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)

Zbl 1161.46029
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References:

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