Cuntz algebra

In functional analysis , the so-called Cuntz algebras (according to Joachim Cuntz ) are a special class of C * algebras that are generated from n pairwise orthogonal isometries on a separable Hilbert space . ${\displaystyle {\mathcal {O}}_{n}}$

definition

Let be a separable infinite-dimensional Hilbert space. For a natural number let isometries on H, i.e. H. it applies to . They should also have the property ${\displaystyle H}$ ${\displaystyle n\geq 2}$ ${\displaystyle S_{1},\dots ,S_{n}\in {\mathcal {L}}(H)}$ ${\displaystyle S_{i}^{*}S_{i}=1}$ ${\displaystyle 1\leq i\leq n}$

${\displaystyle \sum _{i=1}^{n}S_{i}S_{i}^{*}=1}$

meet, the image projectors are orthogonal in pairs. In this case , a sequence of isometrics with the property is required ${\displaystyle n=\infty }$ ${\displaystyle S_{1},S_{2},S_{3},\dots \quad }$

${\displaystyle \sum _{i=1}^{k}S_{i}S_{i}^{*}\leq 1\quad }$ for all ${\displaystyle k\in \mathbb {N} .}$

One defines now

${\displaystyle {\mathcal {O}}_{n}=C^{*}(S_{1},\dots ,S_{n})}$

than the C * subalgebra generated by in . In order to maintain a uniform notation, this notation is retained in the case . ${\displaystyle S_{1},\dots ,S_{n}}$ ${\displaystyle {\mathcal {L}}(H)}$ ${\displaystyle n=\infty }$

properties

Cuntz algebra has a number of remarkable properties, it is an example of a separable , unital and simple C * algebra. ${\displaystyle {\mathcal {O}}_{n}}$

Uniqueness

If there are other isometries , it follows ${\displaystyle {\tilde {S}}_{1},\dots ,{\tilde {S}}_{n}\in {\mathcal {L}}(H)}$ ${\displaystyle \sum _{i=1}^{n}{\tilde {S}}_{i}{\tilde {S}}_{i}^{*}=1}$

${\displaystyle C^{*}(S_{1},\dots ,S_{n})\simeq C^{*}({\tilde {S}}_{1},\dots ,{\tilde {S}}_{n}).}$

The isomorphism class does not depend on the choice of the producer. This justifies the spelling , which does not refer to the producer . ${\displaystyle {\mathcal {O}}_{n}}$ ${\displaystyle S_{1},\dots ,S_{n}}$

A special role in the investigation of playing the C * -Unteralgebra , the elements of the form with is generated. One can show that this is isomorphic to the UHF algebra of the supernatural number . If you define a producer, for example and write , there are images so that each can be represented as ${\displaystyle {\mathcal {O}}_{n}}$ ${\displaystyle {\mathcal {F}}^{n}}$ ${\displaystyle S_{i_{1}}S_{i_{2}}\dots S_{i_{k}}S_{j_{k}}^{*}S_{j_{k-1}}^{*}\dots S_{j_{1}}^{*}}$ ${\displaystyle k\in \mathbb {N} ,1\leq i_{l},j_{l}\leq n}$ ${\displaystyle n^{\infty }}$ ${\displaystyle V=S_{1}}$ ${\displaystyle V^{-1}=S_{1}^{*}}$ ${\displaystyle F_{i}:{\mathcal {O}}_{n}\to {\mathcal {F}}^{n}}$ ${\displaystyle A\in {\mathcal {O}}_{n}}$

${\displaystyle A=\sum _{i=-\infty }^{-1}V^{i}F_{i}(A)+F_{0}(A)+\sum _{i=1}^{\infty }F_{i}(A)V^{i}}$.

An important step in the proof of the above uniqueness property is to interpret it analogously to Fourier coefficients in a Laurent series . This makes it possible to show that on the purely algebraic product of only one C * norm can exist, which shows the claim. ${\displaystyle F_{i}(A)}$ ${\displaystyle S_{1},\dots ,S_{n},S_{1}^{*},\dots ,S_{n}^{*}}$

simplicity

A C * -algebra is called simple if it has no non-trivial closed two-sided ideals . is simple even in the algebraic sense. ${\displaystyle {\mathcal {O}}_{n}}$

Sentence: Be . Then exist with . ${\displaystyle 0\neq X\in {\mathcal {O}}_{n}}$ ${\displaystyle A,B\in {\mathcal {O}}_{n}}$ ${\displaystyle AXB=1}$

In addition, Cuntz algebras are related to simple, unital, infinite C * algebras in the following sense.

Theorem: Let be a simple, infinite, unital C * -algebra. Then there exists a C * subalgebra of that is isomorphic to . For finite there is a C * -subalgebra that contains an ideal such that . ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {O}}_{\infty }}$ ${\displaystyle n\geq 2}$ ${\displaystyle {\mathcal {B}}\subset {\mathcal {A}}}$ ${\displaystyle {\mathcal {J}}}$ ${\displaystyle {\mathcal {O}}_{n}\simeq {\mathcal {B}}/{\mathcal {J}}}$

classification

Let it be like above. If you define , isometrics are also included and it is obvious . ${\displaystyle {\mathcal {O}}_{2}=C^{*}(S_{1},S_{2})}$ ${\displaystyle {\hat {S}}_{1}=S_{1}^{2},{\hat {S}}_{2}=S_{1}S_{2},{\hat {S}}_{3}=S_{2}}$ ${\displaystyle {\hat {S}}_{1},{\hat {S}}_{2},{\hat {S}}_{3}}$ ${\displaystyle {\hat {S}}_{1}{\hat {S}}_{1}^{*}+{\hat {S}}_{2}{\hat {S}}_{2}^{*}+{\hat {S}}_{3}{\hat {S}}_{3}^{*}=1}$ ${\displaystyle C^{*}({\hat {S}}_{1},{\hat {S}}_{2},{\hat {S}}_{3})\subset C^{*}(S_{1},S_{2})}$

In this way the inclusions are obtained

${\displaystyle {\mathcal {O}}_{\infty }\subset {\mathcal {O}}_{n}\subset {\mathcal {O}}_{2}}$.

With K-theoretical methods one shows that and are not isomorphic if . If is finite, then the group is calculated from to . In the event it arises . Since the group is an isomorphism invariant, the claim immediately follows. ${\displaystyle {\mathcal {O}}_{n}}$ ${\displaystyle {\mathcal {O}}_{m}}$ ${\displaystyle n\neq m}$ ${\displaystyle n}$ ${\displaystyle K_{0}}$ ${\displaystyle {\mathcal {O}}_{n}}$ ${\displaystyle \mathbb {Z} _{n-1}}$ ${\displaystyle n=\infty }$ ${\displaystyle K_{0}=\mathbb {Z} }$ ${\displaystyle K_{0}}$

Representation as a cross product

There is an * - automorphism such that . Since as a UHF algebra is nuclear , it follows from this representation as a cross product that it is also nuclear. ${\displaystyle {\mathcal {F}}^{n}}$ ${\displaystyle \Phi }$ ${\displaystyle {\mathcal {O}}_{n}\simeq {\mathcal {F}}^{n}\rtimes _{\Phi }\mathbb {Z} }$ ${\displaystyle {\mathcal {F}}^{n}}$ ${\displaystyle {\mathcal {O}}_{n}}$

literature

• Joachim Cuntz : Simple C * -algebras generated by isometries. (Pdf) In: Comm. Math. Phys. 57. 1977, pp. 173-185 , accessed April 17, 2012 (English). 
• KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-821-80599-1