Stable rank one and real rank zero for crossed products by finite group actions with the tracial Rokhlin property

2009 ◽  
Vol 30 (2) ◽  
pp. 179-186 ◽  
Author(s):  
Qingzhai Fan ◽  
Xiaochun Fang
Keyword(s):  
Finite Group ◽  
Group Actions ◽  
Stable Rank ◽  
Crossed Products ◽  
Real Rank ◽  
Real Rank Zero ◽  
Finite Group Actions ◽  
Rank Zero ◽  
Rank One ◽  
Tracial Rokhlin Property

Ideal Structure of Multiplier Algebras of Simple C*-algebras With Real Rank Zero

2001 ◽  
Vol 53 (3) ◽  
pp. 592-630 ◽  
Author(s):  
Francesc Perera
Keyword(s):  
Stable Rank ◽  
Real Rank ◽  
Ideal Structure ◽  
Real Rank Zero ◽  
Von Neumann ◽  
Rank Zero ◽  
Rank One ◽  
A Chain ◽  
Multiplier Algebras ◽  
The Ideal

AbstractWe give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of σ-unital simple C*-algebras A with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of themultiplier algebra , is therefore analyzed. In important cases it is shown that, if A has finite scale then the quotient of modulo any closed ideal I that properly contains A has stable rank one. The intricacy of the ideal structure of is reflected in the fact that can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.

scholarly journals The Tracial Class Property for Crossed Products by Finite Group Actions

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Xinbing Yang ◽  
Xiaochun Fang
Keyword(s):  
Finite Group ◽  
Group Actions ◽  
Crossed Products ◽  
Finite Group Actions ◽  
Infinite Dimensional ◽  
A Finite Group ◽  
Tracial Rokhlin Property

We define the concept of tracial𝒞-algebra ofC*-algebras, which generalize the concept of local𝒞-algebra ofC*-algebras given by H. Osaka and N. C. Phillips. Let𝒞be any class of separable unitalC*-algebras. LetAbe an infinite dimensional simple unital tracial𝒞-algebra with the (SP)-property, and letα:G→Aut(A)be an action of a finite groupGonAwhich has the tracial Rokhlin property. ThenA  ×α  Gis a simple unital tracial𝒞-algebra.

scholarly journals A Short Return to Simple Ah-Algebras with Real Rank Zero

2014 ◽  
Vol 114 (2) ◽  
pp. 264
Author(s):  
Huaxin Lin
Keyword(s):  
Stable Rank ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
Rank One ◽  
Tracial Rank ◽  
Unital Simple

Let $A$ be a unital simple AH-algebra with stable rank one and real rank zero such that $kx=0$ for all $x\in\operatorname{ker}\rho_A$, the subgroup of infinitesmal elements in $K_0(A)$, and for the same integer $k\ge 1$. We show that $A$ has tracial rank zero and is isomorphic to a unital simple AH-algebra with no dimension growth.

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.

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.

scholarly journals Crossed products by finite group actions with the Rokhlin property

2011 ◽  
Vol 270 (1-2) ◽  
pp. 19-42 ◽  
Author(s):  
Hiroyuki Osaka ◽  
N. Christopher Phillips
Keyword(s):  
Finite Group ◽  
Group Actions ◽  
Crossed Products ◽  
Finite Group Actions

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.

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