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 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 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 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 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 The noncommutative geometry of k-graph C*-algebras

2007 ◽  
Vol 1 (2) ◽  
pp. 259-304 ◽  
Author(s):  
David Pask ◽  
Adam Rennie ◽  
Aidan Sims
Keyword(s):  
Fixed Point ◽  
Isomorphism Class ◽  
Lower Semicontinuous ◽  
Index Theorem ◽  
Graph Algebras ◽  
Gauge Invariant ◽  
Local Index ◽  
C Algebras ◽  
Local Index Theorem ◽  
Fixed Point Algebra

AbstractThis paper is comprised of two related parts. First we discuss which k-graph algebras have faithful traces. We characterise the existence of a faithful semifinite lower-semicontinuous gauge-invariant trace on C* (Λ) in terms of the existence of a faithful graph trace on Λ.Second, for k-graphs with faithful gauge invariant trace, we construct a smooth (k,∞)-summable semifinite spectral triple. We use the semifinite local index theorem to compute the pairing with K-theory. This numerical pairing can be obtained by applying the trace to a KK-pairing with values in the K-theory of the fixed point algebra of the Tk action. As with graph algebras, the index pairing is an invariant for a finer structure than the isomorphism class of the algebra.

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 Hecke C*-algebras and semi-direct products

2009 ◽  
Vol 52 (1) ◽  
pp. 127-153 ◽  
Author(s):  
S. Kaliszewski ◽  
Magnus B. Landstad ◽  
John Quigg
Keyword(s):  
Fixed Point ◽  
Direct Product ◽  
Hecke Algebra ◽  
Quadratic Extension ◽  
Crossed Product ◽  
Direct Products ◽  
C Algebras ◽  
Fixed Point Algebra

AbstractWe analyse Hecke pairs (G,H) and the associated Hecke algebra $\mathcal{H}$ when G is a semi-direct product N ⋊ Q and H = M ⋊ R for subgroups M ⊂ N and R ⊂ Q with M normal in N. Our main result shows that, when (G,H) coincides with its Schlichting completion and R is normal in Q, the closure of $\mathcal{H}$ in C*(G) is Morita–Rieffel equivalent to a crossed product I⋊βQ/R, where I is a certain ideal in the fixed-point algebra C*(N)R. Several concrete examples are given illustrating and applying our techniques, including some involving subgroups of GL(2,K) acting on K2, where K = ℚ or K = ℤ[p−1]. In particular we look at the ax + b group of a quadratic extension of K.

scholarly journals DUALITY OF COMPACT GROUPS AND HILBERT C*-SYSTEMS FOR C*-ALGEBRAS WITH A NONTRIVIAL CENTER

2004 ◽  
Vol 15 (08) ◽  
pp. 759-812 ◽  
Author(s):  
HELLMUT BAUMGÄRTEL ◽  
FERNANDO LLEDÓ
Keyword(s):  
Technical Condition ◽  
Minimality Condition ◽  
Compact Groups ◽  
Character Group ◽  
C Algebra ◽  
C Algebras ◽  
Dual Object ◽  
Nontrivial Center ◽  
Chain Group ◽  
Fixed Point Algebra

In this paper we present duality theory for compact groups in the case when the C*-algebra [Formula: see text], the fixed point algebra of the corresponding Hilbert C*-system [Formula: see text], has a nontrivial center [Formula: see text] and the relative commutant satisfies the minimality condition [Formula: see text] as well as a technical condition called regularity. The abstract characterization of the mentioned Hilbert C*-system is expressed by means of an inclusion of C*-categories [Formula: see text], where [Formula: see text] is a suitable DR-category and [Formula: see text] a full subcategory of the category of endomorphisms of [Formula: see text]. Both categories have the same objects and the arrows of [Formula: see text] can be generated from the arrows of [Formula: see text] and the center [Formula: see text]. A crucial new element that appears in the present analysis is an abelian group [Formula: see text], which we call the chain group of [Formula: see text], and that can be constructed from certain equivalence relation defined on [Formula: see text], the dual object of [Formula: see text]. The chain group, which is isomorphic to the character group of the center of [Formula: see text], determines the action of irreducible endomorphisms of [Formula: see text] when restricted to [Formula: see text]. Moreover, [Formula: see text] encodes the possibility of defining a symmetry ∊ also for the larger category [Formula: see text] of the previous inclusion.

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.

scholarly journals Examples of non connective C *-algebras

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Anna Gąsior ◽  
Andrzej Szczepański
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Blow Up ◽  
Fractional Diffusion ◽  
Uniqueness Of Solutions ◽  
C Algebras ◽  
Space Fractional Diffusion Equation ◽  
Self Similar ◽  
Similar Solutions

Abstract This paper investigates the problem of the existence and uniqueness of solutions under the generalized self-similar forms to the space-fractional diffusion equation. Therefore, through applying the properties of Schauder’s and Banach’s fixed point theorems; we establish several results on the global existence and blow-up of generalized self-similar solutions to this equation.

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