The cone of lower semicontinuous traces on a C*-algebra

By G. Elliott, Leonel Robert, Luis Santiago. 2008

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The cone of lower semicontinuous traces is studied with a view to its use as an invariant. Its properties include compactness, Hausdorffness, and continuity with respect to inductive limits. A suitable notion of dual cone is given. The cone of lower semicontinuous 2-quasitraces on a (non-exact) C*-algebra is considered as well. These results are applied to the study of the Cuntz semigroup. It is shown that if a C*-algebra absorbs the Jiang-Su algebra, then the subsemigroup of its Cuntz semigroup consisting of the purely non-compact elements is isomorphic to the dual cone of the cone of lower semicontinuous 2-quasitraces. This yields a computation of the Cuntz semigroup for the following two classes of C*-algebras: C*-algebras that absorb the Jiang-Su algebra and have no non-zero simple subquotients, and simple C*-algebras that absorb the Jiang-Su algebra. 1. Introduction. The most standard invariants in the classification of nu- clear C*-algebras are their K-groups and their traces. The traces, at least for simple C*-algebras, are assumed to be bounded in the unital case, and lower semicontinuous and densely finite in the non-unital case. If one has in mind the classification of non-simple C*-algebras, it is clear that these two kinds of traces will not suffice and a broader class should be considered. In this paper we study the properties of the cone of all lower semicontinuous traces on a C*-algebra, with the purpose of applying our results to questions in the classification of non- simple C*-algebras. One sign that this is natural is that this cone in particular determines the lattice of closed two-sided ideals, an important invariant in its own right (see (3.4)). We shall also consider lower semicontinuous 2-quasitraces, since these appear naturally as functionals on the Cuntz semigroup of the algebra. If the

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e1ef58b8d4bb7ade3e482540f34774ce38d5496c

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Citations in the corpus

The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension (2016, background): “6 of [19] that the map from Cu(A) to LAff+(T̃(B)) is surjective.”
Nuclear dimension of simple stably projectionless C∗-algebras (2019, methodology): “The existence of such hereditary subalgebra follows from the Cuntz semigroup computation of [26] for Z-stable C∗-algebras, and Brown’s Theorem assures us that A0 is stably isomorphic to A.”
Classification of inductive limits of 1-dimensional NCCW complexes (2010, methodology): “It is known that for a simple Calgebra with strict comparison, the map [a] →  [a] is injective on the complement of the subsemigroup of Cuntz classes [p], with p a projection (see [20,5]).”
Large subalgebras (2014, background): “4 of [14], but we give the easy direct proof here.”
The cone of functionals on the Cuntz semigroup (2011, background): “Indeed, SR coincides with the ordered semigroup L(F(S)) introduced in [6].”
Divisibility properties for C*‐algebras (2011, background): “are both necessary and suﬃcient conditions to ensure that the Cuntz semigroup of a simple, separable, unital, exact C∗-algebra A is (naturally) isomorphic to Aﬀ(T(A)) ⊔ V(A), cf. [PT07], [BPT08], and [ERS]. The existence of simple C∗-algebras with bad comparability properties was discovered by Villa”
On the classification of simple amenable C*-algebras with finite decomposition rank (2015, background): “kφ∗(fˆ)− γ(fˆ)k∞ <ε/2, f∈ F. Note that Aand Chave cancellation for projections, and also K + 0(A) = K 0(C) = Q+ (unital identiﬁcation) and Aﬀ(T(A)) ∼= Aﬀ(T(C)). By Theorem 4.4 and Corollary 6.8 of [8] (see also Theorem 2.6 of [5] and Theorem 5.5 of [4], expressed in terms of W instead of Cu), it ”