Weight (functional analysis)

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 ${\displaystyle A}$ ${\displaystyle A^{+}}$ ${\displaystyle a^{*}a,\,a\in A}$ ${\displaystyle A}$ ${\displaystyle \omega :A^{+}\rightarrow [0,\infty ]}$

• ${\displaystyle \omega (a+b)=\omega (a)+\omega (b)}$ for all ${\displaystyle a,b\in A^{+}}$
• ${\displaystyle \omega (\lambda a)=\lambda \omega (a)}$for everyone and . ${\displaystyle \lambda \in \mathbb {R} _{0}^{+}}$ ${\displaystyle a\in A^{+}}$

The usual calculation rules are used for , i.e. for all , for all and . One defines a weight ${\displaystyle \infty }$ ${\displaystyle x+\infty =\infty +x=\infty }$ ${\displaystyle x\in \mathbb {R} _{0}^{+}}$ ${\displaystyle x\cdot \infty =\infty \cdot x=\infty }$ ${\displaystyle x\in \mathbb {R} ^{+}}$ ${\displaystyle 0\cdot \infty =\infty \cdot 0=0}$ ${\displaystyle \omega }$

${\displaystyle A_{+}^{\omega }:=\{a\in A^{+}|\,\omega (a)<\infty \}}$

${\displaystyle A^{\omega }}$= linear envelope of ${\displaystyle A_{+}^{\omega }}$

${\displaystyle L_{\omega }:=\{a\in A^{+}|\,\omega (a^{*}a)=0\}}$

${\displaystyle A_{2}^{\omega }:=\{a\in A|\,\omega (a^{*}a)<\infty \}}$

Then and are left ideals and is a sub-C * algebra in . ${\displaystyle A_{2}^{\omega }}$ ${\displaystyle L_{\omega }}$ ${\displaystyle A^{\omega }}$ ${\displaystyle A}$

Weights with additional properties

The following properties are considered for weights

• A weight is tightly-defined , if respect. The standard topology tight is. ${\displaystyle \omega }$ ${\displaystyle A_{+}^{\omega }\subset A^{+}}$
• A weight on a Von Neumann algebra is called semi-finite if it is dense with respect to the weak operator topology . ${\displaystyle \omega }$ ${\displaystyle A_{+}^{\omega }\subset A^{+}}$
• A weight is called faithful , if it is. ${\displaystyle \omega }$ ${\displaystyle L_{\omega }=\{0\}}$
• A weight is called semi-steady from below if it is closed for each . ${\displaystyle \omega }$ ${\displaystyle \{a\in A^{+}|\,\omega (a)\leq \alpha \}}$ ${\displaystyle \alpha \in \mathbb {R} _{0}^{+}}$
• 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 . ${\displaystyle \omega }$ ${\displaystyle A}$ ${\displaystyle (a_{i})_{i\in I}}$ ${\displaystyle A^{+}}$ ${\displaystyle a\in A^{+}}$ ${\displaystyle \textstyle \sup _{i\in I}\omega (a_{i})=\omega (a)}$
• A weight is called a track weight , if additional for all unitary elements . ${\displaystyle \omega }$ ${\displaystyle \omega (uau^{*})=\omega (a)}$ ${\displaystyle u\in A}$

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 ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle f(a^{*}a)\geq 0}$ ${\displaystyle a\in A}$ ${\displaystyle f|_{A^{+}}}$ ${\displaystyle [0,\infty )}$ ${\displaystyle \omega }$ ${\displaystyle \mathbb {R} _{0}^{+}}$ ${\displaystyle A_{+}^{\omega }=A^{+}}$ ${\displaystyle f}$ ${\displaystyle \omega =f|_{A^{+}}}$

Sums of functionals

If a family of positive functionals is up , it's done ${\displaystyle (f_{i})_{i\in I}}$ ${\displaystyle A}$

${\displaystyle \omega (a):=\sum _{i\in I}f_{i}(a),\quad a\in A^{+}}$

a weight off declared. ${\displaystyle A}$

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 ${\displaystyle (\xi _{i})_{i\in I}}$ ${\displaystyle H}$ ${\displaystyle L(H)}$ ${\displaystyle H}$

${\displaystyle \omega (a):=\sum _{i\in I}\langle a\xi _{i},\xi _{i}\rangle ,\quad a\in A^{+}}$.

