User:Mct mht/Cuntz
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Classification
[edit]The Cuntz algebras are pairwise non-isomorphic, i.e. and are non-isomorphic to n ≠ m. The K0 group of is Zn - 1, the abelian cyclic group of order n. Since K0 is a (functorial) invariant, and are non-isomorphic.
By extensions
[edit]An extension of the compact operators by is a short exact sequence of C*-algebras
Descending to the Calkin algebra , extensions are in bijective correspondence with *-monomorphisms
Let π be the quotient map onto the Calkin algebra. Two extensions τ 1 and τ2 are said to be equivalent of
for some unitary U. They are weakly equivalent if
for some unitary u in .