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].

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 $S$-regularity and the corona factorization property

2006 ◽  
Vol 99 (2) ◽  
pp. 204 ◽  
Author(s):  
D. Kucerovsky ◽  
P. W. Ng
Keyword(s):  
Fundamental Property ◽  
Sufficient Condition ◽  
Factorization Property ◽  
Stable Property ◽  
C Algebra ◽  
C Algebras ◽  
The Third ◽  
Explicit Reference ◽  
Stable Algebra ◽  
Extension Algebra

Stability is an important and fundamental property of $C^{*}$-algebras. Given a short exact sequence of $C^{*}$-algebras $0\longrightarrow B\longrightarrow E\longrightarrow A\longrightarrow 0$ where the ends are stable, the middle algebra may or may not be stable. We say that the first algebra, $B$, is $S$-regular if every extension of $B$ by a stable algebra $A$ has a stable extension algebra, $E$. Rördam has given a sufficient condition for $S$-regularity. We define a new condition, weaker than Rördam's, which we call the corona factorization property, and we show that the corona factorization property implies $S$-regularity. The corona factorization property originated in a study of the Kasparov $KK^1(A,B)$ group of extensions, however, we obtain our results without explicit reference to $KK$-theory. Our main result is that for a separable stable $C^{*}$-algebra $B$ the first two of the following properties (which we define later) are equivalent, and both imply the third. With additional hypotheses on the $C^{*}$-algebra, all three properties are equivalent. $B$ has the corona factorization property. Stability is a stable property for full hereditary subalgebras of $B$. $B$ is $S$-regular. We also show that extensions of separable stable $C^{*}$-algebras with the corona factorization property give extension algebras with the corona factorization property, extending the results of [9].

scholarly journals Comparison Properties of the Cuntz Semigroup and Applications to C* -algebras

2018 ◽  
Vol 70 (1) ◽  
pp. 26-52
Author(s):  
Joan Bosa ◽  
Henning Petzka
Keyword(s):  
Factorization Property ◽  
C Algebra ◽  
C Algebras ◽  
Algebraic Setting ◽  
Cuntz Semigroup ◽  
Comparison Property

AbstractWe study comparison properties in the category Cu aiming to lift results to the C* -algebraic setting. We introduce a new comparison property and relate it to both the corona factorization property (CFP) and ω-comparison. We show differences of all properties by providing examples that suggest that the corona factorization for C* -algebras might allow for both finite and infinite projections. In addition, we show that Rørdam's simple, nuclear C* -algebra with a finite and an inifnite projection does not have the CFP.

scholarly journals CENTRAL SEQUENCES IN SUBHOMOGENEOUS UNITAL C*-ALGEBRAS

2020 ◽  
pp. 1-8
Author(s):  
DON HADWIN ◽  
HEMANT PENDHARKAR
Keyword(s):  
Irreducible Representations ◽  
C Algebra ◽  
Central Sequence ◽  
C Algebras ◽  
Pointwise Limit ◽  
Central Sequences ◽  
Multiplicity Free

Abstract Suppose that $\mathcal {A}$ is a unital subhomogeneous C*-algebra. We show that every central sequence in $\mathcal {A}$ is hypercentral if and only if every pointwise limit of a sequence of irreducible representations is multiplicity free. We also show that every central sequence in $\mathcal {A}$ is trivial if and only if every pointwise limit of irreducible representations is irreducible. Finally, we give a nice representation of the latter algebras.

scholarly journals A factorization property of positive maps on C*-algebras

2020 ◽  
Vol 18 (05) ◽  
pp. 2050019
Author(s):  
B. V. Rajarama Bhat ◽  
Hiroyuki Osaka
Keyword(s):  
Hilbert Spaces ◽  
Short Paper ◽  
Complete Positivity ◽  
Factorization Property ◽  
Positive Map ◽  
C Algebra ◽  
Matrix Algebras ◽  
C Algebras ◽  
Linear Maps ◽  
Positive Linear Maps

The purpose of this short paper is to clarify and present a general version of an interesting observation by [Piani and Mora, Phys. Rev. A 75 (2007) 012305], linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let [Formula: see text], [Formula: see text] be unital C*-algebras and let [Formula: see text] be positive linear maps from [Formula: see text] to [Formula: see text] [Formula: see text]. We obtain conditions under which any positive map [Formula: see text] from the minimal C*-tensor product [Formula: see text] to [Formula: see text], such that [Formula: see text], factorizes as [Formula: see text] for some positive map [Formula: see text]. In particular, we show that when [Formula: see text] are completely positive (CP) maps for some Hilbert spaces [Formula: see text] [Formula: see text], and [Formula: see text] is a pure CP map and [Formula: see text] is a CP map so that [Formula: see text] is also CP, then [Formula: see text] for some CP map [Formula: see text]. We show that a similar result holds in the context of positive linear maps when [Formula: see text] and [Formula: see text]. As an application, we extend IX Theorem of Ref. 4 (revisited recently by [Huber et al., Phys. Rev. Lett. 121 (2018) 200503]) to show that for any linear map [Formula: see text] from a unital C*-algebra [Formula: see text] to a C*-algebra [Formula: see text], if [Formula: see text] is decomposable for some [Formula: see text], where [Formula: see text] is the identity map on the algebra [Formula: see text] of [Formula: see text] matrices, then [Formula: see text] is CP.

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

On C*-Algebras Associated with Subshifts

1997 ◽  
Vol 08 (03) ◽  
pp. 357-374 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Symbolic Dynamics ◽  
Markov Shift ◽  
C Algebra ◽  
C Algebras ◽  
Markov Shifts ◽  
Universal Properties

We construct and study C*-algebras associated with subshifts in symbolic dynamics as a generalization of Cuntz–Krieger algebras for topological Markov shifts. We prove some universal properties for the C*-algebras and give a criterion for them to be simple and purely infinite. We also present an example of a C*-algebra coming from a subshift which is not conjugate to a Markov shift.

scholarly journals CUNTZ–PIMSNER C*-ALGEBRAS ASSOCIATED WITH SUBSHIFTS

2008 ◽  
Vol 19 (01) ◽  
pp. 47-70 ◽  
Author(s):  
TOKE MEIER CARLSEN
Keyword(s):  
Partial Isometries ◽  
C Algebra ◽  
C Algebras ◽  
Shift Space

By using C*-correspondences and Cuntz–Pimsner algebras, we associate to every subshift (also called a shift space) 𝖷 a C*-algebra [Formula: see text], which is a generalization of the Cuntz–Krieger algebras. We show that [Formula: see text] is the universal C*-algebra generated by partial isometries satisfying relations given by 𝖷. We also show that [Formula: see text] is a one-sided conjugacy invariant of 𝖷.

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