Cuntz algebra

Definitions

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

::\sum_{i=1}^n s_i s_i^* = 1.

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

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

Properties

Classification

The Cuntz algebras are pairwise non-isomorphic, i.e., \mathcal{O}_n and \mathcal{O}_m are non-isomorphic for n\neq m. The K0 group of \mathcal{O}_n is \mathbb{Z}/(n-1)\mathbb{Z}, the cyclic group of order n-1. Since K_0 is a functor, \mathcal{O}_n and \mathcal{O}_m are non-isomorphic.

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

Theorem. The concrete C*-algebra \mathcal{A} is isomorphic to the universal C*-algebra \mathcal{L} generated by n generators s_1,\dots,s_n subject to relations s_i^s_i=1 for all i and \textstyle\sum s_is_i^=1.

The proof of the theorem hinges on the following fact: any C*-algebra generated by n isometries s_1,\dots,s_n with orthogonal ranges contains a copy of the UHF algebra \mathcal{F} type n^\infty. Namely, \mathcal{F} is spanned by words of the form

:s_{i_1}\cdots s_{i_k}s_{j_1}^* \cdots s_{j_k}^*,\quad k \geq 0.

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

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

:P \mathcal{F} P = \mathcal{F'}

is isomorphic to \mathcal{F}. The -endomorphism \phi that maps \mathcal{F} onto \mathcal{F}' is implemented by the isometry s_1, i.e., \phi(\cdot)=s_1(\cdot)s_1^. ;\mathcal{O}_n is in fact the crossed product of \mathcal{F} with the endomorphism \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 a\in A, where a:\rho\to\sigma if a\rho(b)=\sigma(b)a for every b\in A. A unital *-endomorphism \rho:A\to A is the direct sum of endomorphisms \sigma_1, \sigma_2, ..., \sigma_n if there are isometries {S_k}_{k=1}^n satisfying the \mathcal{O}_n relations and

:\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 S_k:\sigma_k\to \rho, and the projection morphisms are 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.

References

References

1. Cuntz, Joachim. (1977). "Simple $C^*$-algebras generated by isometries". Communications in Mathematical Physics.
2. "Analysis and Probability: Wavelets, Signals, Fractals". [[Springer-Verlag]].