The spatial tensor product considered in the mathematical branch of functional analysis offers the possibility of constructing new ones from C * -algebras . In general, there are several ways the algebraic tensor product of two C * -algebras to a C * algebra complete ; the C * -norm treated here on the tensor product turns out to be minimal among these possibilities, which is why one speaks of the minimal tensor product . The construction presented here goes back to M. Takesaki .
Definitions
Let and two C * -algebras. A C * -norm on the algebraic tensor product is a norm such that
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![A \ odot B](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d3ccfd7ba3bf0f696f0922e64610d1dbaa7d16)
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
-
is a normalized algebra
-
for all
If there is such a C * -norm, then the completion marked with is a C * -algebra. If there is a C * -norm that can be defined for every pair of C * -algebras and , then one speaks of a -tensor product.
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
![A \ otimes_ \ alpha B](https://wikimedia.org/api/rest_v1/media/math/render/svg/c937555db77acb5bbe7395f8efddfd8dac5b3312)
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
One can show that C * norms automatically have the cross norm property , that is, it applies
to all .
![\ alpha (a \ otimes b) = \ | a \ | \ cdot \ | b \ |](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c483ac444bfae38e15649fb836896eecca3c954)
![a \ in A, b \ in B](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e763283c99e320f33d1e4d3780253b8560d10a8)
This article by means of Hilbert spaces on which the C * -algebras operate with defined designated C * norms, which because of Hilbert spaces used in spatial (German space should) remember.
![\ sigma](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36)
![\ sigma](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36)
construction
Let and two C * -algebras. According to Gelfand-Neumark's theorem, there are Hilbert spaces and and isometric * - homomorphisms and , that is, we can assume that the C * -algebras are subalgebras of the full operator algebra over suitable Hilbert spaces. One can take the universal representations , for example . The Hilbert space tensor product is now formed and an element of the algebraic tensor product is considered as an operator , which by
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
![A \ rightarrow L (H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/42d54d28456f341c31248cb20f949dfe3abe9400)
![H \ otimes K](https://wikimedia.org/api/rest_v1/media/math/render/svg/92b81ae1b368244915c6606efc3b6013378bf524)
![\ sum_ {i = 1} ^ n a_i \ otimes b_i](https://wikimedia.org/api/rest_v1/media/math/render/svg/65c4eed7f62f6a09d8a1f9db32f6928cb6200006)
![A \ odot B](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d3ccfd7ba3bf0f696f0922e64610d1dbaa7d16)
![H \ otimes K](https://wikimedia.org/api/rest_v1/media/math/render/svg/92b81ae1b368244915c6606efc3b6013378bf524)
is defined, whereby well-definedness is to be shown. Then it is clear that the restriction of the operator norm of on a C norm is *.
![\ sigma](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36)
![L (H \ otimes K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ad35c6e622572cbe00399264a8110ece106f47)
![A \ odot B](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d3ccfd7ba3bf0f696f0922e64610d1dbaa7d16)
Independence from the Hilbert rooms
The above construction initially depends on the choice of Hilbert spaces. Here a formula is set up for the spatial norm that is independent of the Hilbert spaces. If and are states on or , then there is exactly one state labeled open with for all and , the so-called product state from and . For an element of the algebraic tensor product we now have
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![f \ otimes g](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c61cd98c51da2ab230a9183da0e52217702e474)
![A \ otimes_ \ sigma B](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3b7f91b5515369120ad7c454862cbe6b02dc11)
![(f \ otimes g) (a \ otimes b) = f (a) g (b)](https://wikimedia.org/api/rest_v1/media/math/render/svg/afc0a645b51bdffd44bc0166a3c7bcf63b02a05d)
![a \ in A](https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5)
![b \ in B](https://wikimedia.org/api/rest_v1/media/math/render/svg/61dbfba9ff608c8700a30596649d98dcc6147d86)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![c = \ sum_ {i = 1} ^ n a_i \ otimes b_i](https://wikimedia.org/api/rest_v1/media/math/render/svg/b037cc266e740bb144278708380e629292a00227)
![A \ odot B](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d3ccfd7ba3bf0f696f0922e64610d1dbaa7d16)
whereby the supremum is formed over all states of , of and with . This formula shows the independence of the choice of Hilbert spaces, because on the right side only data of the abstract C * -algebras and their algebraic tensor product can be found.
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![s \ in A \ odot B](https://wikimedia.org/api/rest_v1/media/math/render/svg/0be8f08018a898093de410b9f8349cd901d0c615)
![(f \ otimes g) (s ^ * s)> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/030742e479a6cac0d5c2ef6eee96f057277d9174)
For the designation: In the textbook by Kadison and Ringrose given below, instead of is written, Murphy uses the notation .
![A \ otimes B](https://wikimedia.org/api/rest_v1/media/math/render/svg/887ed3ce13b460df337cda66497515383ded3e5d)
![A \ otimes_ \ sigma B](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3b7f91b5515369120ad7c454862cbe6b02dc11)
![A \ otimes _ {*} B](https://wikimedia.org/api/rest_v1/media/math/render/svg/10a1d66403b00f354bbcc4244c7d9e924804506b)
properties
- If and * -homomorphisms are between C * -algebras, there is exactly one * -homomorphism marked with , so that for all . If both and are isometric or * -isomophisms, then has the same property.
