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.

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 REALIZATION OF MINIMAL C*-DYNAMICAL SYSTEMS IN TERMS OF CUNTZ–PIMSNER ALGEBRAS

2009 ◽  
Vol 20 (06) ◽  
pp. 751-790 ◽  
Author(s):  
FERNANDO LLEDÓ ◽  
EZIO VASSELLI
Keyword(s):  
Dynamical Systems ◽  
Group Action ◽  
Unitary Irreducible Representation ◽  
Full Spectrum ◽  
C Algebra ◽  
Finite Dimensional ◽  
Relative Commutant ◽  
Chain Group ◽  
Fixed Point Algebra ◽  
Discrete Abelian Group

In the present article, we provide several constructions of C*-dynamical systems [Formula: see text] with a compact group [Formula: see text] in terms of Cuntz–Pimsner algebras. These systems have a minimal relative commutant of the fixed-point algebra [Formula: see text] in [Formula: see text], i.e. [Formula: see text], where [Formula: see text] is the center of [Formula: see text], which is assumed to be non-trivial. In addition, we show in our models that the group action [Formula: see text] has full spectrum, i.e. any unitary irreducible representation of [Formula: see text] is carried by a [Formula: see text]-invariant Hilbert space within [Formula: see text]. First, we give several constructions of minimal C*-dynamical systems in terms of a single Cuntz–Pimsner algebra [Formula: see text] associated to a suitable [Formula: see text]-bimodule ℌ. These examples are labelled by the action of a discrete Abelian group ℭ (which we call the chain group) on [Formula: see text] and by the choice of a suitable class of finite dimensional representations of [Formula: see text]. Second, we present a more elaborate contruction, where now the C*-algebra [Formula: see text] is generated by a family of Cuntz–Pimsner algebras. Here, the product of the elements in different algebras is twisted by the chain group action. We specify the various constructions of C*-dynamical systems for the group [Formula: see text], N ≥ 2.

scholarly journals The Jiang–Su Absorption for Inclusions of Unital C*-algebras

2018 ◽  
Vol 70 (2) ◽  
pp. 400-425 ◽  
Author(s):  
Hiroyuki Osaka ◽  
Tamotsu Teruya
Keyword(s):  
Finite Group ◽  
Conditional Expectation ◽  
C Algebra ◽  
C Algebras ◽  
Crossed Product Algebra ◽  
Fixed Point Algebra ◽  
Stably Finite ◽  
A Finite Group ◽  
Product Algebra ◽  
Tracial Rokhlin Property

AbstractWe introduce the tracial Rokhlin property for a conditional expectation for an inclusion of unital C*-algebras P ⊂ A with index finite, and show that an action α from a finite group G on a simple unital C*- algebra A has the tracial Rokhlin property in the sense of N. C. Phillips if and only if the canonical conditional expectation E: A → AG has the tracial Rokhlin property. Let be a class of infinite dimensional stably finite separable unital C*-algebras that is closed under the following conditions:(1) If A ∊ and B ≅ A, then B ∊ .(2) If A ∊ and n ∊ ℕ, then Mn(A) ∊ .(3) If A ∊ and p ∊ A is a nonzero projection, then pAp ∊ .Suppose that any C*-algebra in is weakly semiprojective. We prove that if A is a local tracial -algebra in the sense of Fan and Fang and a conditional expectation E: A → P is of index-finite type with the tracial Rokhlin property, then P is a unital local tracial -algebra.The main result is that if A is simple, separable, unital nuclear, Jiang–Su absorbing and E: A → P has the tracial Rokhlin property, then P is Jiang–Su absorbing. As an application, when an action α from a finite group G on a simple unital C*-algebra A has the tracial Rokhlin property, then for any subgroup H of G the fixed point algebra AH and the crossed product algebra H is Jiang–Su absorbing. We also show that the strict comparison property for a Cuntz semigroup W(A) is hereditary to W(P) if A is simple, separable, exact, unital, and E: A → P has the tracial Rokhlin property.

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

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.

On C*-Algebras Associated with Subshifts

1997 ◽  
Vol 08 (03) ◽  
pp. 357-374 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Symbolic Dynamics ◽  
Markov Shift ◽  
C Algebra ◽  
C Algebras ◽  
Markov Shifts ◽  
Universal Properties

We construct and study C*-algebras associated with subshifts in symbolic dynamics as a generalization of Cuntz–Krieger algebras for topological Markov shifts. We prove some universal properties for the C*-algebras and give a criterion for them to be simple and purely infinite. We also present an example of a C*-algebra coming from a subshift which is not conjugate to a Markov shift.

scholarly journals CUNTZ–PIMSNER C*-ALGEBRAS ASSOCIATED WITH SUBSHIFTS

2008 ◽  
Vol 19 (01) ◽  
pp. 47-70 ◽  
Author(s):  
TOKE MEIER CARLSEN
Keyword(s):  
Partial Isometries ◽  
C Algebra ◽  
C Algebras ◽  
Shift Space

By using C*-correspondences and Cuntz–Pimsner algebras, we associate to every subshift (also called a shift space) 𝖷 a C*-algebra [Formula: see text], which is a generalization of the Cuntz–Krieger algebras. We show that [Formula: see text] is the universal C*-algebra generated by partial isometries satisfying relations given by 𝖷. We also show that [Formula: see text] is a one-sided conjugacy invariant of 𝖷.

scholarly journals Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

2017 ◽  
Vol 69 (3) ◽  
pp. 548-578 ◽  
Author(s):  
Michael Hartglass
Keyword(s):  
Differential Calculus ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Weighted Graph ◽  
Positive Cone ◽  
Von Neumann ◽  
Planar Algebras ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient

AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

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