scholarly journals Regularity of Villadsen algebras and characters on their central sequence algebras

2018 ◽  
Vol 123 (1) ◽  
pp. 121-141
Author(s):  
Martin S. Christensen
Keyword(s):  
Stable Rank ◽  
Factorization Property ◽  
Central Sequence ◽  
Tracial State ◽  
Finite Dimensional ◽  
Cw Complex ◽  
Sequence Algebra ◽  
Unital Simple

We show that if $A$ is a simple Villadsen algebra of either the first type with seed space a finite dimensional CW complex, or of the second type, then $A$ absorbs the Jiang-Su algebra tensorially if and only if the central sequence algebra of $A$ does not admit characters.Additionally, in a joint appendix with Joan Bosa (see the following paper), we show that the Villadsen algebra of the second type with infinite stable rank fails the Corona Factorization Property, thus providing the first example of a unital, simple, separable and nuclear $C^\ast $-algebra with a unique tracial state which fails to have this property.

scholarly journals When central sequence C*-algebras have characters

2015 ◽  
Vol 26 (07) ◽  
pp. 1550049 ◽  
Author(s):  
Eberhard Kirchberg ◽  
Mikael Rørdam
Keyword(s):  
Factorization Property ◽  
Tensor Power ◽  
C Algebra ◽  
Central Sequence ◽  
C Algebras ◽  
Regularity Properties ◽  
Sequence Algebra ◽  
Divisibility Properties

We investigate C*-algebras whose central sequence algebra has no characters, and we raise the question if such C*-algebras necessarily must absorb the Jiang–Su algebra (provided that they also are separable). We relate this question to a question of Dadarlat and Toms if the Jiang–Su algebra always embeds into the infinite tensor power of any unital C*-algebra without characters. We show that absence of characters of the central sequence algebra implies that the C*-algebra has the so-called strong Corona Factorization Property, and we use this result to exhibit simple nuclear separable unital C*-algebras whose central sequence algebra does admit a character. We show how stronger divisibility properties on the central sequence algebra imply stronger regularity properties of the underlying C*-algebra.

Central Sequence Algebras of a Purely Infinite Simple C*-algebra

2004 ◽  
Vol 56 (6) ◽  
pp. 1237-1258 ◽  
Author(s):  
Akitaka Kishimoto
Keyword(s):  
Fixed Point ◽  
Partial Isometry ◽  
Characterization Result ◽  
C Algebra ◽  
Central Sequence ◽  
Sequence Algebra ◽  
K Theory

AbstractWe are concerned with a unital separable nuclear purely infinite simple C*-algebra A satisfying UCT with a Rohlin flow, as a continuation of [12]. Our first result (which is independent of the Rohlin flow) is to characterize when two central projections in A are equivalent by a central partial isometry. Our second result shows that the K-theory of the central sequence algebra A′ ∩ Aω (for an ω ∈ βN\N) and its fixed point algebra under the flow are the same (incorporating the previous result). We will also complete and supplement the characterization result of the Rohlin property for flows stated in [12].

Appendix to ``Regularity of Villadsen algebras'': The failure of the Corona Factorization Property for the Villadsen algebra $\mathcal{V}_\infty$

2018 ◽  
Vol 123 (1) ◽  
pp. 142-146
Author(s):  
Joan Bosa ◽  
Martin S. Christensen
Keyword(s):  
Factorization Property ◽  
Central Sequence

In this appendix to M. S. Christensen, “Regularity of Villadsen algebras and characters on their central sequence algebras”, Math. Scand. ?? (????), ???--???, we prove that the Villadsen algebra $\mathcal{V} _\infty $ does not satisfy the Corona Factorization Property (CFP).

Extension of Maps to Nilpotent Spaces

2001 ◽  
Vol 44 (3) ◽  
pp. 266-269 ◽  
Author(s):  
M. Cencelj ◽  
A. N. Dranishnikov
Keyword(s):  
Nilpotent Group ◽  
Closed Subset ◽  
Cohomological Dimension ◽  
Dimension Theory ◽  
Homotopy Groups ◽  
Finitely Generated ◽  
Finite Dimensional ◽  
Cw Complex ◽  
Extension Of Maps ◽  
Theory Condition

AbstractWe show that every compactum has cohomological dimension 1 with respect to a finitely generated nilpotent group G whenever it has cohomological dimension 1 with respect to the abelianization of G. This is applied to the extension theory to obtain a cohomological dimension theory condition for a finite-dimensional compactum X for extendability of every map from a closed subset of X into a nilpotent CW-complex M with finitely generated homotopy groups over all of X.

scholarly journals Rotation Algebras and the Exel Trace Formula

2015 ◽  
Vol 67 (2) ◽  
pp. 404-423 ◽  
Author(s):  
Jiajie Hua ◽  
Huaxin Lin
Keyword(s):  
Real Number ◽  
Trace Formula ◽  
Rank Zero ◽  
Tracial State ◽  
Tracial Rank ◽  
Image Position ◽  
Unital Simple ◽  
Tracial States

