W * -dynamic systems are examined in the mathematical sub-area of functional analysis. It is a construction with which one obtains a new Von Neumann algebra from a Von Neumann algebra and a locally compact group that operates in a certain way on the Von Neumann algebra.
definition
A W * -dynamic system is a triple consisting of a Von Neumann algebra over a Hilbert space , a locally compact group and a homomorphism of into the group of * - automorphisms of , which is strongly continuous at points , i.e. all mappings are norm-continuous are.
![(A, G, \ alpha)](https://wikimedia.org/api/rest_v1/media/math/render/svg/031314aecbc23e5c363d302c429a56eb97a0a774)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![\ alpha: G \ rightarrow {\ mathrm {Aut}} (A), s \ mapsto \ alpha _ {s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ada8ef6ba6ecff1ee409468528c1f757669c2b7c)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
You can replace the strong operator topology with the weak or ultra- weak operator topology and get the same term.
Construction of the cross product
For a W * -dynamic system we construct a Von Neumann algebra as follows . Here we reproduce the construction presented in. First we describe the Hilbert space on which the new Von Neumann algebra is supposed to operate.
![(A, G, \ alpha)](https://wikimedia.org/api/rest_v1/media/math/render/svg/031314aecbc23e5c363d302c429a56eb97a0a774)
![A \ ltimes _ {\ alpha} G](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff16896ba8dc4da093cc8ac833a989ed8ce9cbeb)
operate on the Hilbert space and let L 2 (G) be the Hilbert space of the square-integrable functions with respect to the hair measure . The Tensor product of Hilbert spaces can with the space of measurable functions with identification. The mapping that assigns the function to an elementary tensor can become a unitary operator
![H \ otimes L ^ 2 (G)](https://wikimedia.org/api/rest_v1/media/math/render/svg/22fe42921566a245db41f3d43899b849a56688c1)
![L ^ {2} (G, H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/52ba79876c43ce5d49cf7e9da0249625a6d7be8e)
![\ xi: G \ rightarrow H](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fd33d73a3025feab6a430a2a3c308b3931bd408)
![\ textstyle \ int_G \ | \ xi (s) \ | ^ 2 \ mathrm {d} s <\ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fb33bf951404fb94f52ad30087df3ac44fe7f82)
![\ xi_0 \ otimes h](https://wikimedia.org/api/rest_v1/media/math/render/svg/67729045899970389092c3daaf244f560660d282)
![H \ otimes L ^ 2 (G) \ rightarrow L ^ 2 (G, H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/fafb273b07fcdb5292fb83cbb3ab3e281224f8b3)
be continued.
Now to the operators of the Von Neumann algebra to be defined. Since the space of continuous functions with compact support is close to , it is sufficient to state the effect of the operators on . For each we define the operator on through
![C_c (G, H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c54e390f3875986bcdedda8c308cd2f30f59a504)
![L ^ 2 (G, H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/52ba79876c43ce5d49cf7e9da0249625a6d7be8e)
![C_c (G, H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c54e390f3875986bcdedda8c308cd2f30f59a504)
![x \ in A](https://wikimedia.org/api/rest_v1/media/math/render/svg/27bcc9b2afb295d4234bc294860cd0c63bcad2ca)
![\ pi (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a62d9bf7b2ba03b30a95e7059362cc4b3865123)
![L ^ {2} (G, H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/52ba79876c43ce5d49cf7e9da0249625a6d7be8e)
![(\ pi (x) f) (s): = \ alpha_ {s ^ {- 1}} (x) f (s), \ quad f \ in C_c (G, H), s \ in G](https://wikimedia.org/api/rest_v1/media/math/render/svg/58804262afa38488fb71e75f6529fdacec55dbea)
and for each the operator on through
![t \ in G](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d8c2125ac6c76f186eca27c0a7d215269b90838)
![\ lambda (t)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0e1cc2c2f9e6d8b22c0da0bb5fe8f0f47791e4a)
![L ^ {2} (G, H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/52ba79876c43ce5d49cf7e9da0249625a6d7be8e)
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.
