Classification of extensions of certain \(C^*\)-algebras by their six term exact sequences in \(K\)-theory (Q1359497)

The family of \(C^*\)-algebras, which are extensions of \(C^*\)-algebras which fall into the recent classification result by Eberhard Kirchberg (the so-called ``pis(u)n-algebras''), is in this paper proved to be classified by the six term exact sequence in \(K\)-theory induced by the extensions. In more detail, if \[ 0\to A_j\to E_j\to B_j\to 0, \] where \(j=1,2\), are short-exact sequences and if \(A_j\) and \(B_j\) are simple, purely infinite, separable, nuclear \(C^*\)-algebras in the bootstrap category \(\mathcal N\) of Rosenberg and Schochet, then \(E_1\) is stably isomorphic to \(E_2\) if and only if the six term exact sequences induced by the sequences \[ 0\to A_j\to E_j\to B_j\to 0 \] are isomorphic (in a natural way). This is proved by establishing an algebraic result, which yields an equality between certain orbits of Ext-groups from an isomorphism between dix term exact sequences, in combination with results of Kirchberg.