In mathematics, the Cuntz algebra
Contents
Every simple infinite C*-algebra contains, for any given n, a subalgebra that has
Definition and basic properties
Let n ≥ 2 and H be a separable Hilbert space. Consider the C*-algebra
of isometries acting on H satisfying
and
Theorem. The concrete C*-algebra
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
The *-subalgebra
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.
Any simple infinite C*-algebra contains a subalgebra that has
The UHF algebra
is isomorphic to
Classification
The Cuntz algebras are pairwise non-isomorphic, i.e.
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.