Operators ◽  
Locally Compact ◽  
C Algebra ◽  
C Algebras ◽  
Strong Property

We introduce the concept of strong property $(\mathbb{B})$ with a constant for Banach algebras and, by applying a certain analysis on the Fourier algebra of the unit circle, we show that all C*-algebras and group algebras have the strong property $(\mathbb{B})$ with a constant given by $288\unicode[STIX]{x1D70B}(1+\sqrt{2})$. We then use this result to find a concrete upper bound for the hyperreflexivity constant of ${\mathcal{Z}}^{n}(A,X)$, the space of bounded $n$-cocycles from $A$ into $X$, where $A$ is a C*-algebra or the group algebra of a group with an open subgroup of polynomial growth and $X$ is a Banach $A$-bimodule for which ${\mathcal{H}}^{n+1}(A,X)$ is a Banach space. As another application, we show that for a locally compact amenable group $G$ and $1<p<\infty$, the space $CV_{P}(G)$ of convolution operators on $L^{p}(G)$ is hyperreflexive with a constant given by $384\unicode[STIX]{x1D70B}^{2}(1+\sqrt{2})$. This is the generalization of a well-known result of Christensen [‘Extensions of derivations. II’, Math. Scand. 50(1) (1982), 111–122] for $p=2$.

scholarly journals QUOTIENTS OF ÉTALE GROUPOIDS AND THE ABELIANIZATIONS OF GROUPOID C*-ALGEBRAS

2020 ◽  
pp. 1-20
Author(s):  
FUYUTA KOMURA
Keyword(s):  
Uniqueness Theorem ◽  
Simple Proof ◽  
C Algebra ◽  
C Algebras ◽  
Étale Groupoid

In this paper, we introduce quotients of étale groupoids. Using the notion of quotients, we describe the abelianizations of groupoid C*-algebras. As another application, we obtain a simple proof that effectiveness of an étale groupoid is implied by a Cuntz–Krieger uniqueness theorem for a universal groupoid C*-algebra.

COMPUTING CONTINGENCIES FOR STABLE RELATIONS

1999 ◽  
Vol 10 (03) ◽  
pp. 301-326 ◽  
Author(s):  
SØREN EILERS ◽  
TERRY A. LORING
Keyword(s):  
Approximation Properties ◽  
Weak Correlation ◽  
C Algebra ◽  
C Algebras ◽  
Special Cases ◽  
Algebra Theory ◽  
K Theory

Close ties between approximation properties for relations on C*-algebra elements and lifting results for the universal C*-algebras the relations generate is a very widespread and useful phenomenon in C*-algebra theory. In this paper, we explore how to achieve results of this kind when the approximation properties and the lifting results are true only in special cases determined by K-theoretical contingencies. To interpolate between properties of these two basic types, we must investigate C*-algebras given by softened relations, in particular with emphasis on their K-theory. A surprisingly weak correlation between the K-theory of the C*-algebras given by exact and softened relations leads to delicate problems which must be treated with care. As an example of a set of relations which are prone to an analysis of this kind we study the pairs of unitaries commuting up to rational rotation.

scholarly journals K-THEORY FOR THE C*-ALGEBRAS OF THE SOLVABLE BAUMSLAG–SOLITAR GROUPS

2017 ◽  
Vol 60 (2) ◽  
pp. 481-486
Author(s):  
SANAZ POOYA ◽  
ALAIN VALETTE
Keyword(s):  
Exact Sequence ◽  
Direct Product ◽  
C Algebras ◽  
K Theory

AbstractWe provide a new computation of the K-theory of the group C*-algebra of the solvable Baumslag–Solitar group BS(1, n) (n ≠ 1); our computation is based on the Pimsner–Voiculescu 6-terms exact sequence, by viewing BS(1, n) as a semi-direct product ℤ[1/n] ⋊ ℤ. We deduce from it a new proof of the Baum–Connes conjecture with trivial coefficients for BS(1, n).

ON THE K-THEORY OF QUARTER-PLANE TOEPLITZ ALGEBRAS

1991 ◽  
Vol 02 (02) ◽  
pp. 195-204 ◽  
Author(s):  
EFTON PARK ◽  
CLAUDE SCHOCHET
Keyword(s):  
Spectral Sequence ◽  
Operator Theory ◽  
Sufficient Conditions ◽  
Theory Analysis ◽  
C Algebra ◽  
C Algebras ◽  
Quarter Plane ◽  
K Theory ◽  
Necessary And Sufficient

Given a C*-algebra A which is filtered by a collection of closed ideals Ai, there is a spectral sequence which relates the K-theory of A to the K-theory of the various quotient algebras Ai/Ai−1. The d1 differentials in this spectral sequence are familiar index invariants, but the higher differentials are not well-understood. Considering the case of Toeplitz C*-algebras associated with certain cones in Z2, it is shown that a d2 differential in the spectral sequence is non-trivial. This differential turns out to be an obstruction to a classical lifting problem in operator theory. Analysis of this obstruction leads to necessary and sufficient conditions for the lifting problem. It is hoped that this example will illuminate the role of higher differentials in the K-theory spectral sequence.

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