Then there is a Hilbert space representation of and a group representation of on the Hilbert space and it applies
![\pi](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![L ^ {2} (G, H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/52ba79876c43ce5d49cf7e9da0249625a6d7be8e)
-
for everyone .![x \ in A, t \ in G](https://wikimedia.org/api/rest_v1/media/math/render/svg/51c8825bd6b380c114e2c210f6e41dbff675dc05)
Therefore the linear hull of the operators is a partial algebra of , with respect to the involution, closed , of the restricted, linear operators on , whose weak closure is a Von Neumann algebra. This is called the Von Neumann Algebra of the W * -dynamic system or the cross product of and (fortune ) and is denoted by. Alternative names are , or .
![\ pi (x) \ lambda (t)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff2a1d7a14e038e28847578d6900c296927b0c46)
![L (L ^ 2 (G, H))](https://wikimedia.org/api/rest_v1/media/math/render/svg/558312fa18d34269d349fe4f1ea08d4aec675f48)
![L ^ {2} (G, H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/52ba79876c43ce5d49cf7e9da0249625a6d7be8e)
![(A, G, \ alpha)](https://wikimedia.org/api/rest_v1/media/math/render/svg/031314aecbc23e5c363d302c429a56eb97a0a774)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
![A \ ltimes _ {\ alpha} G](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff16896ba8dc4da093cc8ac833a989ed8ce9cbeb)
![A \ otimes_ \ alpha G](https://wikimedia.org/api/rest_v1/media/math/render/svg/744673a643fc295f77136945a67dd0fe1a916ce4)
![A \ times_ \ alpha G](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab641f2925e3f86a56ab0ee921820ea84eb3408)
![W ^ * (A, G, \ alpha)](https://wikimedia.org/api/rest_v1/media/math/render/svg/24e2ea759ccb8fc98756297d01c81b3c26ff336b)
If one observes the isomorphism given above , one can show that is contained in the tensor product .
![H \ otimes L ^ 2 (G) \ cong L ^ 2 (G, H)](https://wikimedia.org/api/rest_v1/media/math/render/svg/88fef24b84d14eab4ad8e9a7374e8dd8973ac535)
![A \ overline {\ otimes} L (L ^ 2 (G))](https://wikimedia.org/api/rest_v1/media/math/render/svg/27959b20a44c184b30d847e9f86a84e90d696cf4)
duality
Be a commutative, locally compact group. Then there is the dual group of continuous group homomorphisms . With the topology of compact convergence, this is again a commutative, locally compact group. For such a group homomorphism we define the unitary operator auf by the formula
![{\ displaystyle G \ rightarrow \ {z \ in \ mathbb {C}; \, | z | = 1 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c425f423266bd687289cb22037973e793737692d)
![\ chi \ in \ hat {G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae3d097ed2500b60d4bf6fb16c354fc0b22c8f7e)
![v_ \ chi](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f38c7f4cf8e7abfcf5ed21065b3608e069696b)
![L ^ {2} (G)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ef0e0fb98c404b6d54cb25eceee8495adbe7aa)
-
.
Then there is a unitary operator over and one can show that it
holds, that is, that through
![1_H \ otimes v_ \ chi](https://wikimedia.org/api/rest_v1/media/math/render/svg/15f2679a652284e5ada0c1e2b8461d4d712b90f5)
![H \ otimes L ^ 2 (G)](https://wikimedia.org/api/rest_v1/media/math/render/svg/22fe42921566a245db41f3d43899b849a56688c1)
![(1 \ otimes v_ \ chi) \, A \ ltimes_ \ alpha G \, (1 \ otimes v_ \ chi) ^ * = A \ ltimes_ \ alpha G](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a570c1e998fd7d7db1c99ddb54be597c0cc5b23)
![\ hat {\ alpha} _ \ chi (y): = (1 \ otimes v_ \ chi) y (1 \ otimes v_ \ chi) ^ *](https://wikimedia.org/api/rest_v1/media/math/render/svg/4320dc5571d8cd1a4c874bc7039242b07233275f)
an automorphism is defined which makes a W * -dynamic system. So you can build the cross product and show that it is isomorphic to .
