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 .
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
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
![n \ geq 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174)
![S_1, \ dots, S_n \ in \ mathcal {L} (H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c246a6725065473687209daec93dea934086804f)
![S_i ^ * S_i = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/541038923d3f98e94e7c56a10a4a9f281187a691)
![1 \ leq i \ leq n](https://wikimedia.org/api/rest_v1/media/math/render/svg/abbe58b9b83f8b6ec0da570e2249323a8930ef1e)
![\ sum_ {i = 1} ^ n S_iS_i ^ * = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac09af26cef1491c2aa5088f197cccb3fb2d4d99)
meet, the image projectors are orthogonal in pairs. In this case , a sequence of isometrics with the property is required
![n = \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/40c93fd8fa11a9b05557492cd993bb04ac63c36c)
![S_1, S_2, S_3, \ dots \ quad](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b76258e5b7585876c732457937a2433cf5a7d61)
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for all
One defines now
![\ mathcal {O} _n = C ^ * (S_1, \ dots, S_n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee9e113c42c3e0d27ce515393e9dac08e3d96bd)
than the C * subalgebra generated by in . In order to maintain a uniform notation, this notation is retained in the case .
![S_1, \ dots, S_n](https://wikimedia.org/api/rest_v1/media/math/render/svg/b37ba787ea2525b5d0b9c387c1683a9af7ce1347)
![\ mathcal {L} (H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d15f755946661f116e406f7969f7f9161b93ecb)
![n = \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/40c93fd8fa11a9b05557492cd993bb04ac63c36c)
properties
Cuntz algebra has a number of remarkable properties, it is an example of a separable , unital and simple C * algebra.
![\ mathcal {O} _n](https://wikimedia.org/api/rest_v1/media/math/render/svg/23c1482aadf309ab3fe04327732c45c8fb8b30c7)
Uniqueness
If there are other isometries , it follows
![\ tilde {S} _1, \ dots, \ tilde {S} _n \ in \ mathcal {L} (H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/da6718a20e7a017eb200af75db8c1565ca5235a2)
![\ sum_ {i = 1} ^ n \ tilde {S} _i \ tilde {S} _i ^ * = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/768ddd317e1cc4c6f833856bac41124b37df7c39)
![C ^ * (S_1, \ dots, S_n) \ simeq C ^ * (\ tilde {S} _1, \ dots, \ tilde {S} _n).](https://wikimedia.org/api/rest_v1/media/math/render/svg/08a09efbd1447d1828778304ca419de5678fc17f)
The isomorphism class does not depend on the choice of the producer. This justifies the spelling , which does not refer to the producer .
![\ mathcal {O} _n](https://wikimedia.org/api/rest_v1/media/math/render/svg/23c1482aadf309ab3fe04327732c45c8fb8b30c7)
![S_1, \ dots, S_n](https://wikimedia.org/api/rest_v1/media/math/render/svg/b37ba787ea2525b5d0b9c387c1683a9af7ce1347)
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
![\ mathcal {O} _n](https://wikimedia.org/api/rest_v1/media/math/render/svg/23c1482aadf309ab3fe04327732c45c8fb8b30c7)
![\ mathcal {F} ^ n](https://wikimedia.org/api/rest_v1/media/math/render/svg/539e0804ce89bbc718c613646a0bd2a8c6a98fda)
![S_ {i_1} S_ {i_2} \ dots S_ {i_k} S_ {j_k} ^ * S_ {j_ {k-1}} ^ * \ dots S_ {j_1} ^ *](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5f0fad512abae332256e130cf44d7d9f7c411ce)
![k \ in \ mathbb {N}, 1 \ leq i_l, j_l \ leq n](https://wikimedia.org/api/rest_v1/media/math/render/svg/b733c74771d8139a6e5dc2f64fa6d1e7c6d65ac0)
![n ^ \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/458436027ee53bb51a944fc381668baf554a7c74)
![V = S_1](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ee7dd3ec649fac5bf85049f9b6a2e0a8077d65)
![V ^ {- 1} = S_1 ^ *](https://wikimedia.org/api/rest_v1/media/math/render/svg/276a625fc069a5e919e08cc11712463d5bff08b2)
![F_i: \ mathcal {O} _n \ to \ mathcal {F} ^ n](https://wikimedia.org/api/rest_v1/media/math/render/svg/357a7f3b3bc80819d09f0454860f312a0619710a)
![A \ in \ mathcal {O} _n](https://wikimedia.org/api/rest_v1/media/math/render/svg/21c9821f13e5798d4aef4bb6095e96792940c037)
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.
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.
