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Classification of inductive limits of 1-dimensional NCCW complexes. (English) Zbl 1268.46041

This paper deals with non-simple \(C^*\)-algebras that are given as inductive limits of certain building blocks. It is shown that such algebras are classified by an invariant that is a variation of the well-known Cuntz semigroup.
The building blocks are one-dimensional NCCW-complexes, as introduced by S. Eilers et al. [J. Reine Angew. Math. 499, 101–143 (1998; Zbl 0897.46056)], with vanishing \(K_1\)-group. Let \(\mathcal{C}\) be the class of \(C^*\)-algebras that are stably isomorphic to an inductive limit of the considered building blocks. It is not known, but quite conceivable, that a \(C^*\)-algebra lies in \(\mathcal{C}\) if it is an inductive limit of one-dimensional NCCW-complexes and its \(K_1\)-group vanishes (one has to show that a trivial \(K_1\)-group of the inductive limit algebra makes it possible to find a suitable inductive limit structure such that each building block has trivial \(K_1\)-group).
The Cuntz semigroup of a \(C^*\)-algebra is an ordered, abelian semigroup given by certain equivalence classes of positive elements (much like the Murray-von Neumann semigroup \(V(A)\) is given by equivalence classes of projections). It was shown that the Cuntz semigroup satisfies many additional properties which were used to define a category \(\mathrm{Cu}\) in [K. T. Coward et al., J. Reine Angew. Math. 623, 161–193 (2008; Zbl 1161.46029)]. The assignment of the Cuntz semigroup \(\mathrm{Cu}(A)\) to a \(C^*\)-algebra \(A\) is a functor that preserves sequential inductive limits. This paper introduces a new functor, denoted by \(\mathrm{Cu}\tilde{\;}\), from stable rank one \(C^*\)-algebras to \(\mathrm{Cu}\).
For simple \(C^*\)-algebras that are inductive limits of one-dimensional NCCW-complexes, it is shown in Section 6 that the new invariant \(\mathrm{Cu}\tilde{\;}\) contains the same information as the ordered \(K_0\)-group, the cone of traces and their pairing. If one additionally assumes that the \(K_1\)-group vanishes (which is, for instance, satisfied for the simple algebras in the class \(\mathcal{C}\)), then \(\mathrm{Cu}\tilde{\;}\) is equivalent to the Elliott invariant. One has to distinguish the two cases that the simple \(C^*\)-algebra \(A\) contains a projection or is projectionless. In the latter case, it is shown that the usual Cuntz semigroup \(\mathrm{Cu}(A)\) contains no information about \(K_0(A)\), which makes apparent the advantage of the new invariant \(\mathrm{Cu}\tilde{\;}(A)\). In the unital case, however, the new invariant is determined by the usual Cuntz semigroup.
Given a \(C^*\)-algebra \(A\) in the class \(\mathcal{C}\) and an arbitrary \(C^*\)-algebra \(B\) of stable rank one, the main result of the paper (Theorem 1.0.1) classifies *-homomorphisms from \(A\) to \(B\) up to approximate unitary equivalence in terms of \(\mathrm{Cu}\tilde{\;}\). Using an approximate intertwining argument, one obtains a classification of the algebras in \(\mathcal{C}\) up to stable isomorphism by the new invariant \(\mathrm{Cu}\tilde{\;}\). By additionally recording the position of a strictly positive element, one achieves classification up to isomorphism.
The article gives two interesting applications. Firstly, it is shown that the Jiang-Su algebra \(\mathcal{Z}\) unitally embeds into the reduced group \(C^*\)-algebra \(C^*(F_\infty)\) of the free group of infinitely many generators. This is done by computing the Cuntz semigroups of both algebras, which turn out to be isomorphic. Using that the Jiang-Su algebra is an element of the class \(\mathcal{C}\) and that \(C^*(F_\infty)\) has stable rank one, the main result allows to lift the map at the level of Cuntz semigroups to a *-homomorphism. Secondly, the paper considers quasi-free actions of the real numbers on the Cuntz algebra \(\mathcal{O}_2\). Such an action is determined by a parameter \(\lambda\), and it is shown that, for a dense set of positive parameters, the resulting crossed products are all pairwise isomorphic.

MSC:

46L35 Classifications of \(C^*\)-algebras
46L80 \(K\)-theory and operator algebras (including cyclic theory)
46L85 Noncommutative topology
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