scholarly journals Duality for compact group actions on operator algebras and applications: Irreducible inclusions and Galois correspondence

2020 ◽  
Vol 31 (09) ◽  
pp. 2050067
Author(s):  
Costel Peligrad
Keyword(s):  
Fixed Point ◽  
Compact Group ◽  
Operator Algebras ◽  
Group Actions ◽  
Normal Subgroups ◽  
Galois Correspondence ◽  
Duality Property ◽  
Relative Commutant ◽  
Fixed Point Algebra

We consider compact group actions on C*- and W*-algebras. We prove results that relate the duality property of the action (as defined in the Introduction) with other relevant properties of the system such as the relative commutant of the fixed point algebras being trivial (called the irreducibility of the inclusion) and also to the Galois correspondence between invariant C*-subalgebras containing the fixed point algebra and the class of closed normal subgroups of the compact group.

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 Discrete coactions on C*-algebras

1996 ◽  
Vol 60 (1) ◽  
pp. 118-127 ◽  
Author(s):  
Chi-Keung Ng
Keyword(s):  
Fixed Point ◽  
Compact Group ◽  
Amenable Group ◽  
Group Actions ◽  
Crossed Product ◽  
Discrete Groups ◽  
C Algebras ◽  
Fixed Point Algebra

AbstractWe will consider coactions of discrete groups on C*-algebras and imitate some of the results about compact group actions on C*-algebras. In particular, the crossed product of a reduced coaction ∈ of a discrete amenable group G on A is liminal (respectively, postliminal) if and only if the fixed point algebra of ∈ is. Moreover, we will also consider ergodic coactions on C*-algebras.

scholarly journals Compact group actions on operator algebras and their spectra

2020 ◽  
Vol 126 (2) ◽  
pp. 276-292
Author(s):  
Costel Peligrad
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Compact Group ◽  
Spectral Properties ◽  
Operator Algebras ◽  
Abelian Groups ◽  
Group Actions ◽  
Algebraic Properties

We consider a class of dynamical systems with compact non-abelian groups that include C*-, W*- and multiplier dynamical systems. We prove results that relate the algebraic properties such as simplicity or primeness of the fixed point algebras to the spectral properties of the action, including the Connes and strong Connes spectra.

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 Rigidity theory for C∗-dynamical systems and the “Pedersen Rigidity Problem”, II

2019 ◽  
Vol 30 (08) ◽  
pp. 1950038
Author(s):  
S. Kaliszewski ◽  
Tron Omland ◽  
John Quigg
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Compact Group ◽  
Related Problem ◽  
Locally Compact Group ◽  
Crossed Products ◽  
Locally Compact ◽  
Fixed Point Algebra ◽  
Multiplier Algebras

This is a follow-up to a paper with the same title and by the same authors. In that paper, all groups were assumed to be abelian, and we are now aiming to generalize the results to nonabelian groups. The motivating point is Pedersen’s theorem, which does hold for an arbitrary locally compact group [Formula: see text], saying that two actions [Formula: see text] and [Formula: see text] of [Formula: see text] are outer conjugate if and only if the dual coactions [Formula: see text] and [Formula: see text] of [Formula: see text] are conjugate via an isomorphism that maps the image of [Formula: see text] onto the image of [Formula: see text] (inside the multiplier algebras of the respective crossed products). We do not know of any examples of a pair of non-outer-conjugate actions such that their dual coactions are conjugate, and our interest is therefore exploring the necessity of latter condition involving the images; and we have decided to use the term “Pedersen rigid” for cases where this condition is indeed redundant. There is also a related problem, concerning the possibility of a so-called equivariant coaction having a unique generalized fixed-point algebra, that we call “fixed-point rigidity”. In particular, if the dual coaction of an action is fixed-point rigid, then the action itself is Pedersen rigid, and no example of non-fixed-point-rigid coaction is known.

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 Analytic functions which operate on homogeneous algebras

1988 ◽  
Vol 45 (1) ◽  
pp. 1-10
Author(s):  
J. A. Ward
Keyword(s):  
Fixed Point ◽  
Compact Group ◽  
Banach Algebra ◽  
Analytic Functions ◽  
Operator Algebras ◽  
Commutative Banach Algebra ◽  
Open Set ◽  
Homogeneous Algebras ◽  
Complex Valued

AbstractIt is well known that a complex-valued function ø, analytic on some open set Ω, extends to any commutative Banach algebra B so that the action of ø on B commutes with the action of the Gelfand transformation. In this paper, it is shown that if B is a homogeneous convolution Banach algebra over any compact group and if 0 ∈ Ω is a fixed point of ø, then a similar result holds, with the Gelfand transformation replaced by the Fourier-Stieltjes transformation. Care is required, in that discussion of this relation usually requires simultaneous consideration of the extension of ø to B and to certain operator algebras.

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.

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