On $C^*$-algebras associated to actions of discrete subgroups of $\operatorname{SL}(2,\mathbb{R})$ on the punctured plane

2020 ◽  
Vol 126 (3) ◽  
pp. 540-558
Author(s):  
Jacopo Bassi
Keyword(s):  
Homogeneous Space ◽  
Transformation Group ◽  
Matrix Multiplication ◽  
Crossed Products ◽  
Discrete Subgroups ◽  
Horocycle Flow ◽  
Discrete Transformation ◽  
Compact Homogeneous Space ◽  
C Algebra ◽  
C Algebras

Dynamical conditions that guarantee stability for discrete transformation group $C^*$-algebras are determined. The results are applied to the case of some discrete subgroups of $\operatorname{SL} (2,\mathbb{R} )$ acting on the punctured plane by means of matrix multiplication of vectors. In the case of cocompact subgroups, further properties of such crossed products are deduced from properties of the $C^*$-algebra associated to the horocycle flow on the corresponding compact homogeneous space of $\operatorname{SL} (2,\mathbb{R} )$.

scholarly journals Homogeneous C*-algebras whose spectra are tori

1985 ◽  
Vol 38 (1) ◽  
pp. 9-39 ◽  
Author(s):  
Shaun Disney ◽  
Iain Raeburn
Keyword(s):  
Isomorphism Class ◽  
Homotopy Theory ◽  
Transformation Group ◽  
Crossed Products ◽  
C Algebras ◽  
Isomorphism Classes ◽  
Twisted Group

AbstractBy a theorem of Fell and Tomiyama-Takesaki, an N-homogeneous C*-algebra with spectrum X has the form Γ(E) for some bundle E over X with fibre MN(C), and its isomorphism class is determined by that of E and its pull-backs f*E along homeomorphisms f of X. We describe the homogeneous C*-algebras with spectrum T2 or T3 by classifying the MN-bundles over Tk using elementary homotopy theory. We then use our results to determine the isomorphism classes of a variety of transformation group C*-algebras, twisted group C*-algebras and more general crossed products.

scholarly journals K-theory of Furstenberg Transformation Group C*-algebras

2013 ◽  
Vol 65 (6) ◽  
pp. 1287-1319 ◽  
Author(s):  
Kamran Reihani
Keyword(s):  
Asymptotic Behavior ◽  
Explicit Formula ◽  
Transformation Group ◽  
Order Structure ◽  
Crossed Products ◽  
Important Class ◽  
Maximal Degree ◽  
C Algebras ◽  
Torsion Subgroups ◽  
Made In

AbstractThis paper studies the K-theoretic invariants of the crossed product C*-algebras associated with an important family of homeomorphisms of the tori Tn called Furstenberg transformations. Using the Pimsner–Voiculescu theorem, we prove that given n, the K-groups of those crossed products whose corresponding n × n integer matrices are unipotent of maximal degree always have the same rank an. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these K-groups is false. Using the representation theory of the simple Lie algebra sl(2;C), we show that, remarkably, an has a combinatorial significance. For example, every a2n+1 is just the number of ways that 0 can be represented as a sum of integers between–n and n (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdős), a simple explicit formula for the asymptotic behavior of the sequence {an} is given. Finally, we describe the order structure of the K0-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

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].

Property T and Amenable Transformation Group C*-algebras

2015 ◽  
Vol 58 (1) ◽  
pp. 110-114 ◽  
Author(s):  
F. Kamalov
Keyword(s):  
Compact Hausdorff Space ◽  
Discrete Group ◽  
Hausdorff Space ◽  
Transformation Group ◽  
Amenable Group ◽  
Transformation Groups ◽  
C Algebra ◽  
C Algebras ◽  
Property T ◽  
Tracial States

AbstractIt is well known that a discrete group that is both amenable and has Kazhdan’s Property T must be finite. In this note we generalize this statement to the case of transformation groups. We show that if G is a discrete amenable group acting on a compact Hausdorff space X, then the transformation group C*-algebra C*(X; G) has Property T if and only if both X and G are finite. Our approach does not rely on the use of tracial states on C*(X; G).

CROSSED PRODUCTS OF CERTAIN C*-ALGEBRAS

2014 ◽  
Vol 25 (02) ◽  
pp. 1450010 ◽  
Author(s):  
JIAJIE HUA ◽  
YAN WU
Keyword(s):  
Simple Formula ◽  
Continuous Functions ◽  
Cantor Set ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebra ◽  
C Algebras ◽  
Tracial Rank ◽  
Universal Coefficient Theorem ◽  
Unital Simple

Let X be a Cantor set, and let A be a unital separable simple amenable [Formula: see text]-stable C*-algebra with rationally tracial rank no more than one, which satisfies the Universal Coefficient Theorem (UCT). We use C(X, A) to denote the algebra of all continuous functions from X to A. Let α be an automorphism on C(X, A). Suppose that C(X, A) is α-simple, [α|1⊗A] = [ id |1⊗A] in KL(1 ⊗ A, C(X, A)), τ(α(1 ⊗ a)) = τ(1 ⊗ a) for all τ ∈ T(C(X, A)) and all a ∈ A, and [Formula: see text] for all u ∈ U(A) (where α‡ and id‡ are homomorphisms from U(C(X, A))/CU(C(X, A)) → U(C(X, A))/CU(C(X, A)) induced by α and id, respectively, and where CU(C(X, A)) is the closure of the subgroup generated by commutators of the unitary group U(C(X, A)) of C(X, A)), then the corresponding crossed product C(X, A) ⋊α ℤ is a unital simple [Formula: see text]-stable C*-algebra with rationally tracial rank no more than one, which satisfies the UCT. Let X be a Cantor set and 𝕋 be the circle. Let γ : X × 𝕋n → X × 𝕋n be a minimal homeomorphism. It is proved that, as long as the cocycles are rotations, the tracial rank of the corresponding crossed product C*-algebra is always no more than one.

