scholarly journals On pathological properties of fixed point algebras in Kirchberg algebras

2019 ◽  
Vol 150 (6) ◽  
pp. 3087-3096
Author(s):  
Yuhei Suzuki
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Approximation Property ◽  
Infinite Group ◽  
Cuntz Algebra ◽  
C Algebra ◽  
Outer Action ◽  
Reduced Group ◽  
Fixed Point Algebra

AbstractWe investigate how the fixed point algebra of a C*-dynamical system can differ from the underlying C*-algebra. For any exact group Γ and any infinite group Λ, we construct an outer action of Λ on the Cuntz algebra 𝒪2 whose fixed point algebra is almost equal to the reduced group C*-algebra ${\rm C}_{\rm r}^* (\Gamma)$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.

scholarly journals ON AUTOMORPHISMS OF GENERALIZED CUNTZ ALGEBRAS

1998 ◽  
Vol 09 (04) ◽  
pp. 493-512 ◽  
Author(s):  
YOSHIKAZU KATAYAMA ◽  
HIROAKI TAKEHANA
Keyword(s):  
Fixed Point ◽  
Finite Type ◽  
Invertible Operator ◽  
Cuntz Algebra ◽  
Gauge Action ◽  
Cuntz Algebras ◽  
C Algebra ◽  
Fixed Point Algebra

Let X be a full right Hilbert B-bimodule of finite type and [Formula: see text] be its generalized Cuntz algebra. We give a notion that the C*-algebra B is X-aperiodic. We show that the fixed point algebra ℱX for a gauge action is simple if and only if the C*-algebra B is X-aperiodic. For a invertible operator U on X with some properties, a quasi-free automorphism αU of [Formula: see text] is defined. We give some conditions in order that αU is inner in the case that B is X-aperiodic. We apply them to the automorphism αU on Cuntz–Krieger algebras.

scholarly journals Extending Characters of Fixed Point Algebras

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 79
Author(s):  
Stefan Wagner
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Topological Group ◽  
Continuous Action ◽  
Group Of Units ◽  
System A ◽  
Fixed Point Algebra ◽  
Continuous Inverse ◽  
Locally Convex ◽  
Locally Convex Algebra

A dynamical system is a triple ( A , G , α ) consisting of a unital locally convex algebra A, a topological group G, and a group homomorphism α : G → Aut ( A ) that induces a continuous action of G on A. Furthermore, a unital locally convex algebra A is called a continuous inverse algebra, or CIA for short, if its group of units A × is open in A and the inversion map ι : A × → A × , a ↦ a − 1 is continuous at 1 A . Given a dynamical system ( A , G , α ) with a complete commutative CIA A and a compact group G, we show that each character of the corresponding fixed point algebra can be extended to a character of A.

scholarly journals A Short Note on the Continuous Rokhlin Property and the Universal Coefficient Theorem in E-theory

2015 ◽  
Vol 58 (2) ◽  
pp. 374-380 ◽  
Author(s):  
Gábor Szabó
Keyword(s):  
Fixed Point ◽  
Compact Group ◽  
Short Note ◽  
Circle Actions ◽  
Continuous Action ◽  
C Algebra ◽  
C Algebras ◽  
Group A ◽  
Fixed Point Algebra ◽  
Universal Coefficient Theorem

AbstractLet G be a metrizable compact group, A a separable C*-algebra, and α:G → Aut(A) a strongly continuous action. Provided that α satisfies the continuous Rokhlin property, we show that the property of satisfying the UCT in E-theory passes from Ato the crossed product C*-algebra A⋊α G and the ûxed point algebra Aα. This extends a similar result by Gardella for KK-theory in the case of unital C*-algebras but with a shorter and less technical proof. For circle actions on separable unital C*-algebras with the continuous Rokhlin property, we establish a connection between the Etheory equivalence class of A and that of its fixed point algebra Aα.

scholarly journals A COMMENT ON JONES INCLUSIONS WITH INFINITE INDEX

1995 ◽  
Vol 07 (04) ◽  
pp. 599-630 ◽  
Author(s):  
FLORIAN NILL ◽  
HANS-WERNER WIESBROCK
Keyword(s):  
Fixed Point ◽  
General Result ◽  
Conditional Expectation ◽  
Von Neumann Algebras ◽  
Alternative Proof ◽  
Infinite Index ◽  
Von Neumann ◽  
Outer Action ◽  
Fixed Point Algebra

