PAIRS OF SIMPLE DIMENSION GROUPS

1999 ◽  
Vol 10 (06) ◽  
pp. 739-761 ◽  
Author(s):  
A. KISHIMOTO
Keyword(s):  
The State ◽  
Crossed Product ◽  
Real Rank ◽  
Real Rank Zero ◽  
State Spaces ◽  
Rank Zero ◽  
C Algebra ◽  
Finite Dimensional ◽  
Dimension Groups

Given a pair of simple dimension groups with isomorphic state spaces we try to express it as the pair K0(A),K0(A×αR) of K0 groups for a C*-algebra A with an action α of R, where both A and the crossed product A×αR are supposed to be simple AT algebras of real rank zero. We solve this when the state spaces are finite-dimensional.

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 of Real Rank Zero Whose K0's are not Riesz Groups

1996 ◽  
Vol 39 (4) ◽  
pp. 429-437 ◽  
Author(s):  
K. R. Goodearl
Keyword(s):  
Real Rank ◽  
Riesz Decomposition Property ◽  
Real Rank Zero ◽  
Decomposition Property ◽  
Rank Zero ◽  
C Algebras ◽  
Finite Dimensional ◽  
Riesz Decomposition ◽  
Stably Finite ◽  
The Riesz Decomposition Property

AbstractExamples are constructed of stably finite, imitai, separable C* -algebras A of real rank zero such that the partially ordered abelian groups K0(A) do not satisfy the Riesz decomposition property. This contrasts with the result of Zhang that projections in C* -algebras of real rank zero satisfy Riesz decomposition. The construction method also produces a stably finite, unital, separable C* -algebra of real rank zero which has the same K-theory as an approximately finite dimensional C*-algebra, but is not itself approximately finite dimensional.

scholarly journals The crossed product of a UHF algebra by a shift

1993 ◽  
Vol 13 (4) ◽  
pp. 615-626 ◽  
Author(s):  
Ola Bratteli ◽  
Erling Størmer ◽  
Akitaka Kishimoto ◽  
Mikael Rørdam
Keyword(s):  
Inductive Limit ◽  
Crossed Product ◽  
Cuntz Algebra ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
Full Matrix ◽  
Matrix Algebras ◽  
Car Algebra ◽  
Homogeneous Algebras

AbstractWe prove that the crossed product of the CAR algebra M2∞ by the shift is an inductive limit of homogeneous algebras over the circle with fibres full matrix algebras. As a consequence the crossed product has real rank zero, and where is the Cuntz algebra of order 2.

scholarly journals Strict Comparison of Positive Elements in Multiplier Algebras

2017 ◽  
Vol 69 (02) ◽  
pp. 373-407 ◽  
Author(s):  
Victor Kaftal ◽  
Ping Wong Ng ◽  
Shuang Zhang
Keyword(s):  
Positive Element ◽  
Real Rank ◽  
Real Rank Zero ◽  
Multiplier Algebra ◽  
Rank Zero ◽  
C Algebra ◽  
Positive Elements ◽  
Rank One ◽  
Multiplier Algebras ◽  
Positive Linear Combination

AbstractMain result: If a C*-algebrais simple,σ-unital, hasfinitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebraalso has strict comparison of positive elements by traces. The same results holds if finitely many extremal traces is replaced byquasicontinuous scale. A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary σ-unital C* -algebra can be approximated by a bi-diagonal series. As an application of strict comparison, ifis a simple separable stable C* -algebra with real rank zero, stable rank one, and strict comparison of positive elements by traces, then whether a positive element is a positive linear combination of projections is determined by the trace values of its range projection.