![A \ ltimes _ {\ alpha} G](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff16896ba8dc4da093cc8ac833a989ed8ce9cbeb)
![(A \ ltimes_ \ alpha G, \ hat {G}, \ hat {\ alpha})](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d72ddfdcd49839a6e63675968570ef685d40ab6)
![(A \ ltimes_ \ alpha G) \ ltimes _ {\ hat {\ alpha}} \ hat {G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a06d912c62554da3de956b935bf1fae176221f)
![A \ overline {\ otimes} L (L ^ 2 (G))](https://wikimedia.org/api/rest_v1/media/math/render/svg/27959b20a44c184b30d847e9f86a84e90d696cf4)
Applications
Construction of factors
Let it be a Borel space that is Borel isomorphic to [0,1] and a σ-finite measure on without atoms, that is, it is for everyone . We consider injective group homomorphisms
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
![\ mu (\ {t \}) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/818223c32e48b27fe5542e3ec846fb604de2cd7f)
![t \ in T](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856)
![\ alpha ': G \ rightarrow \ mathrm {Iso} (T, \ mu)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f15d5327ae3fd91e4bad5d31cc2a538af5eb74e)
a discrete group in the group of Borel isomorphisms , so that the following applies:
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![(T, \ mu)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc662d6e97ac682ee0106866f4f6067015a6b06)
- From follows for everyone too .
![\ mu (N) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b976c47b5d229e78e08df8c63c14919a9c774e4)
![g \ in G](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1be73903416a0dd94b8cbc2268ce480810c0e62)
![\ mu (\ alpha'_g (N)) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/a72fb0811dbc5e6e0bb5818971877e11c4156d20)
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operate freely , that is, for everyone different from the neutral element .![(T, \ mu)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc662d6e97ac682ee0106866f4f6067015a6b06)
![\ mu (\ {t \ in T; \, \ alpha'_g (t) = t \}) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bfb5be62d94df2b84436a56be15697c621a25e7)
![g \ in G](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1be73903416a0dd94b8cbc2268ce480810c0e62)
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operate ergodically , that is, is with for a different from the neutral element , so is or .![(T, \ mu)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc662d6e97ac682ee0106866f4f6067015a6b06)
![N \ subset T](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f92de359833b54bc6b4449e629171a189315e7c)
![\ mu (\ alpha'_g (N) \ setminus N) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa9e74a9b8c7a5c554d77962df95c9c70b788393)
![g \ in G](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1be73903416a0dd94b8cbc2268ce480810c0e62)
![\ mu (N) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b976c47b5d229e78e08df8c63c14919a9c774e4)
![\ mu (T \ setminus N) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/dee9194dab951c79eb07542198ed0c4f8aa16315)
A group homomorphism is obtained from
![\ alpha ': G \ rightarrow \ mathrm {Iso} (T, \ mu)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f15d5327ae3fd91e4bad5d31cc2a538af5eb74e)
![\ alpha: G \ rightarrow \ mathrm {Aut} (L ^ \ infty (T, \ mu)), \ quad (\ alpha_g (f)) (t): = f (\ alpha '_ {g ^ {- 1 }} (t))](https://wikimedia.org/api/rest_v1/media/math/render/svg/802a35028208d8faaf8da16c332bf80a44f410c1)
into the automorphism group of the Abelian Von Neumann algebra and one obtains a W * -dynamic system . Hence one can form the cross product . For this applies:
![L ^ \ infty (T, \ mu)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8755c2eb186a69739f64422b7367c168b21d507d)
![(L ^ \ infty (T, \ mu), G, \ alpha)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e554b97a718ba710591beeb29c660ea381f514a9)
![L ^ \ infty (T, \ mu) \ ltimes_ \ alpha G](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7427e6c975f99e9f180251918d264cdf3e64a31)
- Is now -invariant, ie for all measurable subsets , it is a type II factor , namely a type II 1 factor if , and otherwise a type II ∞ factor.
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![\ mu (\ alpha'_g (N)) = \ mu (N)](https://wikimedia.org/api/rest_v1/media/math/render/svg/402c68d8a1dc40355de1d6dea431478c46b7f48b)
![N \ subset T](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f92de359833b54bc6b4449e629171a189315e7c)
![L ^ \ infty (T, \ mu) \ ltimes_ \ alpha G](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7427e6c975f99e9f180251918d264cdf3e64a31)
![\ mu (T) <\ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/48e27b5662e3b088e98ab74c4fe23c93f7faea3c)
- If it is not -invariant, but is invariant with regard to a subgroup of , which also operates ergodically , then it is a type III factor.
