Twisted Group C*-Algebras as Compact Quantum Metric Spaces

2016 ◽  
Vol 71 (3-4) ◽  
pp. 911-931 ◽  
Author(s):  
Botao Long ◽  
Wei Wu
Keyword(s):  
Metric Spaces ◽  
C Algebras ◽  
Twisted Group

scholarly journals Nilpotent Group C*-algebras as Compact Quantum Metric Spaces

2017 ◽  
Vol 60 (1) ◽  
pp. 77-94 ◽  
Author(s):  
Michael Christ ◽  
Marc A. Rieòel
Keyword(s):  
Finite Groups ◽  
Word Length ◽  
Polynomial Growth ◽  
Metric Spaces ◽  
Length Function ◽  
C Algebra ◽  
Pointwise Multiplication ◽  
C Algebras ◽  
Definition Of ◽  
Compact Quantum Metric Space

AbstractLet be a length function on a group G, and let M denote the operator of pointwise multiplication by on l2(G). Following Connes, M𝕃 can be used as a “Dirac” operator for the reduced group C*-algebra (G). It deûnes a Lipschitz seminorm on (G), which defines a metric on the state space of (G). We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-* topology (a key property for the definition of a “compact quantum metric space”). In particular, this holds for all word-length functions on ûnitely generated nilpotent-by-finite groups.

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 On the Structure of Twisted Group C ∗ -Algebras

1992 ◽  
Vol 334 (2) ◽  
pp. 685 ◽  
Author(s):  
Judith A. Packer ◽  
Iain Raeburn
Keyword(s):  
C Algebras ◽  
Twisted Group

scholarly journals Gabor duality theory for Morita equivalent C∗-algebras

2020 ◽  
Vol 31 (10) ◽  
pp. 2050073 ◽  
Author(s):  
Are Austad ◽  
Mads S. Jakobsen ◽  
Franz Luef
Keyword(s):  
Duality Theory ◽  
Morita Equivalence ◽  
Gabor Frames ◽  
Duality Principle ◽  
Gabor Analysis ◽  
Matrix Algebras ◽  
C Algebras ◽  
Morita Equivalent ◽  
Twisted Group ◽  
Density Theorems

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalence bimodules with some extra properties. For certain twisted group [Formula: see text]-algebras, the reformulation of the duality principle to the setting of Morita equivalence bimodules reduces to the well-known Gabor duality principle by localizing with respect to a trace. We may lift all results at the module level to matrix algebras and matrix modules, and in doing so, it is natural to introduce [Formula: see text]-matrix Gabor frames, which generalize multi-window super Gabor frames. We are also able to establish density theorems for module frames on equivalence bimodules, and these localize to density theorems for [Formula: see text]-matrix Gabor frames.

scholarly journals Standard homogeneous C*-algebras as compact quantum metric spaces

2020 ◽  
Vol 120 ◽  
pp. 81-110
Author(s):  
Konrad Aguilar ◽  
Tristan Bice
Keyword(s):  
Metric Spaces ◽  
C Algebras

INDUCTIVE LIMITS OF C*-ALGEBRAS AND COMPACT QUANTUM METRIC SPACES

2020 ◽  
pp. 1-24
Author(s):  
KONRAD AGUILAR
Keyword(s):  
Inductive Limit ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Ideal Space ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Af Algebras ◽  
The Ideal

Given a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a Rieffel compact quantum metric, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov–Hausdorff propinquity of Latrémolière on compact quantum metric spaces. This allows us to place new quantum metrics on all unital approximately finite-dimensional (AF) algebras that extend our previous work with Latrémolière on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov–Hausdorff propinquity topology.

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