The Cuntz semigroup as an invariant for C*-algebras

By Kristofer T. Coward, G. Elliott, Cristian Ivanescu. 2007

212 citations recorded by the source corpus.

Abstract A category is described to which the Cuntz semigroup belongs and as a functor into which it preserves inductive limits.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c28760de10769500559ae87b468972cfdb36c295

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. W. Winter
2. Andrew S. Toms
3. Leonel Robert
4. G. Elliott
5. Z. Niu
6. Stuart White
7. Hannes Thiel
8. Yasuhiko Sato

In Mathematical Physics

1. W. Winter
2. Andrew S. Toms
3. Leonel Robert
4. G. Elliott
5. Z. Niu
6. Yasuhiko Sato
7. Stuart White
8. F. Perera

Citations in the corpus

• Classification of inductive limits of 1-dimensional NCCW complexes (2010, methodology): “∞(Si)), L2 for each x,y∈ Sisuch that αi,∞(x) 6 αi,∞(y), and x′ ≪ x, there exists jsuch that αi,j(x′) 6 αi,j(y). (It is easy to see that these properties hold for the inductive limit as constructed in [10], and that they imply in turn the universal property of an inductive limit in the category Cu.) ”
• The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension (2016, background): “Since D ⊗ K has stable rank one, by [8] (see also 1.”
• Pullbacks, $C(X)$-algebras, and their Cuntz semigroup (2011, background): “valence classes of positive elements in the stabilisation of A, in a similar way as the projection semigroup is built out of the Murray-von Neumann equivalence. Coward, Elliott and Ivanescu proved in [9] that the order relation in Cu(A) has additional properties and a new category Cu was defined for ”
• Nuclear dimension and -stability of pure C ∗ -algebras (2010, unknown): “By [7] we also have k · x = sup{k · xl} ≤ 〈a〉.”
• The cone of lower semicontinuous traces on a C*-algebra (2008, background): “If [p] + [a] = [p + a] is the class of a non-zero finite projection, say q, then, by the formulation of Cuntz equivalence given in [7], and since, also by [7], [q] is compact, and sup >0 [(a − )+] = [a], whence sup >0 [p+(a− )+] = [p+a], the Hilbert A-module qA is (by compactness) isomorphic to a su”
• Classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras (2014, background): “The following result is well known to the experts (for example, see [11]).”
• A Remark on Invariants for C*-Algebras of Stable Rank One (2007, background): “erval algebras—sometimes called AI algebras. The purpose of this paper is to point out certain relations between these two invariants, in the case of stable rank one, which follow from the results of [4] (see Theorems 4 and 10). (Roughly speaking, in this case the two invariants coincide.) In partic”
• K-Theory for operator algebras. Classification of C$^*$-algebras (2009, background): “[14], [54]). In our journey we shall dwell on an important recent development of the theory, namely the introduction by Coward, Elliott and Ivanescu [ 11 ] of a stable version of the Cuntz semigroup, denoted Cu(A), that has interesting connections with the theory of Hilbert modules (of which the bas”
• Divisibility properties for C*‐algebras (2011, methodology): “et of Cuntz equivalence classes of positive elements in A ⊗ K endowed with suitable order and addition structures. Given a positive element a in A ⊗ K, we shall denote by hai the Cuntz class of a. In [CEI08], Coward, Elliott and Ivanescu give an alternative picture of the Cuntz semigroup where Cu(A)”
• Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, methodology): “A key tool for establishing approximate unitary equivalence of two positive elements h1 and h2 in a C ∗-algebra C is the Cuntz semigroup, dating back to [16], and the Cuntz category from [15].”
• Ranks of Operators in Simple $$C^*$$-algebras with Stable Rank One (2017, methodology): “aperis the Cuntz semigroup. It provides a convenient way to organize the comparison theory of positive elements 4 HANNES THIEL in a C∗-algebrainto a positively ordered monoid. Coward-Elliott-Ivanescu,[CEI08], initiated a systematic study of Cuntz semigroups by introducing the category Cu of abstract”
• Quasidiagonality of nuclear C*-algebras (2015, methodology): “Lemma 2.6 below) with unital sum, which are the inputs into the “stable uniqueness across the interval” procedure. We start by using Cuntz subequivalence, originating in [18] and further developed in [61, 17] (we refer to [1] for a full account of the Cuntz semigroup), and strict comparison of posit”
• Isomorphism of Hilbert modules over stably finite C*-algebras (2008, methodology): “ar X iv :0 81 1.”