normal state

Normal states are examined in the mathematical sub-area of functional analysis. These are certain continuous, linear functionals on a Von Neumann algebra .

Definitions

Let it be a Von Neumann algebra over a Hilbert space . A state is a linear functional with and for everyone . One can show that states are automatically continuous with , this is true even for every C * -algebra. Other operator topologies are available for Von Neumann algebras and it is therefore only natural to study continuity properties with regard to these topologies. ${\displaystyle A}$ ${\displaystyle H}$ ${\displaystyle f:A\rightarrow \mathbb {C} }$ ${\displaystyle f(1)=1}$ ${\displaystyle f(a^{*}a)\geq 0}$ ${\displaystyle a\in A}$ ${\displaystyle \|f\|=1}$

Furthermore, Von Neumann algebras are closed compared to the supremum formation of upwardly directed families of self-adjoint elements. The order for is defined by the condition for all . One will want to look at conditions that receive Suprema. We therefore define: ${\displaystyle a\leq b}$ ${\displaystyle a,b\in A}$ ${\displaystyle \langle a\xi ,\xi \rangle \leq \langle b\xi ,\xi \rangle }$ ${\displaystyle \xi \in H}$

A state on the Von Neumann algebra is called normal if the following applies: If a monotonically growing network is in with supremum , then applies . ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle (a_{i})_{i\in I}}$ ${\displaystyle A}$ ${\displaystyle a\in A}$ ${\displaystyle \textstyle \sup _{i\in I}f(a_{i})=f(a)}$

A special case of a monotonic network arises from a family of pairwise orthogonal orthogonal projections , that is, of elements with and for all . Then the family of all finite sums of elements of this family is an ascending network of orthogonal projections, the supremum of which is called the sum . ${\displaystyle e_{i}\in A}$ ${\displaystyle e_{i}^{*}=e_{i}=e_{i}^{2}}$ ${\displaystyle e_{i}e_{j}=0}$ ${\displaystyle i\not =j}$ ${\displaystyle \textstyle \sum _{i\in I}e_{i}}$

A state on the Von Neumann algebra is called completely additive if the following applies: If a family of pairwise orthogonal orthogonal projections is in , then is . ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle (e_{i})_{i\in I}}$ ${\displaystyle A}$ ${\displaystyle \textstyle f(\sum _{i\in I}e_{i})=\sum _{i\in I}f(e_{i})}$

Characterization of normal conditions

By definition, complete additivity is weaker than normality, because the former only requires the formation of the supremum of very specific networks. Since orthogonal projections have the norm 1, it is also a condition that is restricted to the unit sphere of . The following applies: ${\displaystyle A_{1}}$ ${\displaystyle A}$

For a state on the Von Neumann algebra , the following statements are equivalent: ${\displaystyle f}$ ${\displaystyle A}$

• ${\displaystyle f}$ is normal.
• ${\displaystyle f}$ is completely additive.
• There is a positive trace class with all ${\displaystyle s}$ ${\displaystyle f(a)=\mathrm {Sp} (sa)}$ ${\displaystyle a\in A}$
• ${\displaystyle f}$is continuous with respect to the ultra-weak topology .
• ${\displaystyle f}$is continuous with respect to the ultra-strong topology .
• ${\displaystyle f|_{A_{1}}}$ is continuous with respect to the weak operator topology .
• ${\displaystyle f|_{A_{1}}}$ is continuous with regard to the strong operator topology .

Here is the restriction of the unit sphere . The first two conditions only refer to the order structure of the Von Neumann algebra and this can even be defined purely algebraically, because it is equivalent to for a . The above sentence shows that these conditions are equivalent to purely topological conditions. ${\displaystyle f|_{A_{1}}}$ ${\displaystyle f}$ ${\displaystyle A_{1}}$ ${\displaystyle a\leq b}$ ${\displaystyle b-a=c^{*}c}$ ${\displaystyle c\in A}$

Predual

The continuity conditions in the above list of equivalent conditions can also be applied to any linear functionals ; one then speaks of normal functional. The normal functionals form a subspace of the dual space of . This is standardized and generated by the normal states; it is designated with . ${\displaystyle A\rightarrow \mathbb {C} }$ ${\displaystyle A^{\#}}$ ${\displaystyle A}$ ${\displaystyle A_{\#}}$

Each element of the Von Neumann algebra defined by means of a continuous linear functional in , and can show that an isometric isomorphism is. In this sense the dual space of ; the latter is therefore called the predual space of . ${\displaystyle a\in A}$ ${\displaystyle \varphi _{a}(f):=f(a)}$ ${\displaystyle \varphi _{a}}$ ${\displaystyle A_{\#}}$ ${\displaystyle \varphi :A\rightarrow (A_{\#})^{\#},\,a\mapsto \varphi _{a}}$ ${\displaystyle A}$ ${\displaystyle A_{\#}}$ ${\displaystyle A}$

These considerations show that every Von Neumann algebra appears as a dual space of a Banach space . According to a theorem by S. Sakai , this characterizes the Von Neumann algebras and the C * algebras.

Representations

As is well known, every state on a C * -algebra defines a Hilbert space representation in the algebra of the operators on a Hilbert space by means of GNS construction . If a Von Neumann algebra and a normal state, then ultra weak is continuous and the image is a Von Neumann algebra. ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle \pi _{f}:A\rightarrow L(H_{f})}$ ${\displaystyle H_{f}}$ ${\displaystyle A}$ ${\displaystyle f}$ ${\displaystyle \pi _{f}}$ ${\displaystyle \pi _{f}(A)}$

See also

• σ-finite Von Neumann algebras are characterized by the fact that there is a faithful, normal state on them.

Individual evidence

1. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-1239-3302-1 , definition 7.1.11
2. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-1239-3302-1 , definition 7.1.11
3. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-1239-3302-1 , Theorem 7.1.12
4. ^ Ola Bratteli, Derek W. Robinson: Operator Algebras and Quantum Statistical Mechanics 1 , Springer-Verlag (1979), ISBN 0-387-09187-4 , Theorem 2.4.21
5. ↑ Ola Bratteli, Derek W. Robinson: Operator Algebras and Quantum Statistical Mechanics 1 , Springer-Verlag (1979), ISBN 0-387-09187-4 , sentence 2.4.18
6. ↑ Ola Bratteli, Derek W. Robinson: Operator Algebras and Quantum Statistical Mechanics 1 , Springer-Verlag (1979), ISBN 0-387-09187-4 , Theorem 2.4.24