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
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for all
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for everyone and .
The usual calculation rules are used for , i.e. for all , for all and . One defines
a weight
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= linear envelope of
Then and are left ideals and is a sub-C * algebra in .
Weights with additional properties
The following properties are considered for weights
- A weight is tightly-defined , if respect. The standard topology tight is.
- A weight on a Von Neumann algebra is called semi-finite if it is dense with respect to the weak operator topology .
- A weight is called faithful , if it is.
- A weight is called semi-steady from below if it is closed for each .
- 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 .
- A weight is called a track weight , if additional for all unitary elements .
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
Sums of functionals
If a family of positive functionals is up , it's done
a weight off declared.
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
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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,
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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
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
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is normal, that is for monotonic networks applies .
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is additive, that is for every family in with valid .
- Is an ultra weak convergent network with limes in is so .
- There is a family of positive, normal functionals for everyone .
- There is a family of positive, normal functionals for everyone .
GNS construction
The GNS construction known for states can essentially also be carried out for weights on a C * -algebra . Through the formula
a scalar product is defined on, the completion is a Hilbert space . By
defined operators on continue to continuous, linear operators on , so that
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.
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