scholarly journals Limit $C^*$-algebras associated with an automorphism

2004 ◽  
Vol 95 (1) ◽  
pp. 101 ◽  
Author(s):  
Baruch Solel
Keyword(s):  
Fock Space ◽  
Direct Limit ◽  
Complete Classification ◽  
C Algebra ◽  
C Algebras ◽  
Product Presentation ◽  
Necessary And Sufficient ◽  
Deddens Algebras ◽  
Creation Operators

We present and study $C^*$-algebras generated by "periodic weighted creation operators" on the Fock space associated with an automorphism $\alpha$ on a $C^*$-algebra $A$. These algebras can be viewed as generalized Bunce-Deddens algebras associated with the automorphism and can be written as a certain direct limit. We prove a crossed product presentation for such an algebra and find a necessary and sufficient condition for it to be simple. In the case where the automorphism is induced by an irrational rotation (on C(T)) we compute the K-theory groups and obtain a complete classification of these algebras.

Strong Morita equivalence for the Denjoy C*-Algebras

1988 ◽  
Vol 31 (4) ◽  
pp. 439-447
Author(s):  
Ian F. Putnam
Keyword(s):  
Periodic Points ◽  
Complete Classification ◽  
Morita Equivalence ◽  
Constant Function ◽  
Necessary And Sufficient Condition ◽  
C Algebras ◽  
Necessary And Sufficient ◽  
Strong Morita Equivalence ◽  
Function Construction

AbstractThe C*-algebras associated with irrational rotations of the circle were classified up to strong Morita equivalence by M. A. Rieffel. As a corollary, he gave a complete classification of the C*-algebras arising from irrational or Kronecker flows on the 2-torus up to *-isomorphism. Here, we extend the result to the socalled Denjoy homeomorphisms. Specifically, we give a necessary and sufficient condition for the strong Morita equivalence of two C*-algebras arising from homeomorphisms of the circle without periodic points. As a corollary, we show that two C*-algebras arising from flows on the 2-torus obtained from such homeomorphisms by the “flow under constant function” construction are *-isomorphic if and only if the flows themselves are topologically conjugate.

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.

Allee’s Effect Bifurcation in Generalized Logistic Maps

2019 ◽  
Vol 29 (03) ◽  
pp. 1950039
Author(s):  
J. Leonel Rocha ◽  
Abdel-Kaddous Taha
Keyword(s):  
Allee Effect ◽  
Dynamical Behavior ◽  
Complete Classification ◽  
Unimodal Maps ◽  
New Class ◽  
New Type ◽  
Local And Global Bifurcations ◽  
One Dimensional Maps ◽  
Necessary And Sufficient

This paper concerns the study of the Allee effect on the dynamical behavior of a new class of generalized logistic maps. The fundamentals of the dynamics of this 4-parameter family of one-dimensional maps are presented. A complete classification of the nature and stability of its fixed points is provided. The main results relate to the Allee effect bifurcation: a new type of bifurcation introduced for this class of unimodal maps. A necessary and sufficient condition so that the Allee fixed point is a snap-back repeller is established. In addition, in the parameters space is defined an Allee’s effect region, which determines the existence of an essential extinction for the generalized logistic maps. Local and global bifurcations of generalized logistic maps are investigated.

scholarly journals Construction and pure infiniteness of $C^*$-algebras associated with lambda-graph systems

2005 ◽  
Vol 97 (1) ◽  
pp. 73 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Hilbert Space ◽  
Bratteli Diagram ◽  
C Algebra ◽  
C Algebras ◽  
Shift Transformation ◽  
Labeled Graphs ◽  
Graph System ◽  
Creation Operators

A $\lambda$-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. In [16] the author has introduced a $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ associated with a $\lambda$-graph system $\mathfrak{L}$ by using groupoid method as a generalization of the Cuntz-Krieger algebras. In this paper, we concretely construct the $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ by using both creation operators and projections on a sub Fock Hilbert space associated with $\mathfrak{L}$. We also introduce a new irreducible condition on $\mathfrak{L}$ under which the $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ becomes simple and purely infinite.

scholarly journals Inductive limit of direct sums of simple TAI algebras

2019 ◽  
pp. 1-26
Author(s):  
Bo Cui ◽  
Chunlan Jiang ◽  
Liangqing Li
Keyword(s):  
Inductive Limit ◽  
Direct Sums ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras ◽  
Rank One ◽  
Tracial Rank ◽  
Ideal Property ◽  
Funct Anal