AbstractWe show that ifuandvare any two unitaries in a unitalC*–algebra such that ∥uv−vu∥ < 2 anduvu*v* commutes withuandv, then theC*–subalgebraAu,vgenerated by u and v is isomorphic to a quotient of some rotation algebraAθ, provided thatAu;vhas a unique tracial state. We also show that the Exel trace formula holds in any unitalC*–algebra. Let θ ∊ (−1/2, 1/2) be a real number. For any ∊ > 0; we prove that there exists ζ > 0 satisfying the following: if u and v are two unitaries in any unital simpleC*–algebra A with tracial rank zero such thatfor all tracial states τof A; then there exists a pair of unitariesandin A such that

scholarly journals Extremally rich C*-crossed products and the cancellation property

1998 ◽  
Vol 64 (3) ◽  
pp. 285-301 ◽  
Author(s):  
Ja A Jeong ◽  
Hiroyuki Osaka
Keyword(s):  
Finite Abelian Group ◽  
Extreme Points ◽  
Stable Rank ◽  
Crossed Product ◽  
Closed Unit Ball ◽  
Crossed Products ◽  
Cancellation Property ◽  
Rank One ◽  
Invertible Elements ◽  
Unital Simple

AbstractA unital C*-algebra A is called extremally rich if the set of quasi-invertible elements A-1 ex (A)A-1 (= A-1q) is dense in A, where ex(A) is the set of extreme points in the closed unit ball A1 of A. In [7, 8] Brown and Pedersen introduced this notion and showed that A is extremally rich if and only if conv(ex(A)) = A1. Any unital simple C*-algebra with extremal richness is either purely infinite or has stable rank one (sr(A) = 1). In this note we investigate the extremal richness of C*-crossed products of extremally rich C*-algebras by finite groups. It is shown that if A is purely infinite simple and unital then A xα, G is extremally rich for any finite group G. But this is not true in general when G is an infinite discrete group. If A is simple with sr(A) =, and has the SP-property, then it is shown that any crossed product A xαG by a finite abelian group G has cancellation. Moreover if this crossed product has real rank zero, it has stable rank one and hence is extremally rich.

On the Dixmier property of simple C*-algebras

1982 ◽  
Vol 91 (1) ◽  
pp. 75-78 ◽  
Author(s):  
Norbert Riedel
Keyword(s):  
Convex Hull ◽  
Von Neumann Algebras ◽  
Present Note ◽  
Linear Span ◽  
Closed Convex Hull ◽  
Von Neumann ◽  
Tracial State ◽  
C Algebras ◽  
Unital Simple

A unital C*-algebra is said to satisfy the Dixmier property if for each element x in the closed convex hull of all elements of the form u*xu, u being a unitary in , intersects the centre of ((2), 2·7). The von Neumann algebras and also some other classes of C*-algebras are known to satisfy the Dixmier property (cf. (2), (3), (4), (6)). If is a simple C*-algebra which satisfies the Dixmier property then has at most one tracial state. In (3) Archbold raised the question whether there exists a unital simple C*-algebra which has at most one tracial state without satisfying the Dixmier property. In the present note we characterize the unital simple C*-algebras with at most one tracial state in terms of a condition which is similar to the Dixmier property, but is in fact formally weaker in the framework of simple C*-algebras. This characterization relies on the method used by Pedersen in (5) in order to show that for a unital simple C*-algebra which has at most one tracial state and at least one non-trivial projection the linear span of all projections in is dense in As an application we characterize those unital simple C*-algebras with a unique tracial state which satisfy the Dixmier property.

scholarly journals CERTAIN APERIODIC AUTOMORPHISMS OF UNITAL SIMPLE PROJECTIONLESS C*-ALGEBRAS

2009 ◽  
Vol 20 (10) ◽  
pp. 1233-1261 ◽  
Author(s):  
YASUHIKO SATO
Keyword(s):  
Inductive Limit ◽  
Crossed Products ◽  
Inductive Limits ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
Tracial State ◽  
C Algebras ◽  
Cyclic Groups ◽  
Unital Simple

Let G be an inductive limit of finite cyclic groups, and A be a unital simple projectionless C*-algebra with K1(A) ≅ G and a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in Aut (A)/ WInn (A) are conjugate, where WInn (A) means the subgroup of Aut (A) consisting of automorphisms which are inner in the tracial representation.In the second part of this paper, we consider a class of unital simple C*-algebras with a unique tracial state which contains the class of unital simple A𝕋-algebras of real rank zero with a unique tracial state. This class is closed under inductive limits and crossed products by actions of ℤ with the Rohlin property. Let A be a TAF-algebra in this class. We show that for any automorphism α of A there exists an automorphism ᾶ of A with the Rohlin property such that ᾶ and α are asymptotically unitarily equivalent. For the proof we use an aperiodic automorphism of the Jiang-Su algebra.

LIMITS OF HOMOMORPHISMS WITH FINITE-DIMENSIONAL RANGE

2005 ◽  
Vol 16 (07) ◽  
pp. 807-821 ◽  
Author(s):  
SHANWEN HU ◽  
HUAXIN LIN ◽  
YIFENG XUE
Keyword(s):  
Metric Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
C Algebra ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Unital Simple ◽  
Dimensional Range

Let X be a compact metric space and A be a unital simple C*-algebra with TR (A)=0. Suppose that ϕ : C(X) → A is a unital monomorphism. We study the problem when ϕ can be approximated by homomorphisms with finite-dimensional range. We give a K-theoretical necessary and sufficient condition for ϕ being approximated by homomorphisms with finite-dimensional range.

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