C*-ALGEBRAS GENERATED BY A PROJECTION AND THE REDUCED GROUP C*-ALGEBRAS ${C^*_r\Bbb Z*\Bbb Z_n}$ AND ${C^*_r\Bbb Z_m*\Bbb Z_n}$

2005 ◽  
Vol 16 (05) ◽  
pp. 533-554
Author(s):  
SHUANG ZHANG
Keyword(s):  
Closed Ideal ◽  
Quotient Algebra ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
Ideal Formula ◽  
C Algebras ◽  
Reduced Word ◽  
Reduced Group

Let Γ=ℤm * ℤn or ℤ * ℤn, and let Γ(h) be the subtree consisting of all reduced words starting with any reduced word h ∈ Γ\{e}. We prove that the C*-algebra [Formula: see text] generated by [Formula: see text] and the projection Ph onto the subspace ℓ2(Γ(h)) has a unique nontrivial closed ideal ℐ, ℐ is *-isomorphic to [Formula: see text], and the quotient algebra [Formula: see text] is *-isomorphic to either [Formula: see text] or [Formula: see text] depending on the last letter of h. We also prove that [Formula: see text] is a purely infinite, simple C*-algebra if the last letter of h is a generator of ℤ, and that [Formula: see text] has a unique nontrivial closed ideal [Formula: see text] if the last letter of h is a generator of ℤn; furthermore, [Formula: see text] is *-isomorphic to [Formula: see text] and [Formula: see text] is again a purely infinite, simple C*-algebra. As consequences, all the C*-algebras above have real rank zero, and [Formula: see text] is nuclear for any h ≠ e.

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.

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*-ALGEBRAS WITH REAL RANK ZERO AND THEIR CORONA AND MULTIPLIER ALGEBRAS PART IV

1992 ◽  
Vol 03 (02) ◽  
pp. 309-330 ◽  
Author(s):  
SHUANG ZHANG
Keyword(s):  
Real Rank ◽  
Real Rank Zero ◽  
Multiplier Algebra ◽  
Von Neumann ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras ◽  
Neumann Theorem ◽  
Von Neumann Theorem ◽  
Deddens Algebras

By proving various equivalent versions of the generalized Weyl-von Neumann theorem, we investigate the structure of projections in the multiplier algebra [Formula: see text] of certain C*-algebra [Formula: see text] with real rank zero. For example, we prove that [Formula: see text] if and only if any two projections in [Formula: see text] are simultaneously quasidiagonal. In case [Formula: see text] is a purely infinite simple C*-algebra, [Formula: see text] if and only if any two projections in [Formula: see text] are simultaneously quasidiagonal. If [Formula: see text] is one of the Cuntz algebras, or one of finite factors or type III factors, then any two projections in [Formula: see text] are simultaneously quasidiagonal. On the other hand, if [Formula: see text] is one of the Bunce-Deddens algebras or one of the irrational rotation algebras of real rank zero, then there exist two projections in [Formula: see text] which are not simultaneously quasidiagonal.

scholarly journals UNIVERSAL C*-ALGEBRA OF REAL RANK ZERO

2000 ◽  
Vol 03 (03) ◽  
pp. 445-452 ◽  
Author(s):  
ALEX CHIGOGIDZE
Keyword(s):  
Continuous Functions ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras

It is well known that every commutative separable unital C*-algebra of real rank zero is a quotient of the C*-algebra of all compex continuous functions defined on the Cantor cube. We prove a non-commutative version of this result by showing that the class of all separable unital C*-algebras of real rank zero coincides with the class of quotients of a certain separable unital C*-algebra of real rank zero.

EXTENSIONS BY C*-ALGEBRAS WITH REAL RANK ZERO

1993 ◽  
Vol 04 (02) ◽  
pp. 231-252 ◽  
Author(s):  
HUAXIN LIN
Keyword(s):  
Unit Circle ◽  
Compact Subset ◽  
Real Line ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
The Real ◽  
C Algebra ◽  
C Algebras ◽  
Essential Extensions

We show that all trivial (unital and essential) extensions of C (X) by a σ-unital purely infinite simple C*-algebra A with K1(A) = 0 are unitarily equivalent, provided that X is homeomorphic to a compact subset of the real line or the unit circle. Therefore all (unital and essential) extensions of such can be completely determined by Ext(B, A). An invariant is introduced to classify all such trivial (unital and essential) extensions of C (X) by a σ-unital C*-algebra A with the properties that RR (M (A)) = 0 and C (A) is simple.

The Structure of Positive Elements for C*-Algebras with Real Rank Zero

1997 ◽  
Vol 08 (03) ◽  
pp. 383-405 ◽  
Author(s):  
Francesc Perera
Keyword(s):  
Grothendieck Group ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras ◽  
Positive Elements ◽  
Riesz Decomposition ◽  
Riesz Decomposition Properties ◽  
Rank One

In this paper we give a representation theorem for the Cuntz monoid S(A) of a σ-unital C*-algebra A with real rank zero and stable rank one, which allows to prove several Riesz decomposition properties on the monoid. As a consequence, it is proved that the comparability conditions (FCQ), stable (FCQ) and (FCQ+) are equivalent for simple C*-algebras with real rank zero. It is also shown that the Grothendieck group [Formula: see text] of S(A) is a Riesz group, and lattice-ordered under some additional assumptions on A.

Linear Maps on C*-Algebras Preserving the Set of Operators that are Invertible in

2011 ◽  
Vol 54 (1) ◽  
pp. 141-146
Author(s):  
Sang Og Kim ◽  
Choonkil Park
Keyword(s):  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
Calkin Algebra ◽  
C Algebras ◽  
Linear Maps

AbstractFor C*-algebras of real rank zero, we describe linear maps ϕ on that are surjective up to ideals , and π(A) is invertible in if and only if π(ϕ(A)) is invertible in , where A ∈ and π : → is the quotient map. We also consider similar linear maps preserving zero products on the Calkin algebra.

scholarly journals On the classification of C*-algebras of real rank zero, III: The infinite case

1998 ◽  
pp. 11-72 ◽  
Author(s):  
Ola Bratteli ◽  
George Elliott ◽  
David Evans ◽  
Akitaka Kishimoto
Keyword(s):  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebras ◽  
Infinite Case
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