Cuntz algebra

In mathematics, the Cuntz algebra ${\displaystyle {\mathcal {O}}_{n}}$, named after Joachim Cuntz, is the universal C*-algebra generated by ${\displaystyle n}$ isometries of an infinite-dimensional Hilbert space ${\displaystyle {\mathcal {H}}}$ satisfying certain relations.^[1] These algebras were introduced as the first concrete examples of a separable infinite simple C*-algebra, meaning that as a Hilbert space, ${\displaystyle {\mathcal {O}}_{n}}$ is isometric to the sequence space ${\displaystyle l^{2}(\mathbb {N} )}$, and it has no non-trivial closed ideals.

These algebras are fundamental to the study of simple infinite C*-algebras since any such algebra contains, for any given ${\displaystyle n}$, a subalgebra that has ${\displaystyle {\mathcal {O}}_{n}}$ as quotient.

Definitions

Let ${\displaystyle n\geq 2}$ and ${\displaystyle {\mathcal {H}}}$ be a separable Hilbert space. Consider the C*-algebra ${\displaystyle {\mathcal {A}}}$ generated by a set ${\displaystyle \{s_{i}\}_{i=1}^{n}}$ of isometries (i.e., ${\displaystyle s_{i}^{*}s_{i}=1}$) acting on ${\displaystyle {\mathcal {H}}}$ satisfying

${\displaystyle \sum _{i=1}^{n}s_{i}s_{i}^{*}=1.}$

This universal C*-algebra is called the Cuntz algebra, denoted by ${\displaystyle {\mathcal {O}}_{n}}$.

A simple C*-algebra is said to be purely infinite if every hereditary C*-subalgebra of it is infinite. ${\displaystyle {\mathcal {O}}_{n}}$ is a separable, simple, purely infinite C*-algebra. Any simple infinite C*-algebra contains a subalgebra that has ${\displaystyle {\mathcal {O}}_{n}}$ as a quotient.

Properties

Classification

The Cuntz algebras are pairwise non-isomorphic, i.e., ${\displaystyle {\mathcal {O}}_{n}}$ and ${\displaystyle {\mathcal {O}}_{m}}$ are non-isomorphic for ${\displaystyle n\neq m}$. The K₀ group of ${\displaystyle {\mathcal {O}}_{n}}$ is ${\displaystyle \mathbb {Z} /(n-1)\mathbb {Z} }$, the cyclic group of order ${\displaystyle n-1}$. Since ${\displaystyle K_{0}}$ 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 ${\displaystyle n}$ generators ${\displaystyle s_{1},\dots ,s_{n}}$ subject to relations ${\displaystyle s_{i}^{*}s_{i}=1}$ for all ${\displaystyle i}$ and ${\displaystyle \textstyle \sum s_{i}s_{i}^{*}=1}$.

The proof of the theorem hinges on the following fact: any C*-algebra generated by ${\displaystyle n}$ isometries ${\displaystyle s_{1},\dots ,s_{n}}$ with orthogonal ranges contains a copy of the UHF algebra ${\displaystyle {\mathcal {F}}}$ type ${\displaystyle n^{\infty }}$. Namely, ${\displaystyle {\mathcal {F}}}$ is spanned by words of the form

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

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 ${\displaystyle M_{n}}$ 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 ${\displaystyle M_{n^{k}}}$ stage of the direct system, one has a rank ${\displaystyle n^{k-1}}$ projection. In the direct limit, this gives a projection ${\displaystyle P}$ in ${\displaystyle {\mathcal {F}}}$. The corner

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

is isomorphic to ${\displaystyle {\mathcal {F}}}$. The *-endomorphism ${\displaystyle \phi }$ that maps ${\displaystyle {\mathcal {F}}}$ onto ${\displaystyle {\mathcal {F}}'}$ is implemented by the isometry ${\displaystyle s_{1}}$, i.e., ${\displaystyle \phi (\cdot )=s_{1}(\cdot )s_{1}^{*}}$. ${\displaystyle \;{\mathcal {O}}_{n}}$ is in fact the crossed product of ${\displaystyle {\mathcal {F}}}$ with the endomorphism ${\displaystyle \phi }$.

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.

Applied mathematics

In signal processing, a subband filter with exact reconstruction give rise to representations of a Cuntz algebra. The same filter also comes from the multiresolution analysis construction in wavelet theory.^[2]

See also

• Approximately finite-dimensional C*-algebra
• Hilbert C*-module