On Extensions of Stably FiniteC*-Algebras (II)

2016 ◽  
Vol 59 (2) ◽  
pp. 435-439
Author(s):  
Hongliang Yao
Keyword(s):  
Necessary Condition ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
Approximate Unit ◽  
K Theory ◽  
Stably Finite ◽  
Sufficient And Necessary Condition ◽  
Smallest Ideal

AbstractFor any C*-algebra A with an approximate unit of projections, there is a smallest ideal I of A such that the quotient A/I is stably finite. In this paper a sufficient and necessary condition for an ideal of a C*-algebra with real rank zero to be this smallest ideal is obtained by using K-theory

scholarly journals AF FLOWS AND CONTINUOUS SYMMETRIES

2001 ◽  
Vol 13 (12) ◽  
pp. 1505-1528 ◽  
Author(s):  
O. BRATTELI ◽  
A. KISHIMOTO
Keyword(s):  
Symmetry Breaking ◽  
Fixed Points ◽  
Automorphism Groups ◽  
Real Rank ◽  
Continuous Symmetry ◽  
Real Rank Zero ◽  
Rank Zero ◽  
Finite Dimensional ◽  
Continuous Symmetries ◽  
Unital Simple

We consider AF flows, i.e. one-parameter automorphism groups of a unital simple AF C*-algebra which leave invariant the dense union of an increasing sequence of finite-dimensional *-subalgebras, and derive two properties for these; an absence of continuous symmetry breaking and a kind of real rank zero property for the almost fixed points.

ON GENERALIZED UNIVERSAL IRRATIONAL ROTATION ALGEBRAS AND ASSOCIATED STRONGLY IRREDUCIBLE OPERATORS

2013 ◽  
Vol 24 (08) ◽  
pp. 1350059
Author(s):  
JUNSHENG FANG ◽  
CHUNLAN JIANG ◽  
HUAXIN LIN ◽  
FENG XU
Keyword(s):  
Irrational Number ◽  
Positive Function ◽  
Irrational Rotation ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
Order Isomorphism ◽  
Strongly Irreducible ◽  
Zero Points

We introduce a class of generalized universal irrational rotation C*-algebras Aθ, γ = C*(x, w) which is characterized by the relations w*w = ww* = 1, x*x = γ(w), xx* = γ(e-2πiθw), and xw = e-2πiθwx, where θ is an irrational number and γ(z) ∈ C(𝕋) is a positive function. We characterize tracial linear functionals, simplicity, and K-groups of Aθ, γ in terms of zero points of γ(z). We show that if Aθ, γ is simple then Aθ, γ is an A𝕋-algebra of real rank zero. We classify Aθ, γ in terms of θ and zero points of γ(z). Let Aθ = C*(u, v) be the universal irrational rotation C*-algebra with vu = e2πiθuv. Then C*(u + v) ≅ Aθ,|1+z|2. As an application, we show that C*(u + v) is a proper simple C*-subalgebra of Aθ which has a unique trace, K1(C*(u + v)) ≅ ℤ, and there is an order isomorphism of K0(C*(u + v)) onto ℤ + ℤθ. Moreover, C*(u + v) is a unital simple A𝕋-algebra of real rank zero. We also show that u + v is strongly irreducible relative to the hyperfinite type II 1 factor.

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.

scholarly journals Tracial Rokhlin property for automorphisms on simple -algebras

2008 ◽  
Vol 28 (4) ◽  
pp. 1215-1241
Author(s):  
HUAXIN LIN ◽  
HIROYUKI OSAKA
Keyword(s):  
Metric Space ◽  
Crossed Product ◽  
Real Rank ◽  
Compact Metric Space ◽  
Real Rank Zero ◽  
Rank Zero ◽  
Simple Algebras ◽  
Minimal Homeomorphisms ◽  
Unital Simple ◽  
Tracial Rokhlin Property

AbstractLet A be a unital simple $A\mathbb {T}$-algebra of real rank zero. Given an isomorphismγ1:K1(A)→K1(A), we show that there is an automorphism α:A→A such that α*1=γ1 and α has the tracial Rokhlin property. Consequently, the crossed product $A\rtimes _{\alpha }\mathbb {Z}$ is a simple unital AH-algebra with real rank zero. We also show that automorphisms with the Rokhlin property can be constructed from minimal homeomorphisms on a connected compact metric space.

Sign in / Sign up

Export Citation Format

Share Document