Cuntz algebra

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In mathematics, the Cuntz algebra , named after Joachim Cuntz, is the universal C*-algebra generated by n isometries satisfying certain relations.[1] It is the first concrete example of a separable infinite simple C*-algebra.

Every simple infinite C*-algebra contains, for any given n, a subalgebra that has as quotient.

Definition and basic properties[edit]

Let n ≥ 2 and H be a separable Hilbert space. Consider the C*-algebra generated by a set

of isometries (i.e. ) acting on H satisfying

Theorem. The concrete C*-algebra is isomorphic to the universal C*-algebra generated by n generators s1... sn subject to relations si*si = 1 for all i and ∑ sisi* = 1.

The proof of the theorem hinges on the following fact: any C*-algebra generated by n isometries s1... sn with orthogonal ranges contains a copy of the UHF algebra type n. Namely is spanned by words of the form

The *-subalgebra , being approximately finite-dimensional, has a unique C*-norm. The subalgebra 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 to is injective, which proves the theorem.

This universal C*-algebra is called the Cuntz algebra, denoted by .

A simple C*-algebra is said to be purely infinite if every hereditary C*-subalgebra of it is infinite. is a separable, simple, purely infinite C*-algebra.

Any simple infinite C*-algebra contains a subalgebra that has as a quotient.

The UHF algebra has a non-unital subalgebra that is canonically isomorphic to itself: In the Mn stage of the direct system defining , 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 Mnk stage of the direct system, one has a rank nk - 1 projection. In the direct limit, this gives a projection P in . The corner

is isomorphic to . The *-endomorphism Φ that maps onto is implemented by the isometry s1, i.e. Φ(·) = s1(·)s1*. is in fact the crossed product of with the endomorphism Φ.


The Cuntz algebras are pairwise non-isomorphic, i.e. and are non-isomorphic for nm. The K0 group of is Zn − 1, the cyclic group of order n − 1. Since K0 is a functor, and are non-isomorphic.


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[edit]

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.[2]


  1. ^ Cuntz, Joachim (1977). "Simple C*-algebras generated by isometries". Comm. Math. Phys. 57. pp. 173–185. Bibcode:1977CMaPh..57..173C. doi:10.1007/bf01625776. Zbl 0399.46045.
  2. ^ Jørgensen, Palle E. T.; Treadway, Brian. Analysis and Probability: Wavelets, Signals, Fractals. Graduate Texts in Mathematics. 234. Springer-Verlag. ISBN 0-387-29519-4.