Groupoid dynamical systems and crossed product

1985 ◽  
pp. 350-361
Author(s):  
Tetsuya Masuda
Keyword(s):  
Dynamical Systems ◽  
Crossed Product

On simplicity of intermediate -algebras

2019 ◽  
Vol 40 (12) ◽  
pp. 3181-3187
Author(s):  
TATTWAMASI AMRUTAM ◽  
MEHRDAD KALANTAR
Keyword(s):  
Dynamical Systems ◽  
Simple Group ◽  
Crossed Product ◽  
Simple Groups ◽  
Stationary States ◽  
Compact Spaces ◽  
Link Type ◽  
One To One

We prove simplicity of all intermediate $C^{\ast }$-algebras $C_{r}^{\ast }(\unicode[STIX]{x1D6E4})\subseteq {\mathcal{B}}\subseteq \unicode[STIX]{x1D6E4}\ltimes _{r}C(X)$ in the case of minimal actions of $C^{\ast }$-simple groups $\unicode[STIX]{x1D6E4}$ on compact spaces $X$. For this, we use the notion of stationary states, recently introduced by Hartman and Kalantar [Stationary $C^{\ast }$-dynamical systems. Preprint, 2017, arXiv:1712.10133]. We show that the Powers’ averaging property holds for the reduced crossed product $\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$ for any action $\unicode[STIX]{x1D6E4}\curvearrowright {\mathcal{A}}$ of a $C^{\ast }$-simple group $\unicode[STIX]{x1D6E4}$ on a unital $C^{\ast }$-algebra ${\mathcal{A}}$, and use it to prove a one-to-one correspondence between stationary states on ${\mathcal{A}}$ and those on $\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$.

Crossed Product Algebras and the Homology of Certain p-Adic and Adélic Dynamical Systems

K-Theory ◽  
2000 ◽  
Vol 21 (1) ◽  
pp. 1-23 ◽  
Author(s):  
Eric Leichtnam ◽  
Victor Nistor
Keyword(s):  
Dynamical Systems ◽  
Crossed Product ◽  
Crossed Product Algebras

GROUP ACTION PRESERVING THE HAAGERUP PROPERTY OF -ALGEBRAS

2015 ◽  
Vol 93 (2) ◽  
pp. 295-300 ◽  
Author(s):  
CHAO YOU
Keyword(s):  
Dynamical Systems ◽  
Group Action ◽  
Crossed Product ◽  
Weak Version ◽  
Haagerup Property ◽  
Tracial State

From the viewpoint of $C^{\ast }$-dynamical systems, we define a weak version of the Haagerup property for the group action on a $C^{\ast }$-algebra. We prove that this group action preserves the Haagerup property of $C^{\ast }$-algebras in the sense of Dong [‘Haagerup property for $C^{\ast }$-algebras’, J. Math. Anal. Appl.377 (2011), 631–644], that is, the reduced crossed product $C^{\ast }$-algebra $A\rtimes _{{\it\alpha},\text{r}}{\rm\Gamma}$ has the Haagerup property with respect to the induced faithful tracial state $\widetilde{{\it\tau}}$ if $A$ has the Haagerup property with respect to ${\it\tau}$.

scholarly journals Spectral triples on irreversible C∗-dynamical systems

2021 ◽  
Author(s):  
Valeriano Aiello ◽  
Daniele Guido ◽  
Tommaso Isola
Keyword(s):  
Dynamical Systems ◽  
Product Formula ◽  
Crossed Product ◽  
Spectral Triple ◽  
Crossed Products ◽  
Spectral Triples ◽  
Main Assumption ◽  
Noncommutative Spaces ◽  
Covering Projection ◽  
Injective Endomorphism

Given a spectral triple on a [Formula: see text]-algebra [Formula: see text] together with a unital injective endomorphism [Formula: see text], the problem of defining a suitable crossed product [Formula: see text]-algebra endowed with a spectral triple is addressed. The proposed construction is mainly based on the works of Cuntz and [A. Hawkins, A. Skalski, S. White and J. Zacharias, On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262–291], and on our previous papers [V. Aiello, D. Guido and T. Isola, Spectral triples for noncommutative solenoidal spaces from self-coverings, J. Math. Anal. Appl. 448(2) (2017) 1378–1412; V. Aiello, D. Guido and T. Isola, A spectral triple for a solenoid based on the Sierpinski gasket, SIGMA Symmetry Integrability Geom. Methods Appl. 17(20) (2021) 21]. The embedding of [Formula: see text] in [Formula: see text] can be considered as the dual form of a covering projection between noncommutative spaces. A main assumption is the expansiveness of the endomorphism, which takes the form of the local isometricity of the covering projection, and is expressed via the compatibility of the Lip-norms on [Formula: see text] and [Formula: see text].

scholarly journals The crossed-product structure of C*-algebras arising from topological dynamical systems

2103 ◽  
Vol 70 (1) ◽  
pp. 191-210
Author(s):  
Cynthia Farthing ◽  
Nura Patani ◽  
Paulette N. Willis
Keyword(s):  
Dynamical Systems ◽  
Product Structure ◽  
Crossed Product ◽  
C Algebras ◽  
Topological Dynamical Systems

The Local Product Structure of Expansive Automorphisms of Solenoids and Their Associated C*-Algebras

1996 ◽  
Vol 48 (4) ◽  
pp. 692-709 ◽  
Author(s):  
Berndt Brenken
Keyword(s):  
Abelian Group ◽  
Dynamical Systems ◽  
Coordinate System ◽  
Finite Type ◽  
Crossed Product ◽  
C Algebras ◽  
Local Product ◽  
Subshifts Of Finite Type ◽  
Canonical Coordinate System ◽  
Canonical Coordinate

AbstractAn explicit description of a hyperbolic canonical coordinate system for an expansive automorphism of a compact connected abelian group is given. These dynamical systems are factors of subshifts of finite type. Some properties of the associated crossed product C*-algebra are discussed. In these examples, the C* -algebras of Ruelle are crossed product algebras.

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