Finite Von Neumann Algebra

Finite Von Neumann algebras are examined in the mathematical sub-area of functional analysis. These are Von-Neumann algebras , the projections of which satisfy a certain finiteness condition.

Definitions

Let it be a Von Neumann algebra over a Hilbert space . Projections are elements made with the property . The work of Murray and von Neumann on the so-called Von Neumann algebras today was based on the idea of ​​examining projections in analogy to sets. The equivalence of two projections is defined in analogy to the uniformity of sets: and are called equivalent if there is a with and ; one writes . The subset relationship corresponds to the subset relationship of the projected spaces, that is, it is defined as . Since a set is finite if and only if it is not equal to any real subset, one defines in the sense of the analogy followed here: ${\displaystyle A}$ ${\displaystyle H}$ ${\displaystyle A}$ ${\displaystyle e=e^{*}=e^{2}}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$ ${\displaystyle v\in A}$ ${\displaystyle e_{1}=v^{*}v}$ ${\displaystyle e_{2}=vv^{*}}$ ${\displaystyle e_{1}\sim e_{2}}$ ${\displaystyle e_{1}\leq e_{2}}$ ${\displaystyle e_{1}(H)\subset e_{2}(H)}$

A projection is called finite , if only for is possible. Note that this concept of finitude depends on, since the concept of equivalence depends on. ${\displaystyle e\in A}$ ${\displaystyle e\sim e_{1}\leq e}$ ${\displaystyle e=e_{1}}$ ${\displaystyle A}$ ${\displaystyle A}$

A Von Neumann algebra is called finite if the one element is finite as a projection from . ${\displaystyle A}$ ${\displaystyle 1=\mathrm {id} _{H}}$ ${\displaystyle A}$

Examples

• Abelian Von Neumann algebras are finite, because for these the equivalence of projections is synonymous with their equality.
• The finite-dimensional algebras over a finite-dimensional Hilbert space are finite, because equivalent projections have the same dimension . ${\displaystyle A=L(\mathbb {C} ^{n})}$
• The algebra over the sequence space is not finite, because if the shift operator is , then is . ${\displaystyle L(\ell ^{2})}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle s\in L(\ell ^{2})}$ ${\displaystyle 1=s^{*}s\sim ss^{*}<1}$
• It's a discreet group . Each element operates as a left operator and as a right operator on the Hilbert space in which and is defined. Let and be the Von Neumann algebras generated by or . Then, and finally and mutual Kommutanten . ${\displaystyle G}$ ${\displaystyle g\in G}$ ${\displaystyle l_{g}}$ ${\displaystyle r_{g}}$ ${\displaystyle \ell ^{2}(G)}$ ${\displaystyle (l_{g}(x))(h):=x(g^{-1}h)}$ ${\displaystyle (r_{g}(x))(h):=x(hg^{-1})}$ ${\displaystyle L_{G}}$ ${\displaystyle R_{G}}$ ${\displaystyle \{l_{g};\,g\in G\}}$ ${\displaystyle \{r_{g};\,g\in G\}}$ ${\displaystyle L_{G}}$ ${\displaystyle R_{G}}$

The trace on a finite Von Neumann algebra

If there is a finite Von Neumann algebra with a center , there is exactly one linear mapping with the following properties: ${\displaystyle A}$ ${\displaystyle Z}$ ${\displaystyle \tau :A\rightarrow Z}$

• ${\displaystyle \tau }$is positive, that is, it follows from ${\displaystyle a\geq 0}$ ${\displaystyle \tau (a)\geq 0}$
• ${\displaystyle \tau }$is a track, that is for everyone ${\displaystyle \tau (ab)=\tau (ba)}$ ${\displaystyle a,b\in A}$
• ${\displaystyle \tau }$is a projection on , that is, for everyone . ${\displaystyle Z}$ ${\displaystyle \tau (z)=z}$ ${\displaystyle z\in Z}$