![F_i (A)](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc48b635903d736111a21f514ac3b13671ad87b0)
![S_1, \ dots, S_n, S_1 ^ *, \ dots, S_n ^ *](https://wikimedia.org/api/rest_v1/media/math/render/svg/227a3b2be288870d099191053e4698e60597daf2)
simplicity
A C * -algebra is called simple if it has no non-trivial closed two-sided ideals . is simple even in the algebraic sense.
![\ mathcal {O} _n](https://wikimedia.org/api/rest_v1/media/math/render/svg/23c1482aadf309ab3fe04327732c45c8fb8b30c7)
Sentence: Be . Then exist with .
![0 \ neq X \ in \ mathcal {O} _n](https://wikimedia.org/api/rest_v1/media/math/render/svg/c62f7910d637e221fb707e4660c363eb79bf5b60)
![A, B \ in \ mathcal {O} _n](https://wikimedia.org/api/rest_v1/media/math/render/svg/006faa91182b8d6b794b8e4cf3f0289319d1d1af)
![AXB = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e8529d2e59a3ade9224f8ebe55fd0de8a78a3a)
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 .
![{\ mathcal {A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8)
![{\ mathcal {A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8)
![\ mathcal {O} _ \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed491206e2588a497dc8038dfc12b1a681aff7b)
![n \ geq 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174)
![\ mathcal {B} \ subset \ mathcal {A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/060c5f93ff243b7b03eaba14b36184b7ac7918a8)
![\ mathcal {J}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8fb0b896b1b2a45546779ecafc567f4f1688714)
![\ mathcal {O} _n \ simeq \ mathcal {B} / \ mathcal {J}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e323af252aa0b6986262d2f359c4ce40d4a284d1)
classification
Let it be like above. If you define , isometrics are also included and it is obvious .
![\ mathcal {O} _2 = C ^ * (S_1, S_2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/17f49e955214d327327b304be3ed31fd5abb2882)
![\ hat {S} _1 = S_1 ^ 2, \ hat {S} _2 = S_1S_2, \ hat {S} _3 = S_2](https://wikimedia.org/api/rest_v1/media/math/render/svg/221e9fd872651f7a8391b20a81fb32c1e9e7f697)
![\ has {S} _1, \ has {S} _2, \ has {S} _3](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a4c0d33132ff3e1bd49282d25579d069f8a4b27)
![\ hat {S} _1 \ hat {S} _1 ^ * + \ hat {S} _2 \ hat {S} _2 ^ * + \ hat {S} _3 \ hat {S} _3 ^ * = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9901fc39a0f520d24526fa852b0bc6109ec7aed)
![C ^ * (\ hat {S} _1, \ hat {S} _2, \ hat {S} _3) \ subset C ^ * (S_1, S_2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a901102bbe2d5970b66221e26bf6ed5b8e93601a)
In this way the inclusions are obtained
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.
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.
![\ mathcal {O} _n](https://wikimedia.org/api/rest_v1/media/math/render/svg/23c1482aadf309ab3fe04327732c45c8fb8b30c7)
![\ mathcal {O} _m](https://wikimedia.org/api/rest_v1/media/math/render/svg/23c152e895fbdb7467badbf1bc050e87b7a2f722)
![n \ neq m](https://wikimedia.org/api/rest_v1/media/math/render/svg/3994a24401e2dfabca26e4f36e53097a07a57af5)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![K_0](https://wikimedia.org/api/rest_v1/media/math/render/svg/44b0af6cafb690d3dbb0f3f30a032631338dc476)
![\ mathcal {O} _n](https://wikimedia.org/api/rest_v1/media/math/render/svg/23c1482aadf309ab3fe04327732c45c8fb8b30c7)
![\ mathbb {Z} _ {n-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c10c3f846f94ac880b57a9bdae329af7018e22)
![n = \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/40c93fd8fa11a9b05557492cd993bb04ac63c36c)
![K_0 = \ mathbb {Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44680aa5f03b6688805737084899e21c41beb959)
![K_0](https://wikimedia.org/api/rest_v1/media/math/render/svg/44b0af6cafb690d3dbb0f3f30a032631338dc476)
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.
![\ Phi](https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24)
![\ mathcal {O} _n \ simeq \ mathcal {F} ^ n \ rtimes_ \ Phi \ mathbb {Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ed394bed510d5ee2492f5644f315806d2bbbac)
![\ mathcal {F} ^ n](https://wikimedia.org/api/rest_v1/media/math/render/svg/539e0804ce89bbc718c613646a0bd2a8c6a98fda)
![\ mathcal {O} _n](https://wikimedia.org/api/rest_v1/media/math/render/svg/23c1482aadf309ab3fe04327732c45c8fb8b30c7)
literature