σ-finite Von Neumann algebra

σ-finite von Neumann algebras of are in the mathematical branch functional analysis examined Von Neumann algebras with an additional Abzählbarkeitseigenschaft . The designation σ-finite is motivated by mass theory, some authors also speak of countably decomposable Von Neumann algebras. These Von Neumann algebras play an important role in the Tomita-Takesaki theory .

Definitions

A Von Neumann algebra is called σ-finite if every family of pairwise orthogonal projections contains at most countably many elements different from 0. Projections are elements with and two such projections are called orthogonal if their product is 0. ${\displaystyle A}$ ${\displaystyle (e_{i})_{i\in I}}$ ${\displaystyle e_{i}\in A}$ ${\displaystyle e\in A}$ ${\displaystyle e=e^{*}=e^{2}}$ ${\displaystyle e_{1},e_{2}\in A}$

More generally, a projection is called σ-finite if every family of pairwise orthogonal projections with at most countably many elements different from 0 contains. Stands for . According to this, a Von Neumann algebra is σ-finite if and only if its unity element is σ-finite as a projection. ${\displaystyle e\in A}$ ${\displaystyle (e_{i})_{i\in I}}$ ${\displaystyle e_{i}\in A}$ ${\displaystyle e_{i}\leq e}$ ${\displaystyle e_{i}\leq e}$ ${\displaystyle e_{i}e=e_{i}}$

Examples

• A projection of a Von Neumann algebra over a Hilbert space is called cyclic if there is such that the orthogonal projection is onto the closed subspace generated by , where denotes the commutant of . Cyclic projections are σ-finite. ${\displaystyle e\in A}$ ${\displaystyle H}$ ${\displaystyle \xi \in H}$ ${\displaystyle e}$ ${\displaystyle \{a'\xi ;\xi \in A'\}}$ ${\displaystyle A'}$ ${\displaystyle A}$
• Every projection of a separable Hilbert space is σ-finite. In particular, every Von Neumann algebra is σ-finite over a separable Hilbert space.
• The concept of σ-finiteness of a projection depends by definition on a Von Neumann algebra. Is z. If, for example, a non-separable Hilbert space, such as the sequence space , the unit element is not σ-finite with regard to the full operator algebra , but it is with regard to the Von-Neumann algebra . Therefore, in case of doubt, the Von Neumann algebra under consideration must be given. ${\displaystyle H}$ ${\displaystyle H=\ell ^{2}(\mathbb {R} )}$ ${\displaystyle 1=\mathrm {id} _{H}}$ ${\displaystyle A=L(H)}$ ${\displaystyle A=\mathbb {C} \cdot \mathrm {id} _{H}}$

characterization

For the following characterization of σ-finite Von Neumann algebras we need the notion of the generating and separating vector. If a Von Neumann algebra is over a Hilbert space , then a subset is called generating if it is generated as a closed subspace of . A single vector is called generating if the singular set is generating. Separating a subset , if this already follows from and for all . A single vector is called separating if the one-element set is separating. Note that these terms are always to be understood relative to a Von Neumann algebra. With them, σ-finite Von Neumann algebras can be characterized as follows: ${\displaystyle A}$ ${\displaystyle H}$ ${\displaystyle X\subset H}$ ${\displaystyle H}$ ${\displaystyle \{a'\xi ;\,a'\in A',\xi \in X\}}$ ${\displaystyle \xi \in H}$ ${\displaystyle \{\xi \}}$ ${\displaystyle X\subset H}$ ${\displaystyle a\in A}$ ${\displaystyle a\xi =0}$ ${\displaystyle \xi \in X}$ ${\displaystyle a=0}$ ${\displaystyle \xi \in H}$ ${\displaystyle \{\xi \}}$

For a Von Neumann algebra over a Hilbert space , the following statements are equivalent: ${\displaystyle A}$ ${\displaystyle H}$

• ${\displaystyle A}$ is σ-finite.
• ${\displaystyle H}$contains a countable subset that is discrete for . ${\displaystyle A}$
• There is a faithful normal state on , ie is ultra weakly continuous , , for all and is only possible. ${\displaystyle f:A\rightarrow \mathbb {C} }$ ${\displaystyle f}$ ${\displaystyle f(1)=1}$ ${\displaystyle f(a^{*}a)\geq 0}$ ${\displaystyle a\in A}$ ${\displaystyle f(a^{*}a)=0}$ ${\displaystyle a=0}$
• ${\displaystyle A}$is isomorphic to a von Neumann algebra , over a possibly different Hilbert space , making it a vector is that for both divisive and generating. ${\displaystyle \pi (A)}$ ${\displaystyle H_{\pi }}$ ${\displaystyle \xi \in H_{\pi }}$ ${\displaystyle \pi (A)}$

The existence of the vector in the last condition is the starting point of the Tomita-Takesaki theory . ${\displaystyle \xi \in H_{\pi }}$

Individual evidence

1. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras I , Academic Press (1983), ISBN 0-12-393301-3 , definition 5.5.14
2. ^ Ola Bratteli, Derek W. Robinson: Operator Algebras and Quantum Statistical Mechanics 1 , Springer-Verlag (1979), ISBN 0-387-09187-4 , definition 2.5.1
3. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras , Volume I, 1983, ISBN 0-12-393301-3 , sentence 5.5.15
4. ↑ Ola Bratteli, Derek W. Robinson: Operator Algebras and Quantum Statistical Mechanics 1 , Springer-Verlag (1979), ISBN 0-387-09187-4 , Theorem 2.4.24