Weights are examined in the mathematical sub-area of functional analysis. It is a generalization of a state on a C * algebra . In the theory of Von Neumann algebras in particular , the Tomita-Takesaki theory can be extended beyond the case of σ-finite Von Neumann algebras by means of certain weights .
definition
Let it be a C * -algebra, the positive cone, that is, the set of all elements of the form . A weight on is a figure with
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A ^ +](https://wikimedia.org/api/rest_v1/media/math/render/svg/b380a5ff4e2d7d22a0dc1aea46e7ecba61f95fe6)
![{\ displaystyle a ^ {*} a, \, a \ in A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32284897e248a8b749252ec1b12a1eaf4f61f3f9)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle \ omega: A ^ {+} \ rightarrow [0, \ infty]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48574ab0198460c5cd81b119102d2e79e0fd7de5)
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for all
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for everyone and .![{\ displaystyle \ lambda \ in \ mathbb {R} _ {0} ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d8869e29a29ffec09a8e3518ebab863275d5e00)
![a \ in A ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f063c1ebaede89fe5d57aa7b6681f692eead675d)
The usual calculation rules are used for , i.e. for all , for all and . One defines
a weight![\ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21)
![{\ displaystyle x + \ infty = \ infty + x = \ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1125c5ac36160cea79d380d2cd6237972d716012)
![{\ displaystyle x \ in \ mathbb {R} _ {0} ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50b2b5aee6088ba812af1159006476bc6d22d91a)
![{\ displaystyle x \ cdot \ infty = \ infty \ cdot x = \ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8bcd4a7c85e247c5e4f650d3b199605755fd38c)
![{\ displaystyle x \ in \ mathbb {R} ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f3ab5fc33241badf9bc8f08b9bee1a3616e357)
![{\ displaystyle 0 \ cdot \ infty = \ infty \ cdot 0 = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2160a8072a92fd29cae700a17322252f05d7dc7d)
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![{\ displaystyle A _ {+} ^ {\ omega}: = \ {a \ in A ^ {+} | \, \ omega (a) <\ infty \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bae2771b009edd70d4aecdf1a0d2e66c7ee241a)
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= linear envelope of
![{\ displaystyle L _ {\ omega}: = \ {a \ in A ^ {+} | \, \ omega (a ^ {*} a) = 0 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a783bf9cebf357c96ac2d687ceefb30a48c145)
![{\ displaystyle A_ {2} ^ {\ omega}: = \ {a \ in A | \, \ omega (a ^ {*} a) <\ infty \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d14ed470c627d4510d2f4f075214ff510d02e8)
Then and are left ideals and is a sub-C * algebra in .
![{\ displaystyle A_ {2} ^ {\ omega}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be714ae7b68d2ec240c16e62015db0a7066c30fc)
![{\ displaystyle A ^ {\ omega}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8dfea9d081099367f6bc0772f229c94846f50e1)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
Weights with additional properties
The following properties are considered for weights
- A weight is tightly-defined , if respect. The standard topology tight is.
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
- A weight on a Von Neumann algebra is called semi-finite if it is dense with respect to the weak operator topology .
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![{\ displaystyle A _ {+} ^ {\ omega} \ subset A ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31ebad97abb9757ea49174da5e89c890fde89453)
- A weight is called faithful , if it is.
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![{\ displaystyle L _ {\ omega} = \ {0 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74f4922a9f0460f692a33c2b3faa834cad2dea2d)
- A weight is called semi-steady from below if it is closed for each .
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![{\ displaystyle \ {a \ in A ^ {+} | \, \ omega (a) \ leq \ alpha \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6894810e0ad05cddd2476b65651cc322ba2a99a)
![{\ displaystyle \ alpha \ in \ mathbb {R} _ {0} ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0ca3dd5b32bbe69d6c09d3db139d361291cde9f)
- A weight on a Von Neumann algebra is called normal if the following applies: If a monotonically growing network is in with supremum , then applies .
