AF-algebras and Ranks of C*-algebras

2001 ◽  
pp. 113-164
Keyword(s):  
C Algebras ◽  
Af Algebras

scholarly journals THE UNITAL EXT-GROUPS AND CLASSIFICATION OF C*-ALGEBRAS

2019 ◽  
Vol 62 (1) ◽  
pp. 201-231 ◽  
Author(s):  
JAMES GABE ◽  
EFREN RUIZ
Keyword(s):  
Weak Version ◽  
C Algebras ◽  
Ext Groups ◽  
Theoretic Classification ◽  
Invertible Elements ◽  
Af Algebras

AbstractThe semigroups of unital extensions of separable C*-algebras come in two flavours: a strong and a weak version. By the unital Ext-groups, we mean the groups of invertible elements in these semigroups. We use the unital Ext-groups to obtain K-theoretic classification of both unital and non-unital extensions of C*-algebras, and in particular we obtain a complete K-theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras.

scholarly journals HOMOGENEITY OF THE PURE STATE SPACE FOR SEPARABLE C*-ALGEBRAS

2001 ◽  
Vol 12 (07) ◽  
pp. 813-845 ◽  
Author(s):  
HAJIME FUTAMURA ◽  
NOBUHIRO KATAOKA ◽  
AKITAKA KISHIMOTO
Keyword(s):  
Automorphism Group ◽  
State Space ◽  
Large Class ◽  
Pure State ◽  
C Algebras ◽  
Inner Automorphisms ◽  
Af Algebras

We prove that the pure state space is homogeneous under the action of the automorphism group (or a certain smaller group of approximately inner automorphisms) for a fairly large class of simple separable nuclear C*-algebras, including the approximately homogeneous C*-algebras and the class of purely infinite C*-algebras which has been recently classified by Kirchberg and Phillips. This extends the known results for UHF algebras and AF algebras by Powers and Bratteli.

scholarly journals FRAÏSSÉ LIMITS OF C*-ALGEBRAS

2016 ◽  
Vol 81 (2) ◽  
pp. 755-773 ◽  
Author(s):  
CHRISTOPHER J. EAGLE ◽  
ILIJAS FARAH ◽  
BRADD HART ◽  
BORIS KADETS ◽  
VLADYSLAV KALASHNYK ◽  
...  
Keyword(s):  
Full Matrix ◽  
Matrix Algebras ◽  
C Algebras ◽  
Af Algebras

AbstractWe realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II1 factor as Fraïssé limits of suitable classes of structures. Moreover by means of Fraïssé theory we provide new examples of AF algebras with strong homogeneity properties. As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.

scholarly journals MORPHISMS OF SIMPLE TRACIALLY AF ALGEBRAS

2004 ◽  
Vol 15 (09) ◽  
pp. 919-957 ◽  
Author(s):  
MARIUS DADARLAT
Keyword(s):  
Existence And Uniqueness ◽  
Residually Finite ◽  
C Algebras ◽  
Finite Dimensional ◽  
K Theory ◽  
Inner Automorphisms ◽  
Af Algebras ◽  
Universal Coefficient Theorem

Let A, B be separable simple unital tracially AF C*-algebras. Assuming that A is exact and satisfies the Universal Coefficient Theorem (UCT) in KK-theory, we prove the existence, and uniqueness modulo approximately inner automorphisms, of nuclear *-homomorphisms from A to B with prescribed K-theory data. This implies the AF-embeddability of separable exact residually finite-dimensional C*-algebras satisfying the UCT and reproves Huaxin Lin's theorem on the classification of nuclear tracially AF C*-algebras.

scholarly journals Noncommutative Lattices and the Algebras of Their Continuous Functions

1998 ◽  
Vol 10 (04) ◽  
pp. 439-466 ◽  
Author(s):  
Elisa Ercolessi ◽  
Giovanni Landi ◽  
Paulo Teotonio-Sobrinho
Keyword(s):  
Quantum Physics ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Hausdorff Topology ◽  
Topological Information ◽  
C Algebras ◽  
Finite Dimensional ◽  
Noncommutative Algebras ◽  
Af Algebras ◽  
The Continuum

Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physics. Posets are truly noncommutative spaces, or noncommutative lattices, since they can be realized as structure spaces of noncommutative C*-algebras. These noncommutative algebras play the same rôle as the algebra of continuous functions [Formula: see text] on a Hausdorff topological space M and can be thought of as algebras of operator valued functions on posets. In this article, we will review some mathematical results that establish a duality between finite posets and a certain class of C*-algebras. We will see that the algebras in question are all postliminal approximately finite dimensional (AF) algebras.

