scholarly journals On the fixed point algebra of a UHF algebra under a periodic automorphism of product type

1977 ◽  
Vol 13 (3) ◽  
pp. 777-791 ◽  
Author(s):  
Akitaka Kishimoto
Keyword(s):  
Fixed Point ◽  
Product Type ◽  
Fixed Point Algebra

Ergodic actions of group extensions on von Neumann algebras

1989 ◽  
Vol 112 (1-2) ◽  
pp. 71-112
Author(s):  
Klaus Thomsen
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Fixed Point Algebra ◽  
Atomic Centre

SynopsisWe consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

Morita equivalences between fixed point algebras and crossed products

1999 ◽  
Vol 125 (1) ◽  
pp. 43-52 ◽  
Author(s):  
CHI-KEUNG NG
Keyword(s):  
Fixed Point ◽  
Quantum Group ◽  
Type Theorem ◽  
Compact Quantum Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Morita Equivalent ◽  
Fixed Point Algebra

In this paper, we will prove that if A is a C*-algebra with an effective coaction ε by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an imprimitivity type theorem for crossed products of coactions by discrete Kac C*-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 Rigidity theory for C∗-dynamical systems and the “Pedersen rigidity problem”

2018 ◽  
Vol 29 (03) ◽  
pp. 1850016 ◽  
Author(s):  
S. Kaliszewski ◽  
Tron Omland ◽  
John Quigg
Keyword(s):  
Fixed Point ◽  
Abelian Group ◽  
Dynamical Systems ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Alternative Formulation ◽  
Crossed Products ◽  
Locally Compact ◽  
Fixed Point Algebra ◽  
Multiplier Algebras

Let [Formula: see text] be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of [Formula: see text] on [Formula: see text]-algebras [Formula: see text] and [Formula: see text] are outer conjugate if and only if there is an isomorphism of the crossed products that is equivariant for the dual actions and preserves the images of [Formula: see text] and [Formula: see text] in the multiplier algebras of the crossed products. The rigidity problem discussed in this paper deals with the necessity of the last condition concerning the images of [Formula: see text] and [Formula: see text]. There is an alternative formulation of the problem: an action of the dual group [Formula: see text] together with a suitably equivariant unitary homomorphism of [Formula: see text] give rise to a generalized fixed-point algebra via Landstad’s theorem, and a problem related to the above is to produce an action of [Formula: see text] and two such equivariant unitary homomorphisms of [Formula: see text] that give distinct generalized fixed-point algebras. We present several situations where the condition on the images of [Formula: see text] and [Formula: see text] is redundant, and where having distinct generalized fixed-point algebras is impossible. For example, if [Formula: see text] is discrete, this will be the case for all actions of [Formula: see text].

Remarks on the fixed-point algebras of product type actions

1980 ◽  
Vol 89 (3) ◽  
pp. 235-242 ◽  
Author(s):  
Norboet Riedel
Keyword(s):  
Fixed Point ◽  
Product Type

scholarly journals Superselection Structures for C*-Algebras with Nontrivial Center

1997 ◽  
Vol 09 (07) ◽  
pp. 785-819 ◽  
Author(s):  
Hellmut Baumgärtel ◽  
Fernando Lledó
Keyword(s):  
Fixed Point ◽  
Compact Group ◽  
C Algebras ◽  
Nontrivial Center ◽  
Relative Commutant ◽  
Fixed Point Algebra

We present and prove some results within the framework of Hilbert C*-systems [Formula: see text] with a compact group [Formula: see text]. We assume that the fixed point algebra [Formula: see text] of [Formula: see text] has a nontrivial center [Formula: see text] and its relative commutant w.r.t. ℱ coincides with [Formula: see text], i.e. we have [Formula: see text]. In this context we propose a generalization of the notion of an irreducible endomorphism and study the behaviour of such irreducibles w.r.t. [Formula: see text]. Finally, we give several characterizations of the stabilizer of [Formula: see text].

scholarly journals Fell bundles and imprimitivity theorems: towards a universal generalized fixed point algebra

2013 ◽  
Vol 62 (6) ◽  
pp. 1691-1716 ◽  
Author(s):  
S. Kaliszewski ◽  
Paul S. Muhly ◽  
John Quigg ◽  
Dana P. Williams
Keyword(s):  
Fixed Point ◽  
Fixed Point Algebra

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 Applications of the Gauge-invariant uniqueness theorem for graph algebras

2002 ◽  
Vol 66 (1) ◽  
pp. 57-67 ◽  
Author(s):  
Teresa Bates
Keyword(s):  
Fixed Point ◽  
Uniqueness Theorem ◽  
Directed Graphs ◽  
Cartesian Product ◽  
Graph Algebras ◽  
Gauge Invariant ◽  
Shift Equivalence ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Fixed Point Algebra

We give applications of the gauge-invariant uniqueness theorem, which states that the Cuntz-Krieger algebras of directed graphs are characterised by the existence of a canonical action of . We classify the C*-algebras of higher order graphs, identify the C*-algebras of cartesian product graphs with a certain fixed point algebra and investigate a relation called elementary shift equivalence on graphs and its effect on the associated graph C*-algebras.

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