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Pullbacks, \(C(X)\)-algebras, and their Cuntz semigroup. (English) Zbl 1255.46030

This paper studies the Cuntz semigroup of \(C(X)\)-algebras where \(X\) is a one-dimensional space and the fibers are separable \(C^{*}\)-algebras of stable rank one such that \(K_1\) vanishes for each ideal of the fiber.
The Cuntz semigroup \(Cu(A)\) of a \(C^{*}\)-algebra \(A\) plays an important role in the structure theory of \(C^{*}\)-algebras and the related Elliott classification program. It is defined analogously to the Murray-von Neumann semigroup \(V(A)\) by using equivalence classes of positive elements instead of projections. In general, the semigroup \(Cu(A)\) contains much more information than \(V(A)\), and it is therefore also more difficult to compute. The Cuntz semigroup is usually considered as an object in the category \(\mathrm{Cu}\) as introduced in [K. T. Coward, G. A. Elliott and C. Ivanescu, J. Reine Angew. Math. 623, 161–193 (2008; Zbl 1161.46029)].
The main result of the paper is Theorem 3.4, which computes the Cuntz semigroup of \(C_0(X,A)\) for a one-dimensional, second-countable, locally compact Hausdorff space \(X\) and a separable \(C^{*}\)-algebra \(A\) with stable rank one and such that \(K_1(I)=0\) for every closed, two-sided ideal \(I\) of \(A\). It is shown that there is a natural isomorphism between \(Cu(C_0(X,A))\) and the semigroup of lower semicontinuous functions \(Lsc(X,Cu(A))\).
The class of \(C^{*}\)-algebras covered by Theorem 3.4 contains for instance all \(AF\)-algebras and every separable, simple \(C^{*}\)-algebra with stable rank and vanishing \(K_1\)-group.
The results are obtained by first considering the special case \(X=[0,1]\) in Section 2. By studying general pullbacks of \(C^{*}\)-algebras and the associated Cuntz semigroups, the results for \([0,1]\) are extended in Section 3 to the case where \(X\) is a finite graph. Using the sequential continuity of the involved invariants, this is further extended to general one-dimensional spaces.
Section 5 contains a thorough discussion of semigroups of lower semicontinuous functions from a space \(X\) to a semigroup \(M\), denoted by \(Lsc(X,M)\). If \(M\) is in the category \(\mathrm{Cu}\), it is natural to ask if \(Lsc(X,M)\) belongs to \(\mathrm{Cu}\) as well. A positive answer to this question is given in Theorem 5.15 for the case that \(X\) is compact, metrizable and finite-dimensional. Further, the functorial properties of \(Lsc(X,-)\) and \(Lsc(-,M)\) are studied (for fixed \(X\), respectively \(M\)).
The results of the paper are used in Section 4 to compute the Cuntz semigroups of many examples, e.g. for one-dimensional NCCW-complexes.

MSC:

46L35 Classifications of \(C^*\)-algebras
46L05 General theory of \(C^*\)-algebras
46L80 \(K\)-theory and operator algebras (including cyclic theory)

Citations:

Zbl 1161.46029
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References:

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