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 CUNTZ SEMIGROUP OF CONTINUOUS FUNCTIONS INTO CERTAIN SIMPLE C*-ALGEBRAS

2011 ◽  
Vol 22 (08) ◽  
pp. 1051-1087 ◽  
Author(s):  
AARON TIKUISIS
Keyword(s):  
Simple Formula ◽  
Continuous Functions ◽  
Von Neumann ◽  
C Algebras ◽  
Cuntz Semigroup ◽  
Unital Simple ◽  
Compact Subsets

This paper contains computations of the Cuntz semigroup of separable C*-algebras of the form C0(X, A), where A is a unital, simple, [Formula: see text]-stable ASH algebra. The computations describe the Cuntz semigroup in terms of Murray–von Neumann semigroups of C(K, A) for compact subsets K of X. In particular, the computation shows that the Elliott invariant is functorially equivalent to the invariant given by the Cuntz semigroup of C(𝕋, A). These results are a contribution towards the goal of using the Cuntz semigroup in the classification of well-behaved non-simple C*-algebras.

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.

scholarly journals Fundamental Group of Simple C*-algebras with Unique Trace III

2012 ◽  
Vol 64 (3) ◽  
pp. 573-587 ◽  
Author(s):  
Norio Nawata
Keyword(s):  
Fundamental Group ◽  
Lower Semicontinuous ◽  
Classification Theorem ◽  
Fundamental Groups ◽  
C Algebras ◽  
Scalar Multiple ◽  
Unique Trace ◽  
Densely Defined

Abstract We introduce the fundamental group ℱ(A) of a simple σ-inital C*-algebra A with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of Fundamental Group of Simple C*-algebras with Unique Trace I and II by Nawata andWatatani. Our definition in this paper makes sense for stably projectionless C*-algebras. We show that there exist separable stably projectionless C*-algebras such that their fundamental groups are equal to ℝ×+ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian.

scholarly journals Large Irredundant Sets in Operator Algebras

2019 ◽  
Vol 72 (4) ◽  
pp. 988-1023
Author(s):  
Clayton Suguio Hida ◽  
Piotr Koszmider
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Continuum Hypothesis ◽  
The Other ◽  
Open Problems ◽  
Von Neumann ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
The Continuum

AbstractA subset ${\mathcal{X}}$ of a C*-algebra ${\mathcal{A}}$ is called irredundant if no $A\in {\mathcal{X}}$ belongs to the C*-subalgebra of ${\mathcal{A}}$ generated by ${\mathcal{X}}\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum.There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic axioms, and we investigate here the noncommutative case.Assuming $\diamondsuit$ (an additional axiom stronger than the continuum hypothesis), we prove that there is an AF C*-subalgebra of ${\mathcal{B}}(\ell _{2})$ of density $2^{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{1}$ with no nonseparable commutative C*-subalgebra and with no uncountable irredundant set. On the other hand we also prove that it is consistent that every discrete collection of operators in ${\mathcal{B}}(\ell _{2})$ of cardinality continuum contains an irredundant subcollection of cardinality continuum.Other partial results and more open problems are presented.

APPROXIMATELY INNER FLOWS ON SEPARABLE C*-ALGEBRAS

2002 ◽  
Vol 14 (07n08) ◽  
pp. 649-673 ◽  
Author(s):  
AKITAKA KISHIMOTO
Keyword(s):  
Irreducible Representation ◽  
Von Neumann Algebra ◽  
Automorphism Groups ◽  
Unitary Groups ◽  
Type I ◽  
Von Neumann ◽  
C Algebra ◽  
C Algebras ◽  
Inner Automorphism ◽  
Uniformly Continuous

We present two types of result for approximately inner one-parameter automorphism groups (referred to as AI flows hereafter) of separable C*-algebras. First, if there is an irreducible representation π of a separable C*-algebra A such that π(A) does not contain non-zero compact operators, then there is an AI flow α such that π is α-covariant and α is far from uniformly continuous in the sense that α induces a flow on π(A) which has full Connes spectrum. Second, if α is an AI flow on a separable C*-algebra A and π is an α-covariant irreducible representation, then we can choose a sequence (hn) of self-adjoint elements in A such that αt is the limit of inner flows Ad eithn and the sequence π(eithn) of one-parameter unitary groups (referred to as unitary flows hereafter) converges to a unitary flow which implements α in π. This latter result will be extended to cover the case of weakly inner type I representations. In passing we shall also show that if two representations of a separable simple C*-algebra on a separable Hilbert space generate the same von Neumann algebra of type I, then there is an approximately inner automorphism which sends one into the other up to equivalence.

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.

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.

CROSSED PRODUCTS OF CERTAIN C*-ALGEBRAS

2014 ◽  
Vol 25 (02) ◽  
pp. 1450010 ◽  
Author(s):  
JIAJIE HUA ◽  
YAN WU
Keyword(s):  
Simple Formula ◽  
Continuous Functions ◽  
Cantor Set ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebra ◽  
C Algebras ◽  
Tracial Rank ◽  
Universal Coefficient Theorem ◽  
Unital Simple

Let X be a Cantor set, and let A be a unital separable simple amenable [Formula: see text]-stable C*-algebra with rationally tracial rank no more than one, which satisfies the Universal Coefficient Theorem (UCT). We use C(X, A) to denote the algebra of all continuous functions from X to A. Let α be an automorphism on C(X, A). Suppose that C(X, A) is α-simple, [α|1⊗A] = [ id |1⊗A] in KL(1 ⊗ A, C(X, A)), τ(α(1 ⊗ a)) = τ(1 ⊗ a) for all τ ∈ T(C(X, A)) and all a ∈ A, and [Formula: see text] for all u ∈ U(A) (where α‡ and id‡ are homomorphisms from U(C(X, A))/CU(C(X, A)) → U(C(X, A))/CU(C(X, A)) induced by α and id, respectively, and where CU(C(X, A)) is the closure of the subgroup generated by commutators of the unitary group U(C(X, A)) of C(X, A)), then the corresponding crossed product C(X, A) ⋊α ℤ is a unital simple [Formula: see text]-stable C*-algebra with rationally tracial rank no more than one, which satisfies the UCT. Let X be a Cantor set and 𝕋 be the circle. Let γ : X × 𝕋n → X × 𝕋n be a minimal homeomorphism. It is proved that, as long as the cocycles are rotations, the tracial rank of the corresponding crossed product C*-algebra is always no more than one.

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.

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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