scholarly journals CERTAIN APERIODIC AUTOMORPHISMS OF UNITAL SIMPLE PROJECTIONLESS C*-ALGEBRAS

2009 ◽  
Vol 20 (10) ◽  
pp. 1233-1261 ◽  
Author(s):  
YASUHIKO SATO
Keyword(s):  
Inductive Limit ◽  
Crossed Products ◽  
Inductive Limits ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
Tracial State ◽  
C Algebras ◽  
Cyclic Groups ◽  
Unital Simple

Let G be an inductive limit of finite cyclic groups, and A be a unital simple projectionless C*-algebra with K1(A) ≅ G and a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in Aut (A)/ WInn (A) are conjugate, where WInn (A) means the subgroup of Aut (A) consisting of automorphisms which are inner in the tracial representation.In the second part of this paper, we consider a class of unital simple C*-algebras with a unique tracial state which contains the class of unital simple A𝕋-algebras of real rank zero with a unique tracial state. This class is closed under inductive limits and crossed products by actions of ℤ with the Rohlin property. Let A be a TAF-algebra in this class. We show that for any automorphism α of A there exists an automorphism ᾶ of A with the Rohlin property such that ᾶ and α are asymptotically unitarily equivalent. For the proof we use an aperiodic automorphism of the Jiang-Su algebra.

Analytic Isomorphisms of Transformation Group C*-Algebras

1990 ◽  
Vol 42 (4) ◽  
pp. 709-730
Author(s):  
Michael P. Lamoureux
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Transformation Group ◽  
Convergence Property ◽  
Subspace Lattice ◽  
C Algebra ◽  
C Algebras ◽  
Reflexive Algebra ◽  
Dominated Convergence ◽  
Cech Cohomology

AbstractAn analytic isomorphism of C*-algebras is a C*-isomorphism which maps one distinguished subalgebra, the analytic subalgebra, onto another. A strict partial order of a topological group acting on a topological space determines the analytic subalgebra of the transformation group C*-algebra as a certain non-self-adjoint subalgebra of the C*-algebra. When the group action is free and locally parallel, this analytic subalgebra is locally a subfield of compact operators contained in a reflexive algebra whose subspace lattice is determined by the group order. If in addition the group has the dominated convergence property, an analytic isomorphism of such transformation group C*-algebras induces a homeomorphism of the transformation spaces which maps orbits to orbits. In particular, the C*-algebras for two regular foliations of the plane are analytically isomorphic only if the foliations are topologically conjugate. In the case of parallel actions, a quotient of the group of analytic automorphisms is isomorphic to the second Čech cohomology of a transversal for the action.

On free transformation groups and C*-algebras

1987 ◽  
Vol 107 (3-4) ◽  
pp. 339-347
Author(s):  
Klaus Thomsen
Keyword(s):  
Transformation Group ◽  
Crossed Product ◽  
Transformation Groups ◽  
Connected Space ◽  
Discrete Transformation ◽  
C Algebras ◽  
The Way

SynopsisWe prove that a free discrete transformation group on a compact connected space X is completely determined by the way C(X) lies inside the corresponding crossed product C*-algebra.

scholarly journals The Weak Ideal Property and Topological Dimension Zero

2017 ◽  
Vol 69 (6) ◽  
pp. 1385-1421 ◽  
Author(s):  
Cornel Pasnicu ◽  
N. Christopher Phillips
Keyword(s):  
Hausdorff Space ◽  
Crossed Products ◽  
Topological Dimension ◽  
C Algebra ◽  
Dimension Zero ◽  
C Algebras ◽  
Ideal Property ◽  
Locally Compact Hausdorff Space ◽  
Totally Disconnected ◽  
The Ideal

AbstractFollowing up on previous work, we prove a number of results for C* -algebras with the weak ideal property or topological dimension zero, and some results for C* -algebras with related properties. Some of the more important results include the following:The weak ideal property implies topological dimension zero.For a separable C* -algebra A, topological dimension zero is equivalent to , to D ⊗ A having the ideal property for some (or any) Kirchberg algebra D, and to A being residually hereditarily in the class of all C* -algebras B such that contains a nonzero projection.Extending the known result for , the classes of C* -algebras with residual (SP), which are residually hereditarily (properly) infinite, or which are purely infinite and have the ideal property, are closed under crossed products by arbitrary actions of abelian 2-groups.If A and B are separable, one of them is exact, A has the ideal property, and B has the weak ideal property, then A ⊗ B has the weak ideal property.If X is a totally disconnected locally compact Hausdorff space and A is a C0(X)-algebra all of whose fibers have one of the weak ideal property, topological dimension zero, residual (SP), or the combination of pure infiniteness and the ideal property, then A also has the corresponding property (for topological dimension zero, provided A is separable).Topological dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C* -algebras, including all separable locally AH algebras.The weak ideal property does not imply the ideal property for separable Z-stable C* -algebras.We give other related results, as well as counterexamples to several other statements one might conjecture.

Sign in / Sign up

Export Citation Format

Share Document