## 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

### Citations:

Zbl 0897.46056; Zbl 1161.46029
Full Text:

### References:

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