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 ≠ m. The K₀ group of ${\mathcal {O}}_{n}$ is Z_{n - 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 π be the quotient map onto the Calkin algebra. Two extensions τ ₁ and τ₂ 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 ${\mathcal {Q(H)}}$.