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

${\displaystyle L(H)_{+}^{\omega }:={\mathcal {S}}_{1}(H)^{+}}$the set of positive elements of the trace class ,

${\displaystyle L_{\omega }:=\{0\}}$, that means is faithful, ${\displaystyle \omega }$

${\displaystyle L(H)_{2}^{\omega }:={\mathcal {S}}_{2}(H)}$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 ${\displaystyle \mu }$ ${\displaystyle X}$ ${\displaystyle A=C_{0}(X)}$ ${\displaystyle X}$

${\displaystyle \omega :C_{0}(X)^{+}\rightarrow [0,\infty ],\quad f\mapsto \int _{X}f\mathrm {d} \mu }$

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 ${\displaystyle \omega }$ ${\displaystyle A}$

• ${\displaystyle \omega }$is normal, that is for monotonic networks applies . ${\displaystyle (a_{i})_{i\in I}\to a\in A^{+}}$ ${\displaystyle \omega (a_{i})\to \omega (a)}$
• ${\displaystyle \omega }$is additive, that is for every family in with valid . ${\displaystyle (a_{i})_{i\in I}}$ ${\displaystyle A^{+}}$ ${\displaystyle \textstyle \sum _{i\in I}a_{i}=a\in A^{+}}$ ${\displaystyle \textstyle \omega (\sum _{i\in I}a_{i})=\sum _{i\in I}\omega (a_{i})}$
• Is an ultra weak convergent network with limes in is so . ${\displaystyle (a_{i})_{i\in I}}$ ${\displaystyle a}$ ${\displaystyle A^{+}}$ ${\displaystyle \omega (a)\leq \textstyle \limsup _{i\in I}\omega (a_{i})}$
• There is a family of positive, normal functionals for everyone . ${\displaystyle (\varphi _{i})_{i\in I}}$ ${\displaystyle \textstyle \omega (a)=\sup _{i\in I}\varphi _{i}(a)}$ ${\displaystyle a\in A^{+}}$
• There is a family of positive, normal functionals for everyone . ${\displaystyle (\varphi _{i})_{i\in I}}$ ${\displaystyle \textstyle \omega (a)=\sum _{i\in I}\varphi _{i}(a)}$ ${\displaystyle a\in A^{+}}$

GNS construction

The GNS construction known for states can essentially also be carried out for weights on a C * -algebra . Through the formula ${\displaystyle \omega }$ ${\displaystyle A}$

${\displaystyle \langle a+L_{\omega },b+L_{\omega }\rangle :=\omega (b^{*}a),\quad a,b\in A_{2}^{\omega }}$

a scalar product is defined on, the completion is a Hilbert space . By ${\displaystyle A_{2}^{\omega }/L_{\omega }}$ ${\displaystyle H_{\omega }}$

${\displaystyle b+L_{\omega }\mapsto ab+L_{\omega },\quad a\in A,b\in A_{2}^{\omega }}$

defined operators on continue to continuous, linear operators on , so that ${\displaystyle A_{2}^{\omega }/L_{\omega }}$ ${\displaystyle \pi _{\omega }(a)}$ ${\displaystyle H_{\omega }}$

${\displaystyle \pi _{\omega }:A\rightarrow L(H_{\omega })}$

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. ${\displaystyle \omega }$ ${\displaystyle \pi _{\omega }}$ ${\displaystyle \omega }$ ${\displaystyle \pi _{\omega }(A)}$ ${\displaystyle \pi _{\omega }}$

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

1. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , definition 7.5.1
2. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , definition 5.1.1
3. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , Chapter 5.1: Weights
4. ^ Ola Bratteli, Derek W. Robinson: Operator Algebras and Quantum Statistical Mechanics 1 , Springer-Verlag (1979), ISBN 0-387-09187-4 , theorem 2.7.11
5. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Theorem 7.5.3
6. ^ 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