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![(a_ {i}) _ {i \ in I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e94378b596b3f40f8816a6ad2df7fb0929ad6ae)
![A ^ +](https://wikimedia.org/api/rest_v1/media/math/render/svg/b380a5ff4e2d7d22a0dc1aea46e7ecba61f95fe6)
![a \ in A ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f063c1ebaede89fe5d57aa7b6681f692eead675d)
![{\ displaystyle \ textstyle \ sup _ {i \ in I} \ omega (a_ {i}) = \ omega (a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b056adddc758281806282752edbab0d59a66c7)
- A weight is called a track weight , if additional for all unitary elements .
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![{\ displaystyle \ omega (uau ^ {*}) = \ omega (a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/835c3406bb1635a0a961cdfd565722a2ecc8b307)
![u \ in A](https://wikimedia.org/api/rest_v1/media/math/render/svg/7373fa8afb94205154d40f84f49609144ea2134a)
Examples
Limited weights
A functional on a C * -algebra is called positive if for all . Then the restriction is obviously a weight with the peculiarity that the image lies in. Conversely , if there is a weight different from 0 with image in , that is with , there is a positive functional with![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle f (a ^ {*} a) \ geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f45cebb8a9b5820dbd2138e0abc699300c074789)
![a \ in A](https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5)
![{\ displaystyle f | _ {A ^ {+}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f8c47952b6b9d559e1411f60655dde34900d07f)
![[0, \ infty)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc2d914c2df66bc0f7893bfb8da36766650fe47)
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![\ mathbb {R} _ {0} ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d54edc82900153fe95d7604ca3418e80cff281e5)
![{\ displaystyle A _ {+} ^ {\ omega} = A ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/834cb6cf45c671afb5078c94765d2a1df4e2f170)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
Sums of functionals
If a family of positive functionals is up , it's done
![(f_ {i}) _ {i \ in I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f3dd9718b5e3460904d390a0d092237aec266ce)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle \ omega (a): = \ sum _ {i \ in I} f_ {i} (a), \ quad a \ in A ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9841b54ee6d5f85f23ae9483c8ac1ff2cec13b8)
a weight off declared.
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
For example, if an orthonormal basis is a Hilbert space , then the sum of the associated vector states is a weight on , the Von Neumann algebra of the continuous, linear operators on . By
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
![L (H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ace334e31b512d9648671e17d4de006e0174ada4)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
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.
a normal trace weight is defined and it can be shown that this does not depend on the selection of the orthonormal basis. It is
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the set of positive elements of the trace class ,
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, that means is faithful,![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
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the H * -algebra of the Hilbert-Schmidt operators .
Dimensions
Let it be a positive measure on a locally compact Hausdorff space and the C * -algebra of the continuous functions that vanish at infinity . Then the picture is
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![{\ displaystyle A = C_ {0} (X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e7dd171bbca8a8973b1c6f6b12c83f2eefd141)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![{\ displaystyle \ omega: C_ {0} (X) ^ {+} \ rightarrow [0, \ infty], \ quad f \ mapsto \ int _ {X} f \ mathrm {d} \ mu}](https://wikimedia.org/api/rest_v1/media/math/render/svg/581e8f131148bfcf947b366f1bb90dcbd5480295)
a weight. Bounded measures lead to bounded weights, that is, positive linear functionals.
Applications and properties
normality
As with normal states, there are also different characterizations of normality for weights. For a weight on a Von Neumann algebra are equivalent
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
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is normal, that is for monotonic networks applies .![{\ displaystyle (a_ {i}) _ {i \ in I} \ to a \ in A ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/027d8bf9ee8d111fa01d99dd86ffe2023ef34381)
![{\ displaystyle \ omega (a_ {i}) \ to \ omega (a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45465152ad0bc590929f5ad8676699bc806b5cf5)
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is additive, that is for every family in with valid .![(a_ {i}) _ {i \ in I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e94378b596b3f40f8816a6ad2df7fb0929ad6ae)
![A ^ +](https://wikimedia.org/api/rest_v1/media/math/render/svg/b380a5ff4e2d7d22a0dc1aea46e7ecba61f95fe6)
![{\ displaystyle \ textstyle \ sum _ {i \ in I} a_ {i} = a \ in A ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc1623ee8c1bace79f1dcae2109af5166a7694f4)
![{\ displaystyle \ textstyle \ omega (\ sum _ {i \ in I} a_ {i}) = \ sum _ {i \ in I} \ omega (a_ {i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68828ca8a19d2fc49fee95f74043df7fd882e1af)
- Is an ultra weak convergent network with limes in is so .
