Classification

The Cuntz algebras are pairwise non-isomorphic, i.e. ${\displaystyle {\mathcal {O}}_{n}}$ and ${\displaystyle {\mathcal {O}}_{m}}$ are non-isomorphic for n ≠ m. The K₀ group of ${\displaystyle {\mathcal {O}}_{n}}$ is ${\displaystyle \mathbb {Z} /(n-1)\mathbb {Z} }$, the cyclic group of order n − 1. Since K₀ is a functor, ${\displaystyle {\mathcal {O}}_{n}}$ and ${\displaystyle {\mathcal {O}}_{m}}$ are non-isomorphic.

Relation between concrete C*-algebras and the universal C*-algebra

Theorem. The concrete C*-algebra ${\displaystyle {\mathcal {A}}}$ is isomorphic to the universal C*-algebra ${\displaystyle {\mathcal {L}}}$ generated by n generators s₁... s_n subject to relations s_i*s_i = 1 for all i and ∑ s_is_i* = 1.

The proof of the theorem hinges on the following fact: any C*-algebra generated by n isometries s₁... s_n with orthogonal ranges contains a copy of the UHF algebra ${\displaystyle {\mathcal {F}}}$ type n^∞. Namely ${\displaystyle {\mathcal {F}}}$ is spanned by words of the form

${\displaystyle s_{i_{1}}\cdots s_{i_{k}}s_{j_{1}}^{*}\cdots s_{j_{k}}^{*},k\geq 0.}$

The *-subalgebra ${\displaystyle {\mathcal {F}}}$, being approximately finite-dimensional, has a unique C*-norm. The subalgebra ${\displaystyle {\mathcal {F}}}$ plays role of the space of Fourier coefficients for elements of the algebra. A key technical lemma, due to Cuntz, is that an element in the algebra is zero if and only if all its Fourier coefficients vanish. Using this, one can show that the quotient map from ${\displaystyle {\mathcal {L}}}$ to ${\displaystyle {\mathcal {A}}}$ is injective, which proves the theorem.

The UHF algebra ${\displaystyle {\mathcal {F}}}$ has a non-unital subalgebra ${\displaystyle {\mathcal {F}}'}$ that is canonically isomorphic to ${\displaystyle {\mathcal {F}}}$ itself: In the M_n stage of the direct system defining ${\displaystyle {\mathcal {F}}}$, consider the rank-1 projection e₁₁, the matrix that is 1 in the upper left corner and zero elsewhere. Propagate this projection through the direct system. At the M_n^k stage of the direct system, one has a rank n^{k − 1} projection. In the direct limit, this gives a projection P in ${\displaystyle {\mathcal {F}}}$. The corner

${\displaystyle P{\mathcal {F}}P={\mathcal {F'}}}$

is isomorphic to ${\displaystyle {\mathcal {F}}}$. The *-endomorphism Φ that maps ${\displaystyle {\mathcal {F}}}$ onto ${\displaystyle {\mathcal {F}}'}$ is implemented by the isometry s₁, i.e. Φ(·) = s₁(·)s₁*. ${\displaystyle \;{\mathcal {O}}_{n}}$is in fact the crossed product of ${\displaystyle {\mathcal {F}}}$ with the endomorphism Φ.

Cuntz algebras to represent direct sums

The relations defining the Cuntz algebras align with the definition of the biproduct for preadditive categories. This similarity is made precise in the C*-category of unital *-endomorphisms over C*-algebras. The objects of this category are unital *-endomorphisms, and morphisms are the elements ${\displaystyle a\in A}$, where ${\displaystyle a:\rho \to \sigma }$ if ${\displaystyle a\rho (b)=\sigma (b)a}$ for every ${\displaystyle b\in A}$. A unital *-endomorphism ${\displaystyle \rho :A\to A}$ is the direct sum of endomorphisms ${\displaystyle \sigma _{1},\sigma _{2},...,\sigma _{n}}$ if there are isometries ${\displaystyle \{S_{k}\}_{k=1}^{n}}$ satisfying the ${\displaystyle {\mathcal {O}}_{n}}$ relations and

${\displaystyle \rho (x)=\sum _{k=1}^{n}S_{k}\sigma _{k}(x)S_{k}^{*},\forall x\in A.}$

In this direct sum, the inclusion morphisms are ${\displaystyle S_{k}:\sigma _{k}\to \rho }$, and the projection morphisms are ${\displaystyle S_{k}^{*}:\rho \to \sigma _{k}}$.

Generalisations

Cuntz algebras have been generalised in many ways. Notable amongst which are the Cuntz–Krieger algebras, graph C*-algebras and k-graph C*-algebras.