An ATAI (or ATAF, respectively) algebra, introduced in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (or in [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], respectively) is an inductive limit [Formula: see text], where each [Formula: see text] is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404], the second author classified all ATAI algebras by an invariant consisting orderd total [Formula: see text]-theory and tracial state spaces of cut down algebras under an extra restriction that all element in [Formula: see text] are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic [Formula: see text]-group as an addition to the invariant used in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404]. The theorem is proved by reducing the class to the classification theorem of [Formula: see text] algebras with ideal property which is done in [G. Gong, C. Jiang and L. Li, A classification of inductive limit C*-algebras with ideal property, preprint (2016), arXiv:1607.07681]. Our theorem generalizes the main theorem of [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (see Corollary 4.3).

scholarly journals Projective Modules Over Quantum Projective Line

2017 ◽  
Vol 28 (03) ◽  
pp. 1750022 ◽  
Author(s):  
Albert Jeu-Liang Sheu
Keyword(s):  
Projective Line ◽  
Complete Classification ◽  
Projective Modules ◽  
C Algebra ◽  
Isomorphism Classes ◽  
Complex Projective Spaces ◽  
Algebra Approach ◽  
Quantum Spheres ◽  
Complex Projective

Taking a groupoid C*-algebra approach to the study of the quantum complex projective spaces [Formula: see text] constructed from the multipullback quantum spheres introduced by Hajac and collaborators, we analyze the structure of the C*-algebra [Formula: see text] realized as a concrete groupoid C*-algebra, and find its [Formula: see text]-groups. Furthermore, after a complete classification of the unitary equivalence classes of projections or equivalently the isomorphism classes of finitely generated projective modules over the C*-algebra [Formula: see text], we identify those quantum principal [Formula: see text]-bundles introduced by Hajac and collaborators among the projections classified.

scholarly journals Connectedness of number theoretical tilings

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Shigeki Akiyama ◽  
Nertila Gjini
Keyword(s):  
Complete Classification ◽  
Necessary And Sufficient Condition ◽  
Digit Set ◽  
Number Systems ◽  
Arcwise Connected ◽  
International Audience ◽  
Consecutive Integers ◽  
Pisot Unit ◽  
Necessary And Sufficient

International audience Let T=T(A,D) be a self-affine tile in \reals^n defined by an integral expanding matrix A and a digit set D. In connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers \0,1,..., |det(A)|-1\. It is shown that in \reals^3 and \reals^4, for any integral expanding matrix A, T(A,D) is connected. We also study the connectedness of Pisot dual tilings which play an important role in the study of β -expansion, substitution and symbolic dynamical system. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. This is naturally expected since the digit set consists of consecutive integers as above. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, a complete classification of the β -expansion of 1 for quartic Pisot units is given.

Discontinuous homomorphisms from C*-algebras

1991 ◽  
Vol 110 (1) ◽  
pp. 147-150 ◽  
Author(s):  
D. W. B. Somerset
Keyword(s):  
Banach Algebra ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient

AbstractA necessary and sufficient condition is given for a unital C*-algebra A to admit a discontinuous homomorphism into a Banach algebra which is continuous on its centre. The condition is that A must have a Glimm ideal G such that the C*-algebra A/G admits a discontinuous homomorphism into a Banach algebra.

Homomorphisms From C(X) Into C*-Algebras

1997 ◽  
Vol 49 (5) ◽  
pp. 963-1009 ◽  
Author(s):  
Huaxin Lin
Keyword(s):  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Countable Rank ◽  
Rank One ◽  
Dimensional Range

AbstractLet A be a simple C*-algebra with real rank zero, stable rank one and weakly unperforated K0(A) of countable rank. We show that a monomorphism Φ: C(S2) → A can be approximated pointwise by homomorphisms from C(S2) into A with finite dimensional range if and only if certain index vanishes. In particular,we show that every homomorphism ϕ from C(S2) into a UHF-algebra can be approximated pointwise by homomorphisms from C(S2) into the UHF-algebra with finite dimensional range.As an application, we show that if A is a simple C*-algebra of real rank zero and is an inductive limit of matrices over C(S2) then A is an AF-algebra. Similar results for tori are also obtained. Classification of Hom (C(X), A) for lower dimensional spaces is also studied.

Complete Moment Balancing of Contra Planar 5R Linkages

2012 ◽  
Author(s):  
Brian Moore ◽  
Clément Gosselin
Keyword(s):  
Special Class ◽  
Sufficient Conditions ◽  
Complete Classification ◽  
Necessary And Sufficient Conditions ◽  
Design Parameters ◽  
Balancing Conditions ◽  
Necessary And Sufficient

In this paper, the complete shaking force and moment balancing conditions for a special class of planar 5R linkages, the contra 5R linkage, is considered. Contra 5R linkages are planar 5R linkages in which the two input links are mechanically coupled and rotate at the same speed in opposite directions. A method to derive necessary and sufficient conditions on the design parameters to achieve moment balancing without introducing additional components is presented. Using this method, a complete classification of all shaking force and moment balanced contra 5R linkages is given.

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