$\mathcal {Z}$-stability of transformation group C*-algebras

2021 ◽  
pp. 1
Author(s):  
Zhuang Niu
Keyword(s):  
Transformation Group ◽  
C Algebras

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.

scholarly journals Ext and OrderExt Classes of Certain Automorphisms of C*-Algebras Arising from Cantor Minimal Systems

2001 ◽  
Vol 53 (2) ◽  
pp. 325-354 ◽  
Author(s):  
Hiroki Matui
Keyword(s):  
Transformation Group ◽  
Minimal System ◽  
Full Group ◽  
C Algebras ◽  
Inner Automorphism ◽  
Inner Automorphisms ◽  
Cantor Minimal Systems

AbstractGiordano, Putnam and Skau showed that the transformation group C*-algebra arising from a Cantor minimal system is an AT-algebra, and classified it by its K-theory. For approximately inner automorphisms that preserve C(X), we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto’s result.

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).

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 $\operatorname{SL} (2,\mathbb{R} )$.

THE INDEX OF TOEPLITZ OPERATORS ON FREE TRANSFORMATION GROUP C*-ALGEBRAS

2002 ◽  
Vol 34 (1) ◽  
pp. 84-90 ◽  
Author(s):  
EFTON PARK
Keyword(s):  
Vector Bundle ◽  
Differential Operator ◽  
Differential Form ◽  
Compact Manifold ◽  
Toeplitz Operators ◽  
Discrete Group ◽  
Transformation Group ◽  
Hermitian Vector Bundle ◽  
First Order ◽  
C Algebras

Let Γ be a discrete group acting on a compact manifold X, let V be a Γ-equivalent Hermitian vector bundle over X, and let D be a first-order elliptic self-adjoint Γ-equivalent differential operator acting on sections of V. This data is used to define Toeplitz operators with symbols in the transformation group C*-algebra C(X)[rtimes ]Γ, and it is shown that if the symbol of such a Toeplitz operator is invertible, then the operator is Fredholm. In the case where Γ is finite and acts freely on X, a geometric-topological formula for the index is stated that involves an explicitly constructed differential form associated to the symbol.

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