Cuntz algebra

In mathematics, the Cuntz algebra [math]\displaystyle{ \mathcal{O}_n }[/math], named after Joachim Cuntz, is the universal C*-algebra generated by n isometries satisfying certain relations. The Cuntz algebras provided the first concrete examples of a separable infinite simple C*-algebra. Every simple infinite C*-algebra contains, for any given n, a subalgebra that has [math]\displaystyle{ \mathcal{O}_n }[/math] as quotient.

Definition and basic properties

Let n ≥ 2 and H be a separable Hilbert space. Consider the C*-algebra [math]\displaystyle{ \mathcal{A} }[/math] generated by a set

[math]\displaystyle{ \{ S_i \}_{i=1}^n }[/math]

of isometries (i.e. [math]\displaystyle{ S_i^*S_i=1 }[/math]) acting on H satisfying

[math]\displaystyle{ \sum_{i=1}^n S_i S_i^* = I. }[/math]

Theorem. The concrete C*-algebra [math]\displaystyle{ \mathcal{A} }[/math] is isomorphic to the universal C*-algebra [math]\displaystyle{ \mathcal{L} }[/math] 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 [math]\displaystyle{ \mathcal{F} }[/math] type n^∞. Namely [math]\displaystyle{ \mathcal{F} }[/math] is spanned by words of the form

[math]\displaystyle{ s_{i_1}\cdots s_{i_k}s_{j_1}^* \cdots s_{j_k}^*, k \geq 0. }[/math]

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

This universal C*-algebra is called the Cuntz algebra, denoted by [math]\displaystyle{ \mathcal{O}_n }[/math].

A simple C*-algebra is said to be purely infinite if every hereditary C*-subalgebra of it is infinite. [math]\displaystyle{ \mathcal{O}_n }[/math] is a separable, simple, purely infinite C*-algebra.

Any simple infinite C*-algebra contains a subalgebra that has [math]\displaystyle{ \mathcal{O}_n }[/math] as a quotient.

The UHF algebra [math]\displaystyle{ \mathcal{F} }[/math] has a non-unital subalgebra [math]\displaystyle{ \mathcal{F}' }[/math] that is canonically isomorphic to [math]\displaystyle{ \mathcal{F} }[/math] itself: In the M_n stage of the direct system defining [math]\displaystyle{ \mathcal{F} }[/math], 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 [math]\displaystyle{ \mathcal{F} }[/math]. The corner

[math]\displaystyle{ P \mathcal{F} P = \mathcal{F'} }[/math]

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

Classification

The Cuntz algebras are pairwise non-isomorphic, i.e. [math]\displaystyle{ \mathcal{O}_n }[/math] and [math]\displaystyle{ \mathcal{O}_m }[/math] are non-isomorphic for n ≠ m. The K₀ group of [math]\displaystyle{ \mathcal{O}_n }[/math] is Z_{n − 1}, the cyclic group of order n − 1. Since K₀ is a functor, [math]\displaystyle{ \mathcal{O}_n }[/math] and [math]\displaystyle{ \mathcal{O}_m }[/math] are non-isomorphic.

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, subband filter with exact reconstruction give rise to representations of Cuntz algebra. The same filter also comes from the multiresolution analysis construction in wavelet theory.