Stability of Real C*-Algebras

2011 ◽  
Vol 54 (4) ◽  
pp. 593-606
Author(s):  
Jeffrey L. Boersema ◽  
Efren Ruiz
Keyword(s):  
Factorization Property ◽  
C Algebras ◽  
The One ◽  
Af Algebras

AbstractWe will give a characterization of stable real C*-algebras analogous to the one given for complex C*-algebras by Hjelmborg and Rørdam. Using this result, we will prove that any real C*-algebra satisfying the corona factorization property is stable if and only if its complexification is stable. Real C*-algebras satisfying the corona factorization property include AF-algebras and purely infinite C*-algebras. We will also provide an example of a simple unstable C*-algebra, the complexification of which is stable.

FUSION RULES FROM ALGEBRAIC K-THEORY

1993 ◽  
Vol 08 (07) ◽  
pp. 1345-1357 ◽  
Author(s):  
ANDREAS RECKNAGEL
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Field ◽  
Fusion Rules ◽  
C Algebras ◽  
K Theory ◽  
Af Algebras

We show how quantum field theory fusion rules can be recovered from K0-groups of C*-algebras. Applying this method to certain AF algebras yields the fusion rules of SU(2)-WZW models.

C*-Algebras With Real Rank Zero and The Internal Structure of Their Corona and Multiplier Algebras Part III

1990 ◽  
Vol 42 (1) ◽  
pp. 159-190 ◽  
Author(s):  
Shuang Zhang
Keyword(s):  
Internal Structure ◽  
Closed Ideal ◽  
Real Rank ◽  
Ideal Structure ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebras ◽  
Points Of View ◽  
Af Algebras ◽  
Multiplier Algebras

In this part, we shall be concerned with the structure of projections in a simple σ-unital C*-algebra with the FS property, and in the associated multiplier and corona algebras. We shall also consider the closed ideal structure of the corona algebra. Most of results appear to be new even for separable simple AF algebras, and are technically independent of the previous parts I and II ([37] and [38]). The whole work develops after finding a new property of a σ-unital (nonunital) simple C*-algebra with FS, which was not known even for a separable simple AF algebra. We relate this new property to the structure of the multiplier and corona algebras from vairous points of view.

C*-ALGEBRAIC METHODS IN STATISTICAL MECHANICS AND FIELD THEORY

1990 ◽  
Vol 04 (05) ◽  
pp. 1069-1118 ◽  
Author(s):  
David E. EVANS
Keyword(s):  
Field Theory ◽  
Statistical Mechanics ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Inductive Limits ◽  
Von Neumann ◽  
C Algebras ◽  
Conformal Field ◽  
Finite Dimensional ◽  
Af Algebras

We survey the recent work in non-commutative operator algebras (especially AF-algebras, those which are inductive limits of finite dimensional C*-algebras) and which arise in studying critical phenomena in classical statistical mechanics and conformal field theory, from a C*- or topological viewpoint, rather than a von Neumann algebra/measure theoretic one.

Subquotients of UHF C*-algebras

1994 ◽  
Vol 115 (3) ◽  
pp. 489-500 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
General Structure ◽  
The Other ◽  
Type I ◽  
Car Algebra ◽  
C Algebras ◽  
Structure Theorems ◽  
Af Algebras

Over the last thirty years, the study of C*-algebras has proceeded in a number of directions. On one hand, much effort has been devoted to understanding the structure of particular classes of algebras, such as the approximately finite (AF) algebras. On the other, general structure theorems have been sought. Classes of algebras defined by certain abstract properties have been investigated with a view to obtaining more concrete descriptions of the algebras. One of the earliest results of this type was the theorem of Glimm [13], later extended by Sakai [20] to the inseparable case, characterizing the non-type I C*-algebras as those algebras which contain a subalgebra with a quotient *-isomorphic to the CAR algebra.

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