Given an irreducible inclusion of infinite von-Neumann-algebras [Formula: see text] together with a conditional expectation [Formula: see text] such that the inclusion has depth 2, we show quite explicitly how [Formula: see text] can be viewed as the fixed-point algebra of [Formula: see text] w.r.t. an outer action of a compact Kac algebra acting on [Formula: see text]. This gives an alternative proof, under this special setting of a more general result of M. Enock and R. Nest [6], see also S. Yamagami [28].

scholarly journals On diagonal quasi-free automorphisms of simple Cuntz-Krieger algebras

2019 ◽  
Vol 125 (2) ◽  
pp. 210-226
Author(s):  
Selçuk Barlak ◽  
Gábor Szabó
Keyword(s):  
Fixed Point ◽  
Abelian Group ◽  
Finite Abelian Group ◽  
Outer Action ◽  
Relative Commutant ◽  
Associated Matrix ◽  
Fixed Point Algebra

We show that an outer action of a finite abelian group on a simple Cuntz-Krieger algebra is strongly approximately inner in the sense of Izumi if the action is given by diagonal quasi-free automorphisms and the associated matrix is aperiodic. This is achieved by an approximate cohomology vanishing-type argument for the canonical shift restricted to the relative commutant of the set of domain projections of the canonical generating isometries in the fixed point algebra.

scholarly journals An action of the Klein four-group on the irrational rotation C*-algebra

1997 ◽  
Vol 56 (1) ◽  
pp. 135-148
Author(s):  
P.J. Stacey
Keyword(s):  
Fixed Point ◽  
Irrational Rotation ◽  
C Algebra ◽  
Fixed Point Algebra

Explicit automorphisms of the irrational rotation algebra are constructed which are associated with the two 2 × 2 diagonal integer matrices of determinant −1. The fixed point algebra of the product of these two automorphisms is shown to be isomorphic to the fixed point algebra of the flip.

scholarly journals Equivariant -absorption theorem for exact groups

2021 ◽  
Vol 157 (7) ◽  
pp. 1492-1506
Author(s):  
Yuhei Suzuki
Keyword(s):  
Approximation Property ◽  
Cuntz Algebra ◽  
Outer Action ◽  
Cocycle Conjugacy

We show that, up to strong cocycle conjugacy, every countable exact group admits a unique equivariantly $\mathcal {O}_{2}$ -absorbing, pointwise outer action on the Cuntz algebra $\mathcal {O}_{2}$ with the quasi-central approximation property (QAP). In particular, we establish the equivariant analogue of the Kirchberg $\mathcal {O}_{2}$ -absorption theorem for these groups.

scholarly journals The Cuntz semigroup and the radius of comparison of the crossed product by a finite group

2021 ◽  
pp. 1-52
Author(s):  
M. ALI ASADI-VASFI ◽  
NASSER GOLESTANI ◽  
N. CHRISTOPHER PHILLIPS
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Fixed Points ◽  
Positive Part ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Cuntz Semigroup ◽  
Fixed Point Algebra ◽  
Stably Finite ◽  
A Finite Group

Abstract Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let $\alpha \colon G \to {\text{Aut}} (A)$ be an action of G on A which has the weak tracial Rokhlin property. Let $A^{\alpha}$ be the fixed point algebra. Then the radius of comparison satisfies ${\text{rc}} (A^{\alpha }) \leq {\text{rc}} (A)$ and ${\text{rc}} ( C^* (G, A, \alpha ) ) \leq ({1}/{\text{card} (G))} \cdot {\text{rc}} (A)$ . The inclusion of $A^{\alpha }$ in A induces an isomorphism from the purely positive part of the Cuntz semigroup ${\text{Cu}} (A^{\alpha })$ to the fixed points of the purely positive part of ${\text{Cu}} (A)$ , and the purely positive part of ${\text{Cu}} ( C^* (G, A, \alpha ) )$ is isomorphic to this semigroup. We construct an example in which $G \,{=}\, {\mathbb {Z}} / 2 {\mathbb {Z}}$ , A is a simple unital AH algebra, $\alpha $ has the Rokhlin property, ${\text{rc}} (A)> 0$ , ${\text{rc}} (A^{\alpha }) = {\text{rc}} (A)$ , and ${\text{rc}} (C^* (G, A, \alpha ) = ( {1}/{2}) {\text{rc}} (A)$ .

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