User:Mct mht/Cuntz

Classification
The Cuntz algebras are pairwise non-isomorphic, i.e. $$\mathcal{O}_n $$ and $$\mathcal{O}_m $$ are non-isomorphic to n &ne; m. The K0 group of $$\mathcal{O}_n$$ is Zn - 1, the abelian cyclic group of order n. Since K0 is a (functorial) invariant, $$\mathcal{O}_n $$ and $$\mathcal{O}_m $$ are non-isomorphic.

By extensions
An extension of the compact operators $$\mathcal{K}$$ by $$\mathcal{O}_n $$ is a short exact sequence of C*-algebras


 * $$ 0 \rightarrow \mathcal{K} \rightarrow \mathcal{E} \rightarrow \mathcal{O}_n \rightarrow 0.$$

Descending to the Calkin algebra $$\mathcal{Q(H)}$$, extensions are in bijective correspondence with *-monomorphisms


 * $$\tau: \mathcal{O}_n \rightarrow \mathcal{Q(H)}.$$

Let &pi; be the quotient map onto the Calkin algebra. Two extensions &tau;1 and &tau;2 are said to be equivalent of


 * $$\pi (U) \tau_1 (\mathcal{O}_n) \pi(U^*) = \tau_2$$

for some unitary U. They are weakly equivalent if


 * $$u \tau_1 (\mathcal{O}_n) u^* = \tau_2$$

for some unitary u in $$\mathcal{Q(H)}$$.