A Remark on Invariants for C*-Algebras of Stable Rank One

By A. Ciuperca, G. Elliott. 2007

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It is shown that, for a C*-algebra of stable rank one (i.e. in which the invertible elements are dense), two well-known invariants, the Cuntz semigroup and the Thomsen semigroup, contain the same information. More precisely, these two invariants, viewed appropriately, determine each other in a natural way.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/be7c08f7c93d038f49f0644c5b08a2f7819d7f62

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Comparison Theory and Smooth Minimal C*-Dynamics (2008, background): “. It is also known that positive elements a,b∈ Aare approximately unitarily equivalent if and only if the canonical maps from C 0(0,1] into Ainduced by aand bagree at the level of the Cuntz semigroup([5]). Thus,totheextentthatoneknows thestructureoftheCuntzsemigroup,one alsoknows what the isomorphis”
Nuclear dimension and $$\mathcal Z$$Z-stability (2014, methodology): “We then use the extra properties of the map φ to obtain order zero maps (0), (1) : (U ⊗ V )ω → Z such that ∑1 i=0 (i)(φ(1A)⊗ 1V ) = 1Zω (for this we use a uniqueness result for certain positive contractions in Zω proved using recent Cuntz-semigroup classification theorems [6,27]); thus the resulting”
C*-algebras of stable rank one and their Cuntz semigroups (2018, unknown): “Yet another use of the Cuntz semigroup, and of the functor associated to it, is as a classiﬁcation invariant for nonsimple C*-algebras; see [CE08, Rob12].”
The cone of lower semicontinuous traces on a C*-algebra (2008, background): “Note that, in [5], the Cuntz semigroup, together with the special element consisting of the class of the strictly positive elements, was shown to be a complete invariant for arbitrary AI algebras, or ideals of AI algebras.”
Pullbacks, $C(X)$-algebras, and their Cuntz semigroup (2011, result): “dsimplealgebras, thissemigroup canbe recovered from the classical Elliott invariant (see[5], [6]), and in the non-simple case it has already been used to prove actual classiﬁcation results (see, e.g. [7], [8], [25], [26], [30]). However, due to its complexity, this semigroup becomes an object very d”
STRUCTURE OF NUCLEAR C*-ALGEBRAS: FROM QUASIDIAGONALITY TO CLASSIFICATION AND BACK AGAIN (2017, background): “l is symmetric under ﬂipping the interval, we see that it agrees with both maps τQω ◦ ´λ and τQ ω ◦ `λ ◦ ﬂip (where ﬂip denotes the canonical isomorphism between C0((0,1]) and C0([0,1)). Moreover, by [9] this is enough to make the maps ´λ ◦ﬂip and `λ approximately unitarily equivalent — and again by”
Regularity properties in the classification program for separable amenable C*-algebras (2007, background): “t comparison (more on this property in section 3) ([5], [6]), and to the classiﬁcation of closed unitary orbits of positive operators in A⊗K through recent work of Ciuperca and the ﬁrst named author ([7]). Essentially, W(A) and Cu(A) contain the same information, but we have chosen to maintain separ”
Classification of inductive limits of 1-dimensional NCCW complexes (2010, methodology): “sts a homomorphism φ: A→ B such that Cu∼(φ) = α. Moreover, φis unique up to approximate unitary equivalence. The above classiﬁcation result continues, and extends, the work of Ciuperca and Elliott in [7], and of Ciuperca, Elliott, and Santiago in [8]. In [7] and [8] the classifying functor is the Cu”
The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension (2016, unknown): “Recall the pseudo-metric dw introduced in [7] dw(α, β) = inf{r ∈ R + : α(〈et+r〉) ≤ β(〈et〉) and β(〈et+r〉) ≤ α(〈et〉), t ∈ R }, (A.”