The clearly determined trace is called the canonical trace . It also has the following properties: ${\displaystyle A}$

• ${\displaystyle \tau }$is strictly positive, that is, follows ${\displaystyle a>0}$ ${\displaystyle \tau (a)>0}$
• ${\displaystyle \tau }$is morphism, that is, for everyone . ${\displaystyle Z}$ ${\displaystyle \tau (za)=z\tau (a)}$ ${\displaystyle a\in A,z\in Z}$
• ${\displaystyle \tau }$is a contraction, that is, for everyone ${\displaystyle \|\tau (a)\|\leq \|a\|}$ ${\displaystyle a\in A}$
• ${\displaystyle \tau }$is ultra weak steady .

Conversely, if a Von Neumann algebra with a center and a strictly positive trace , then is finite. Indeed , there is with and . It follows from this and because of the tracing quality and then because of the strict positivity. Hence every projection in is finite, from which the finiteness of results. ${\displaystyle A}$ ${\displaystyle Z}$ ${\displaystyle \tau :A\rightarrow Z}$ ${\displaystyle A}$ ${\displaystyle e_{1}\sim e_{2}\leq e_{1}}$ ${\displaystyle v\in A}$ ${\displaystyle e_{1}=v^{*}v}$ ${\displaystyle e_{2}=vv^{*}}$ ${\displaystyle e_{2}-e_{1}\geq 0}$ ${\displaystyle \tau (e_{2}-e_{1})=\tau (vv^{*})-\tau (v^{*}v)=0}$ ${\displaystyle e_{2}-e_{1}=0}$ ${\displaystyle A}$ ${\displaystyle A}$

Further characterizations

Types of finite Von Neumann algebras

In the type classification of the Von Neumann algebras, precisely the type I _n algebras with and the type II ₁ algebras are finite. ${\displaystyle n<\infty }$

Unitary equivalence of projections

Two projections of a Von Neumann algebra are called unitary equivalent if there is a unitary element (i.e. ) with . From the unitary equivalence follows the usual equivalence defined above, because from the defining equation follows and . The converse is generally wrong. ${\displaystyle e_{1},e_{2}}$ ${\displaystyle A}$ ${\displaystyle u\in A}$ ${\displaystyle 1=u^{*}u=uu^{*}}$ ${\displaystyle e_{1}=u^{*}e_{2}u}$ ${\displaystyle e_{1}=u^{*}e_{2}u=(e_{2}u)^{*}(e_{2}u)}$ ${\displaystyle (e_{2}u)(e_{2}u)^{*}=e_{2}uu^{*}e_{2}=e_{2}1e_{2}=e_{2}^{2}=e_{2}}$

A Von Neumann algebra is finite if and only if equivalence and unitary equivalence coincide.

Continuity of involution

The involution on a Von Neumann algebra is generally not continuous wrt. The strong operator topology how the example of the unilateral shift operator can show, because for all true , but what other than 0 does not converge to 0. Such a thing cannot happen in finite Von Neumann algebras. ${\displaystyle s\in L(\ell ^{2})}$ ${\displaystyle \xi =(\xi _{n})_{n}\in \ell ^{2}}$ ${\displaystyle \|{s^{n}}^{*}\xi \|=\|(\xi _{n+1},\xi _{n+2},\ldots )\|\rightarrow 0}$ ${\displaystyle \|s^{n}\xi \|=\|\xi \|}$ ${\displaystyle \xi }$

A Von Neumann algebra is finite if and only if the involution is strongly continuous on all bounded sets.

Individual evidence

1. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , 6.7.2 - 6.7.4
2. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Theorem 8.2.8
3. ^ Jacques Dixmier : Von Neumann algebras. North-Holland, Amsterdam 1981, ISBN 0-444-86308-7 , III.4 Existence and uniqueness theorems for operator traces
4. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , 9/6/11.
5. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , Corollary 5.4.13