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 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.

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 The Cone of Functionals on the Cuntz Semigroup

2013 ◽  
Vol 113 (2) ◽  
pp. 161 ◽  
Author(s):  
Leonel Robert
Keyword(s):  
Distributive Lattice ◽  
Ordered Semigroup ◽  
Riesz Decomposition Property ◽  
Decomposition Property ◽  
C Algebra ◽  
C Algebras ◽  
Riesz Decomposition ◽  
Cuntz Semigroup

The functionals on an ordered semigroup $S$ in the category $\mathbf{Cu}$ - a category to which the Cuntz semigroup of a C*-algebra naturally belongs - are investigated. After appending a new axiom to the category $\mathbf{Cu}$, it is shown that the "realification" $S_{\mathsf{R}}$ of $S$ has the same functionals as $S$ and, moreover, is recovered functorially from the cone of functionals of $S$. Furthermore, if $S$ has a weak Riesz decomposition property, then $S_{\mathsf{R}}$ has refinement and interpolation properties which imply that the cone of functionals on $S$ is a complete distributive lattice. These results apply to the Cuntz semigroup of a C*-algebra. At the level of C*-algebras, the operation of realification is matched by tensoring with a certain stably projectionless C*-algebra.

scholarly journals On the Structure of Cuntz Semigroups in (Possibly) Nonunital C*-algebras

2015 ◽  
Vol 58 (2) ◽  
pp. 402-414 ◽  
Author(s):  
Aaron Peter Tikuisis ◽  
Andrew Toms
Keyword(s):  
Lower Semicontinuous ◽  
Positive Operators ◽  
Von Neumann ◽  
C Algebra ◽  
C Algebras ◽  
Simple Algebras ◽  
Cuntz Semigroup ◽  
Unital Simple ◽  
Densely Defined

AbstractWe examine the ranks of operators in semi-finite C*-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple C*-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray–von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are satisfied by not-necessarily-unital approximately subhomogeneous algebras of slow dimension growth. Combined with results of the first author, this shows that slow dimension growth coincides with Z-stability for approximately subhomogeneous algebras.

scholarly journals The Corona Factorization Property, Stability, and the Cuntz Semigroup of a C*-algebra

2011 ◽  
Vol 2012 (1) ◽  
pp. 34-66 ◽  
Author(s):  
Eduard Ortega ◽  
Francesc Perera ◽  
Mikael Rørdam
Keyword(s):  
Factorization Property ◽  
C Algebra ◽  
Cuntz Semigroup

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.

scholarly journals Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

2017 ◽  
Vol 69 (3) ◽  
pp. 548-578 ◽  
Author(s):  
Michael Hartglass
Keyword(s):  
Differential Calculus ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Weighted Graph ◽  
Positive Cone ◽  
Von Neumann ◽  
Planar Algebras ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient

AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

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