![\ pi_1: A_1 \ rightarrow B_1](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3635c96aa4efe40f9b690c5fb58fb52f4c3133f)
![\ pi_2: A_2 \ rightarrow B_2](https://wikimedia.org/api/rest_v1/media/math/render/svg/4572872aad7c8cac0c5c19d2e6ad049459ec976e)
![\ pi_1 \ otimes \ pi_2](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b66a83592c503cc6308822838de275bd313986f)
![A_1 \ otimes A_2 \ rightarrow B_1 \ otimes B_2](https://wikimedia.org/api/rest_v1/media/math/render/svg/69db9a793c2357aa2040fd3a5761e74cf10573ca)
![\ pi_1 \ otimes \ pi_2 (a_1 \ otimes a_2) = \ pi_1 (a_1) \ otimes \ pi_2 (a_2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c548947f092457ab5b744b1a7f80f900f5e26b1)
![a_i \ in A_i](https://wikimedia.org/api/rest_v1/media/math/render/svg/9498f2f918d0b59e82c9346aa62925f06ff9fae8)
![\ pi _ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/542cbd3dacd0a061d666ed7fc4ed7ad15b47444b)
![\ pi _ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b4d39c450ad33d7c407aec6fff9f225463ac1f0)
![\ pi_1 \ otimes \ pi_2](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b66a83592c503cc6308822838de275bd313986f)
- If there is a C * -norm on the algebraic tensor product , then . For this reason the spatial tensor product is also called the minimal tensor product, and the notation is sometimes found .
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
![A \ odot B](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d3ccfd7ba3bf0f696f0922e64610d1dbaa7d16)
![\ sigma \ le \ alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/40b228f05ab218952d980bc7c957f86f641a2462)
![A \ otimes _ {\ mathrm {min}} B](https://wikimedia.org/api/rest_v1/media/math/render/svg/25f64d4b18618c7292f58ed9d59f6d05c0c5fb88)
Examples
Let be a C * -algebra and a compact Hausdorff space . be the set of all continuous functions . For , and define:
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![C (X, A)](https://wikimedia.org/api/rest_v1/media/math/render/svg/06977f3e3b7d3f6b40ba6b8a91085326918da4d0)
![X \ rightarrow A](https://wikimedia.org/api/rest_v1/media/math/render/svg/27183334a6bcb30003cce703e62c1d8899d8fa7d)
![f, g \ in C (X, A)](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a5e1860317e981f4959f020342a320f1a581d00)
![{\ displaystyle \ lambda \ in \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11a6d1585381827bdf73529c2a418bc14098567c)
![x \ in X](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d)
-
.
This becomes a C * -algebra and you have an isometric isomorphism .
![C (X, A)](https://wikimedia.org/api/rest_v1/media/math/render/svg/06977f3e3b7d3f6b40ba6b8a91085326918da4d0)
![C (X) \ otimes_ \ sigma A \ rightarrow C (X, A), f \ otimes a \ mapsto f (\ cdot) a](https://wikimedia.org/api/rest_v1/media/math/render/svg/71fab3d117cd45e7c2e692647053629d7ec086f0)
Let be the C * -algebra of the complex matrices and a C * -algebra operating on a Hilbert space . Let the algebra of the matrices with entries be off ; this operates in the usual way on , that is
![M_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8656f32ad5c50e679b491b361a423727491496a0)
![n \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
![M_n (A)](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f339552c4cad90e21bb3f340da8954d8e83aba0)
![n \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![H ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c813df15dbb76f2e02b3bceb3f16b83a69d9c6)
This carries the norm of and shows that , where is mapped to.
![M_n (A)](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f339552c4cad90e21bb3f340da8954d8e83aba0)
![L (H ^ n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/49e2133485f6362e05de15e2299d8f9059ded73a)
![M_n \ otimes A \ cong M_n (A)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec1b3d7bbf1f5da516da44f1971d229f55b6d5b3)
![(c_ {i, j}) _ {i, j} \ otimes a](https://wikimedia.org/api/rest_v1/media/math/render/svg/39751620ad88c99bc60defd3d0976e4f705956a4)
![(c_ {i, j} \ cdot a) _ {i, j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48e0b9e5ea6b080125eefafa704c21c88ff19282)
See also
Individual evidence
-
↑ M. Takesaki: On the cross-norm of the direct product of C * -algebras , Tohoku Mathematical Journal, Volume 10 (1958), pages 111-122
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^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , §11.3
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^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , Lemma 11.3.3
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^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , sentence 11.1.2 and §11.3.1
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^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , Theorem 11.1.3
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↑ Gerald. J. Murphy: C * -Algebras and Operator Theory , Academic Press Inc. (1990), ISBN 0-12-511360-9 , Theorem 6.4.18
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^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , Theorem 11.3.9
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^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , example 11.1.7
literature
- Gerald. J. Murphy: C * -Algebras and Operator Theory , Academic Press Inc. (1990), ISBN 0-12-511360-9
- RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1