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![(T, \ mu)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc662d6e97ac682ee0106866f4f6067015a6b06)
![L ^ \ infty (T, \ mu) \ ltimes_ \ alpha G](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7427e6c975f99e9f180251918d264cdf3e64a31)
The following specific examples can be given for this:
Concrete examples
(i) Let be the circular line with the hair measure . It be and
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
![G = \ Q / \ Z](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb235ca938a98518b9d2c67633653678c636f3c)
![\ alpha ': G \ rightarrow \ mathrm {Iso} (\ mathbb {T}, \ mu), \ quad \ alpha' _ {q + \ Z} (e ^ {2 \ pi it}): = e ^ {2 \ pi i (t + q)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/513021caecde198e8e636be98db18e83a298fb00)
This fulfills the requirements of the above sentence, and it follows that a Type II 1 factor is.
![L ^ \ infty (\ mathbb {T}, \ mu) \ ltimes _ {\ alpha} \ Q / \ Z](https://wikimedia.org/api/rest_v1/media/math/render/svg/e145841e764717dd5113300b82f9c674dea4ded1)
(ii) Be with the Lebesgue measure .
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![\ alpha ': \ Q \ rightarrow \ mathrm {Iso} (\ R, \ lambda), \ quad \ alpha'_q (t): = t + q](https://wikimedia.org/api/rest_v1/media/math/render/svg/7156e95b1d71e92104552b44c2fa15cb84f2aab3)
This fulfills the requirements of the above theorem, and it follows that there is a type II ∞ -factor.
![L ^ \ infty (\ mathbb {R}, \ lambda) \ ltimes _ {\ alpha} \ Q](https://wikimedia.org/api/rest_v1/media/math/render/svg/de3bbc658ba1163ef9ad42cba00cd11c5786878b)
(iii) Let be with the Lebesgue measure and be the multiplicative matrix group . For be . Then meets the requirements of the above sentence, and it follows that is a type III factor.
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![G = \ {\ begin {pmatrix} a & b \\ 0 & 1 \ end {pmatrix}; \, a, b \ in \ Q \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/220aede3c171545f96dc0a3eb107d0dfef5f6412)
![g = \ begin {pmatrix} a & b \\ 0 & 1 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/712736149723d646d2e0b63425d4756097a67e7d)
![\ alpha'_g (t): = at + b](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a8ebc58d8bfb9154e2d74524a3e1e265716b3f)
![\ alpha ': G \ rightarrow \ mathrm {Iso} (\ R, \ lambda)](https://wikimedia.org/api/rest_v1/media/math/render/svg/67c292de3caf45ff98743690a37a37b64b353921)
![L ^ \ infty (\ mathbb {R}, \ lambda) \ ltimes _ {\ alpha} G](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a984bb0f461c48bd8767e11071f3f3f940e909f)
The modular group
For σ-finite Von Neumann algebras , the Tomita-Takesaki theory yields a W * -dynamic system for every true, normal state . The dependence on the state is described by a so-called Connes cocycle , from which it follows that the cross products of the W * -dynamic systems are isomorphic to different states. One can therefore speak of the cross product with the modular group.
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![(A, \ mathbb {R}, \ sigma)](https://wikimedia.org/api/rest_v1/media/math/render/svg/52e850f17e24ce292860dee54fc3a30ce41ad5b9)
![A \ ltimes_ \ sigma \ R](https://wikimedia.org/api/rest_v1/media/math/render/svg/053c83aa9b7a27e9c536998585277fda2c5e4a6f)
Duality plays an important role in Takesaki's theorem on the structure of Type III Von Neumann algebras .
![(A \ ltimes_ \ sigma \ R) \ ltimes _ {\ hat {\ sigma}} \ R \ cong A \ otimes L (L ^ 2 (\ R))](https://wikimedia.org/api/rest_v1/media/math/render/svg/d75ad13e97e04c696a7b5da71cbc4c256e8b57cc)
See also
Individual evidence
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^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , 7.4.2
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^ A van Daele: Continuous crossed products and type III von Neumann algebras , Cambridge University Press (1978), ISBN 0-521-21975-2
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^ A van Daele: Continuous crossed products and type III von Neumann algebras , Cambridge University Press (1978), ISBN 0-521-21975-2 , Theorem 4.11
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^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , November 7 , 2016
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^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-1239-3302-1 , Theorem 8.6.10