In mathematics, the Cuntz algebra
, named after Joachim Cuntz, is the universal C*-algebra generated by
isometries of an infinite-dimensional Hilbert space
[1] These algebras were introduced as the first concrete examples of a separable infinite simple C*-algebra, meaning as a Hilbert space,
is isometric to the sequence space and it has no nontrivial closed ideals.
be a separable Hilbert space.
generated by a set of isometries (i.e.
satisfying This universal C*-algebra is called the Cuntz algebra, denoted by
is a separable, simple, purely infinite C*-algebra.
Any simple infinite C*-algebra contains a subalgebra that has
The Cuntz algebras are pairwise non-isomorphic, i.e.
, the cyclic group of order n − 1.
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
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.
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 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
is the direct sum of endomorphisms
relations and In this direct sum, the inclusion morphisms are
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.