Cuntz algebra
In mathematics, the Cuntz algebra formula_1, named after Joachim Cuntz, is the universal C*-algebra generated by formula_2 isometries of an infinite-dimensional Hilbert space formula_3 satisfying certain relations. These algebras were introduced as the first concrete examples of a separable infinite simple C*-algebra, meaning as a Hilbert space, formula_1 is isometric to the sequence space and it has no nontrivial closed ideals. These algebras are fundamental to the study of simple infinite C*-algebras since any such algebra contains, for any given "n", a subalgebra that has formula_1 as quotient. Definitions. Let "n" ≥ 2 and formula_3 be a  separable Hilbert space. Consider the C*-algebra formula_8 generated by a set of isometries (i.e. formula_10) acting on formula_3 satisfying This universal C*-algebra is called the Cuntz algebra, denoted by formula_1. A simple C*-algebra is said to be purely infinite if every hereditary C*-subalgebra of it is infinite. formula_1 is a separable, simple, purely infinite C*-algebra. Any simple infinite C*-algebra contains a subalgebra that has formula_15 as a quotient. Properties. Classification. The Cuntz algebras are pairwise non-isomorphic, i.e. formula_1 and formula_17 are non-isomorphic for "n" ≠ "m". The "K" group of formula_15 is formula_19, the cyclic group of order  n − 1. Since "K" is a functor, formula_1 and formula_17 are non-isomorphic. Relation between concrete C*-algebras and the universal C*-algebra. Theorem. The concrete C*-algebra formula_8 is isomorphic to the universal C*-algebra formula_23 generated by "n" generators "s"... "s" subject to relations "s*s" = 1 for all "i" and ∑ "ss"* = 1. The proof of the theorem hinges on the following fact: any C*-algebra generated by "n" isometries "s"... "s" with orthogonal ranges contains a copy of the UHF algebra formula_24 type "n". Namely formula_24 is spanned by words of the form The *-subalgebra formula_24, being approximately finite -dimensional, has a unique C*-norm. The subalgebra formula_24 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 formula_23 to formula_8 is injective, which proves the theorem. The UHF algebra formula_24 has a non-unital subalgebra formula_32 that is canonically isomorphic to formula_24 itself: In the M"" stage of the direct system defining formula_24, consider the rank-1 projection "e", the matrix that  is 1 in the upper left corner and zero elsewhere. Propagate this projection through the direct system. At the M"" stage of the direct system, one has a rank "n" projection. In the direct limit, this gives a projection "P" in formula_24. The corner is isomorphic to formula_24. The *-endomorphism Φ that maps formula_24 onto formula_32 is implemented by the isometry "s", i.e. Φ(·) = "s"(·)"s"*. formula_40is in fact the crossed product of formula_24 with the endomorphism Φ. 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.