Cuntz algebra

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

::$\backslash sum\_\{i=1\}^n\; s\_i\; s\_i^*\; =\; 1.$

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

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

The Cuntz algebras are pairwise non-isomorphic, i.e., $\backslash mathcal\{O\}\_n$ and $\backslash mathcal\{O\}\_m$ are non-isomorphic for $n\backslash neq\; m$. The K₀ group of $\backslash mathcal\{O\}\_n$ is $\backslash mathbb\{Z\}/(n-1)\backslash mathbb\{Z\}$, the cyclic group of order $n-1$. Since $K\_0$ is a functor, $\backslash mathcal\{O\}\_n$ and $\backslash mathcal\{O\}\_m$ are non-isomorphic.

Theorem. The concrete C*-algebra $\backslash mathcal\{A\}$ is isomorphic to the universal C*-algebra $\backslash mathcal\{L\}$ generated by $n$ generators $s\_1,\backslash dots,s\_n$ subject to relations $s\_i^*s\_i=1$ for all $i$ and $\backslash textstyle\backslash sum\; s\_is\_i^*=1$.

The proof of the theorem hinges on the following fact: any C*-algebra generated by $n$ isometries $s\_1,\backslash dots,s\_n$ with orthogonal ranges contains a copy of the UHF algebra $\backslash mathcal\{F\}$ type $n^\backslash infty$. Namely, $\backslash mathcal\{F\}$ is spanned by words of the form

:$s\_\{i\_1\}\backslash cdots\; s\_\{i\_k\}s\_\{j\_1\}^*\; \backslash cdots\; s\_\{j\_k\}^*,\backslash quad\; k\; \backslash geq\; 0.$

The *-subalgebra $\backslash mathcal\{F\}$, being approximately finite-dimensional, has a unique C*-norm. The subalgebra $\backslash mathcal\{F\}$ 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 $\backslash mathcal\{L\}$ to $\backslash mathcal\{A\}$ is injective, which proves the theorem.

The UHF algebra $\backslash mathcal\{F\}$ has a non-unital subalgebra $\backslash mathcal\{F\}\text{'}$ that is canonically isomorphic to $\backslash mathcal\{F\}$ itself: in the $M\_n$ stage of the direct system defining $\backslash mathcal\{F\}$, 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 $\backslash mathcal\{F\}$. The corner

:$P\; \backslash mathcal\{F\}\; P\; =\; \backslash mathcal\{F\text{'}\}$

is isomorphic to $\backslash mathcal\{F\}$. The *-endomorphism $\backslash phi$ that maps $\backslash mathcal\{F\}$ onto $\backslash mathcal\{F\}\text{'}$ is implemented by the isometry $s\_1$, i.e., $\backslash phi(\backslash cdot)=s\_1(\backslash cdot)s\_1^*$. $\backslash ;\backslash mathcal\{O\}\_n$ is in fact the crossed product of $\backslash mathcal\{F\}$ with the endomorphism $\backslash phi$.

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 $a\backslash in\; A$, where $a:\backslash rho\backslash to\backslash sigma$ if $a\backslash rho(b)=\backslash sigma(b)a$ for every $b\backslash in\; A$. A unital *-endomorphism $\backslash rho:A\backslash to\; A$ is the direct sum of endomorphisms $\backslash sigma\_1,\; \backslash sigma\_2,\; ...,\; \backslash sigma\_n$ if there are isometries $\backslash \{S\_k\backslash \}\_\{k=1\}^n$ satisfying the $\backslash mathcal\{O\}\_n$ relations and

:$\backslash rho(x)\; =\; \backslash sum\_\{k=1\}^n\; S\_k\backslash sigma\_k(x)S\_k^*,\; \backslash forall\; x\backslash in\; A.$

In this direct sum, the inclusion morphisms are $S\_k:\backslash sigma\_k\backslash to\; \backslash rho$, and the projection morphisms are $S\_k^*:\backslash rho\backslash to\backslash sigma\_k$.

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

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.{{cite book|title=Analysis and Probability: Wavelets, Signals, Fractals|volume=234|series=Graduate Texts in Mathematics |first1=Palle E. T.|last1=Jørgensen|first2=Brian|last2=Treadway |publisher=Springer-Verlag|isbn=0-387-29519-4 }}

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

Category:C*-algebras