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| In mathematics, the '''Cuntz algebra''' <math>\mathcal{O}_n </math> (after [[Joachim Cuntz]]) is the [[universal C*-algebra]] generated by ''n'' isometries satisfying certain relations. It is the first concrete example of a [[Separable space|separable]] infinite simple C*-algebra.
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| Every simple infinite C*-algebra contains, for any given ''n'', a subalgebra that has <math>\mathcal{O}_n </math> as quotient.
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| == Definition and basic properties ==
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| Let ''n'' ≥ 2 and ''H'' be a [[separable space|separable]] [[Hilbert space]]. Consider the [[C*-algebra]] <math>\mathcal{A}</math> generated by a set
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| ::<math> \{ S_i \}_{i=1}^n </math>
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| of isometries acting on ''H'' satisfying
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| ::<math> \sum_{i=1}^n S_i S_i^* = I.</math>
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| Note that, in particular, the ''S''<sub>''i''</sub>'s have the property that
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| ::<math> S_i^* S_j = \delta_{ij} I.</math>
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| '''Theorem.''' The concrete C*-algebra <math>\mathcal{A}</math> is isomorphic to the universal C*-algebra <math>\mathcal{L}</math> generated by ''n'' generators ''s''<sub>1</sub>... ''s''<sub>''n''</sub> subject to relations ''s<sub>i</sub>*s<sub>i</sub>'' = 1 for all ''i'' and ∑ ''s<sub>i</sub>s<sub>i</sub>''* = 1.
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| The proof of the theorem hinges on the following fact: any C*-algebra generated by ''n'' isometries ''s''<sub>1</sub>... ''s''<sub>''n''</sub> with orthogonal ranges contains a copy of the [[UHF algebra]] <math>\mathcal{F}</math> type ''n''<sup>∞</sup>. Namely <math>\mathcal{F}</math> is spanned by words of the form
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| :<math>s_{i_1}\cdots s_{i_k}s_{j_1}^* \cdots s_{j_k}^*, k \geq 0.</math> | |
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| The *-subalgebra <math>\mathcal{F}</math>, being [[Approximately finite dimensional C*-algebra|approximately finite dimensional]], has a unique C*-norm.
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| The subalgebra <math>\mathcal{F}</math> plays role of the space of ''[[Fourier coefficient]]s'' 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 <math>\mathcal{L}</math> to <math>\mathcal{A}</math> is injective, which proves the theorem.
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| This universal C*-algebra is called the '''Cuntz algebra''', denoted by <math>\mathcal{O}_n </math>.
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| A C*-algebra is said to be '''purely infinite''' if every [[hereditary C*-subalgebra]] of it is infinite. <math>\mathcal{O}_n </math> is a separable, [[simple algebra|simple]], purely infinite C*-algebra.
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| Any simple infinite C*-algebra contains a subalgebra that has <math>\mathcal{O}_n</math> as a quotient.
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| The UHF algebra <math>\mathcal{F}</math> has a non-unital subalgebra <math>\mathcal{F}'</math> that is canonically isomorphic to <math>\mathcal{F}</math> itself: In the M''<sub>n</sub>'' stage of the direct system defining <math>\mathcal{F}</math>, consider the rank-1 projection ''e''<sub>11</sub>, the matrix that is 1 in the upper left corner and zero elsewhere. Propagate this projection through the direct system. At the M''<sub>n<sup>k</sup></sub>'' stage of the direct system, one has a rank ''n''<sup>''k'' - 1</sup> projection. In the [[direct limit]], this gives a projection ''P'' in <math>\mathcal{F}</math>. The corner
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| :<math>P \mathcal{F} P = \mathcal{F'} </math> | |
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| is isomorphic to <math>\mathcal{F}</math>. The *-endomorphism Φ that maps <math>\mathcal{F}</math> onto <math>\mathcal{F}'</math> is implemented by the isometry ''s''<sub>1</sub>, i.e. Φ(·) = ''s''<sub>1</sub>(·)''s''<sub>1</sub>*. <math>\;\mathcal{O}_n </math>is in fact the [[crossed product]] of <math>\mathcal{F}</math> with the endomorphism Φ.
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| == Classification ==
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| The Cuntz algebras are pairwise non-isomorphic, i.e. <math>\mathcal{O}_n </math> and <math>\mathcal{O}_m </math> are non-isomorphic for ''n'' ≠ ''m''. The [[operator K-theory|''K''<sub>0</sub>]] group of <math>\mathcal{O}_n</math> is '''Z'''<sub>''n'' − 1</sub>, the abelian cyclic group of order ''n'' − 1. Since ''K''<sub>0</sub> is a (functorial) invariant, <math>\mathcal{O}_n </math> and <math>\mathcal{O}_m </math> are non-isomorphic.
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| == Generalisations ==
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| Cuntz algebras have been generalised in many ways. Notable amongst which are the [[Cuntz–Krieger]] algebras, [[graph C*-algebra]]s and [[k-graph C*-algebra]]s.
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| == Applied mathematics ==
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| In [[signal processing]], [[subband coding|subband filter]] with exact reconstruction give rise to representations of Cuntz algebra. The same filters also comes from the [[multiresolution analysis]] construction in [[wavelet]] theory.
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| ==References==
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| * {{citation |first=J.|last=Cuntz |title=Simple C*-algebras generated by isometries |journal=[[Comm. Math. Phys.]] | volume=57 |pages=173–185 | year=1977 }}
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| * {{citation|title=Analysis and probability: wavelets, signals, fractals |volume=234 |series=[[Graduate texts in mathematics]] |first1=Palle E. T. |last1=Jørgensen |first2=Brian |last2=Treadway |publisher=[[Springer-Verlag]] isbn=0-387-29519-4 }}
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| [[Category:C*-algebras]]
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