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.

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.

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 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 Ideal structure and $C^*$-algebras of low rank

2007 ◽  
Vol 100 (1) ◽  
pp. 5 ◽  
Author(s):  
Lawrence G. Brown ◽  
Gert K. Pedersen
Keyword(s):  
Stable Rank ◽  
Low Rank ◽  
Real Rank ◽  
Ideal Structure ◽  
C Algebra ◽  
C Algebras ◽  
Rank One ◽  
The Relationship

We explore various constructions with ideals in a $C^*$-algebra $A$ in relation to the notions of real rank, stable rank and extremal richness. In particular we investigate the maximum ideals of low rank. And we investigate the relationship between existence of infinite or properly infinite projections in an extremally rich $C^*$-algebra and non-existence of ideals or quotients of stable rank one.

scholarly journals Rokhlin dimension: duality, tracial properties, and crossed products

2019 ◽  
Vol 41 (2) ◽  
pp. 408-460
Author(s):  
EUSEBIO GARDELLA ◽  
ILAN HIRSHBERG ◽  
LUIS SANTIAGO
Keyword(s):  
Compact Group ◽  
Local Approximation ◽  
Stable Rank ◽  
Crossed Product ◽  
Crossed Products ◽  
Real Rank ◽  
Ideal Structure ◽  
The Core ◽  
Universal Coefficient Theorem ◽  
The Ideal

We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of $C^{\ast }$-algebras, including: $D$-absorbing $C^{\ast }$-algebras, where $D$ is a strongly self-absorbing $C^{\ast }$-algebra; stable $C^{\ast }$-algebras; $C^{\ast }$-algebras with finite nuclear dimension (or decomposition rank); $C^{\ast }$-algebras with finite stable rank (or real rank); and $C^{\ast }$-algebras whose $K$-theory is either trivial, rational, or $n$-divisible for $n\in \mathbb{N}$. The combination of nuclearity and the universal coefficient theorem (UCT) is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous $C(X)$-algebra with fibers that are stably isomorphic to the underlying algebra. The space $X$ is computed in some cases of interest, and we use its description to construct a $\mathbb{Z}_{2}$-action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.

AFD MULTIPLIER ALGEBRAS

2006 ◽  
Vol 17 (09) ◽  
pp. 1091-1102
Author(s):  
P. W. NG
Keyword(s):  
Von Neumann Algebra ◽  
Real Rank Zero ◽  
Multiplier Algebra ◽  
Von Neumann ◽  
Rank Zero ◽  
C Algebra ◽  
Unique Trace ◽  
Type Property ◽  
Unital Simple ◽  
Multiplier Algebras

We show that the multiplier algebra of a simple stable nuclear C*-algebra has a property similar to that of an AFD or hyperfinite von Neumann algebra. Specifically, we prove the following: Theorem 0.1. Let [Formula: see text] be a unital simple separable C*-algebra. Let [Formula: see text] be the multiplier algebra of the stabilization of [Formula: see text]. Then [Formula: see text] is nuclear if and only if [Formula: see text] has the AFD-type property. We also study a stronger property called the "strong AFD-type property". We show that if [Formula: see text] is a unital simple real rank zero AT-algebra with unique trace, then the multiplier algebra [Formula: see text] of the stabilization of [Formula: see text] has the strong AFD-type property, and we raise the question of whether this is true more generally.

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