Geodesic
The Bockstein Map is Necessary
Canadian mathematical bulletin, Tome 42 (1999) no. 3, pp. 274-284

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DOI

We construct two non-isomorphic nuclear, stably finite, real rank zero ${{C}^{*}}$ -algebras $E$ and ${{E}^{'}}$ for which there is an isomorphism of ordered groups $\Theta :{{\oplus }_{n\ge 0}}{{K}_{\bullet }}(E;\mathbb{Z}/n)\to {{\oplus }_{n\ge 0}}{{K}_{\bullet }}({E}';\mathbb{Z}/n)$ which is compatible with all the coefficient transformations. The ${{C}^{*}}$ -algebras $E$ and ${{E}^{'}}$ are not isomorphic since there is no $\Theta$ as above which is also compatible with the Bockstein operations. By tensoring with Cuntz’s algebra ${{\mathcal{O}}_{\infty }}$ one obtains a pair of non-isomorphic, real rank zero, purely infinite ${{C}^{*}}$ -algebras with similar properties.
DOI : 10.4153/CMB-1999-033-0
Mots-clés : 46L35, 46L80, 19K14, K-theory, torsion coefficients, natural transformations, Bockstein maps, C*-algebras, real rank zero, purely infinite, classification 
Dădărlat, Marius; Eilers, Søren. The Bockstein Map is Necessary. Canadian mathematical bulletin, Tome 42 (1999) no. 3, pp. 274-284. doi: 10.4153/CMB-1999-033-0
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     title = {The {Bockstein} {Map} is {Necessary}},
     journal = {Canadian mathematical bulletin},
     pages = {274--284},
     year = {1999},
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     number = {3},
     doi = {10.4153/CMB-1999-033-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-033-0/}
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