![(a_ {i}) _ {i \ in I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e94378b596b3f40f8816a6ad2df7fb0929ad6ae)
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![A ^ +](https://wikimedia.org/api/rest_v1/media/math/render/svg/b380a5ff4e2d7d22a0dc1aea46e7ecba61f95fe6)
![{\ displaystyle \ omega (a) \ leq \ textstyle \ limsup _ {i \ in I} \ omega (a_ {i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b47d2d3b75ed33b7135b01dd0b573762ba5e434)
- There is a family of positive, normal functionals for everyone .
![{\ displaystyle (\ varphi _ {i}) _ {i \ in I}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8785a015533a23dffa44b3d70a2f8e882a76433)
![{\ displaystyle \ textstyle \ omega (a) = \ sup _ {i \ in I} \ varphi _ {i} (a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4830d75c8d79c482baf20419c3c9cc6416ba16fb)
![a \ in A ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f063c1ebaede89fe5d57aa7b6681f692eead675d)
- There is a family of positive, normal functionals for everyone .
![{\ displaystyle (\ varphi _ {i}) _ {i \ in I}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8785a015533a23dffa44b3d70a2f8e882a76433)
![{\ displaystyle \ textstyle \ omega (a) = \ sum _ {i \ in I} \ varphi _ {i} (a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb121c01cb6638e44f076fe7414048b71079f2ee)
![a \ in A ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f063c1ebaede89fe5d57aa7b6681f692eead675d)
GNS construction
The GNS construction known for states can essentially also be carried out for weights on a C * -algebra . Through the formula
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle \ langle a + L _ {\ omega}, b + L _ {\ omega} \ rangle: = \ omega (b ^ {*} a), \ quad a, b \ in A_ {2} ^ {\ omega }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96e93b980fa0e0600f19d2168785d0082bf5ad78)
a scalar product is defined on, the completion is a Hilbert space . By
![{\ displaystyle A_ {2} ^ {\ omega} / L _ {\ omega}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef573ba663dcb8651b212a5d2a0b95481b1c1e65)
![H_ \ omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8c3d14d71a9150bff074381c3efaac910408882)
![{\ displaystyle b + L _ {\ omega} \ mapsto ab + L _ {\ omega}, \ quad a \ in A, b \ in A_ {2} ^ {\ omega}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/926df98a0b68dc0ebe8719c0fd0ecff90b5feaa9)
defined operators on continue to continuous, linear operators on , so that
![{\ displaystyle A_ {2} ^ {\ omega} / L _ {\ omega}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef573ba663dcb8651b212a5d2a0b95481b1c1e65)
![{\ displaystyle \ pi _ {\ omega} (a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d20ce27484c382a3dcf2b35189dbedfb10e968db)
![H_ \ omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8c3d14d71a9150bff074381c3efaac910408882)
![{\ displaystyle \ pi _ {\ omega}: A \ rightarrow L (H _ {\ omega})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d412dd89f2a112d45687371184f67b4fc9d7628)
defines a Hilbert space representation . Is faithful and semi-finite, so is faithful. If a normal weight is on a Von Neumann algebra, then it is also a Von Neumann algebra and the representation is normal.
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![{\ displaystyle \ pi _ {\ omega}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/770772bf574c55dadde22e969364f91caf235c73)
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![{\ displaystyle \ pi _ {\ omega} (A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8406f4953b733fa21d3423dbc705045d7d9393ce)
![{\ displaystyle \ pi _ {\ omega}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/770772bf574c55dadde22e969364f91caf235c73)
Tomita Takesaki Theory
On a Von Neumann algebra there are always faithful, normal and semi-finite weights. On the picture of the associated GNS representation, certain automorphisms can be defined that lead to the Tomita-Takesaki theory .
Individual evidence
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^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , definition 7.5.1
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^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , definition 5.1.1
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^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , Chapter 5.1: Weights
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^ Ola Bratteli, Derek W. Robinson: Operator Algebras and Quantum Statistical Mechanics 1 , Springer-Verlag (1979), ISBN 0-387-09187-4 , theorem 2.7.11
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^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Theorem 7.5.3
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^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , A further extension of modular theory , from page 639