Document Zbl 0946.46047 - zbMATH Open
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The Bockstein map is necessary. (English) Zbl 0946.46047
In the classification of nuclear \(C^\ast\)-algebras initiated by G. A. Elliott in [J. Reine Angew. Math. 443, 179-219 (1993; Zbl 0809.46067)] the ordered \(K\)-groups play a prominent role. From recent work of several authors it turned out that a more refined invariant is provided by the total group \(\bigoplus_{n\geq 0, i\in\mathbb{Z}_2}K_i(A;\mathbb{Z}_n)\) of a \(C^\ast\)-algebra \(A\) together with the various natural coefficient transformations and the Bockstein maps. The authors present several examples which show that one cannot do without some information about the coefficient transformations or the Bockstein maps. Such information is also lost by tensoring with the Cuntz algebra \({\mathcal O}_\infty\) leading to real rank zero purely infinite \(C^\ast\)-algebras.