## Decomposable approximations and approximately finite dimensional $$C^*$$-algebras.(English)Zbl 1442.46046

A $$C^*$$-algebra $$A$$ is said to be nuclear if, for every other $$C^*$$-algebra $$B$$, there is a unique $$C^*$$-norm on the algebraic tensor product $$A\odot B$$. This important property is equivalent to the completely positive approximation property (CPAP): the identity on $$A$$ can be approximately factored as the composition of completely positive, contractive (cpc) maps $$A\to F$$ and $$F\to A$$ with $$F$$ a finite-dimensional $$C^*$$-algebra.
One obtains interesting subclasses of nuclear $$C^*$$-algebras by considering strengthened versions of the CPAP where one requires additional properties of the map $$F\to A$$. For example, one says that $$A$$ has decomposition rank at most $$n$$ if the identity on $$A$$ can be approximately factored through cpc-maps $$A\to F$$ and $$\varphi\colon F\to A$$ such that $$F$$ decomposes as a direct sum of $$n+1$$ summands and the restriction of $$\varphi$$ to each summand is orthogonality-preserving. The decomposition rank is a noncommutative generalization of covering dimension introduced in [E. Kirchberg and W. Winter, Int. J. Math. 15, No. 1, 63–85 (2004; Zbl 1065.46053)], and it plays an important role in the current structure and classification theory of simple, nuclear $$C^*$$-algebras.
In [I. Hirshberg et al., Adv. Math. 230, No. 3, 1029–1039 (2012; Zbl 1256.46019)] it was shown that nuclear $$C^*$$-algebras automatically enjoy a strengthened versions of the CPAP: one may additionally assume that the cpc-map $$F\to A$$ is a convex combination of orthogonality-preserving cpc-maps, but the number of summands may grow with increasing precision of the approximation.
It is thus natural to consider the combination of these strengthenings of the CPAP: Given $$n$$, when does a $$C^*$$-algebra $$A$$ admit approximate factorizations of the identity through cpc-maps $$A\to F$$ and $$\varphi\colon F\to A$$ such that $$\varphi$$ is a convex combination of at most $$n$$ orthogonality-preserving cpc-maps? The present paper shows that this happens if and only if $$A$$ has decomposition rank zero, which for separable $$C^*$$-algebras means precisely that $$A$$ is an AF-algebra.

### MSC:

 46L35 Classifications of $$C^*$$-algebras 46B28 Spaces of operators; tensor products; approximation properties

### Citations:

Zbl 1065.46053; Zbl 1256.46019
Full Text:

### References:

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