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.


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