Type classification (Von Neumann algebra)

The type classification presented here divides the Von Neumann algebras examined in mathematics into classes, which are called types. This classification , which goes back to Francis J. Murray and John von Neumann , is based on an analysis of the structure of the orthogonal projections contained in a Von Neumann algebra . While any Von Neumann algebras can have components of different types, a factor is always of exactly one type. Therefore, these concept formations play an important role in examining the factors.

motivation

The basic idea is to compare projections of a Von Neumann algebra on a Hilbert space according to size. If such a projection (projection is always meant here as an orthogonal projection), then the projected space belongs to it and, conversely, there is exactly one projection onto this subspace for every closed subspace. It therefore makes sense to use the quantities to compare sizes. Applies to two projections and , one will want to call it the larger one. As with general sets, it may happen that two projections are not directly comparable with one another in this way, as there is no inclusion relationship between the projected spaces. In the case of two sets, comparability can be established by mapping one of the sets bijectively onto a subset of the other. If one pursues this analogy between sets and projections further, and this view proves to be very fruitful, one easily comes to the following concept formation: ${\displaystyle A\subset L(H)}$ ${\displaystyle H}$ ${\displaystyle e\in A}$ ${\displaystyle e(H)}$ ${\displaystyle H}$ ${\displaystyle e(H)}$ ${\displaystyle e(H)\subset f(H)}$ ${\displaystyle e}$ ${\displaystyle f}$ ${\displaystyle f}$

Definition: Two projections are called equivalent , in signs if there is one with and , then such is a partial isometry . They say be weaker than , in signs , if there is a projection with and . ${\displaystyle e,f\in A}$ ${\displaystyle e\,\sim \,f}$ ${\displaystyle u\in A}$ ${\displaystyle e=u^{*}u}$ ${\displaystyle f=uu^{*}}$ ${\displaystyle u}$ ${\displaystyle e}$ ${\displaystyle f}$ ${\displaystyle e\preceq f}$ ${\displaystyle e_{0}\in A}$ ${\displaystyle e\,\sim \,e_{0}}$ ${\displaystyle e_{0}(H)\subset f(H)}$

Equivalence and comparability depend on the Von Neumann algebra , because it is required that the partial isometry of the above definition is also in . In a commutative Von Neumann algebra , equivalent projections are the same (because from and follows because of commutativity ), in the larger Von Neumann algebra this is not the case. ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A\subset L(H)}$ ${\displaystyle e=u^{*}u}$ ${\displaystyle f=uu^{*}}$ ${\displaystyle e=f}$ ${\displaystyle L(H)}$

One can show that is an equivalence relation and induces a partial order on the set of equivalence classes. In particular , if and what is the more difficult part of the proof. ${\displaystyle \sim }$ ${\displaystyle \preceq }$ ${\displaystyle e\,\sim \,f}$ ${\displaystyle e\preceq f}$ ${\displaystyle f\preceq e}$

Projections in Von Neumann Algebras

Projections in a Von Neumann algebra can have a number of properties:

• A projection is called central if it is in the center of . Here the commutant of . ${\displaystyle A\cap A'}$ ${\displaystyle A}$ ${\displaystyle A'}$ ${\displaystyle A}$
• A variety of projection 0 is minimal if for each projection with either or true. ${\displaystyle e\in A}$ ${\displaystyle f\in A}$ ${\displaystyle f(H)\subset e(H)}$ ${\displaystyle f=0}$ ${\displaystyle f=e}$
• A projection is called finite if for every projection with and already follows. Note the analogy to set theory: A set is finite if and only if it is not of equal power to a real subset . Minimal projections are finite and, in analogy to set theory, these correspond to the one-element sets. ${\displaystyle e\in A}$ ${\displaystyle f\in A}$ ${\displaystyle e\,\sim \,f}$ ${\displaystyle f(H)\subset e(H)}$ ${\displaystyle e\,=\,f}$
• Non-finite projections are called infinite . A projection is really infinite if for every central projection either or is infinite. ${\displaystyle e\in A}$ ${\displaystyle c\in A}$ ${\displaystyle ce=0}$ ${\displaystyle ce}$
• A projection is pure infinity if for any finite projection with already follows. Purely infinite projections are really infinite. ${\displaystyle e\in A}$ ${\displaystyle f\in A}$ ${\displaystyle f(H)\subset e(H)}$ ${\displaystyle f=0}$
• A projection is Abelian if an Abelian von Neumann algebra is. To note that with usually the algebra of all operators , is referred to, what is always one again Von Neumann algebra. Abelian projections are finite. ${\displaystyle e\in A}$ ${\displaystyle eAe}$ ${\displaystyle e(H)}$ ${\displaystyle eAe}$ ${\displaystyle ea|_{e(H)}:e(H)\rightarrow e(H)}$ ${\displaystyle a\in A}$
• At any projection there is a smallest central projection with , that is for any other central projection with true . This projection is called the central support of and is denoted by. ${\displaystyle e\in A}$ ${\displaystyle p\in A}$ ${\displaystyle e=ep}$ ${\displaystyle q\in A}$ ${\displaystyle e=eq}$ ${\displaystyle p(H)\subset q(H)}$ ${\displaystyle p}$ ${\displaystyle e}$ ${\displaystyle C_{e}}$

Accordingly, a Von Neumann algebra is called finite , infinite , genuinely infinite , or purely infinite , if these properties apply to the one element . The same relation obviously holds for the property abelian : A Von Neumann algebra is abelian (that is, commutative) if and only if is an abelian projection. ${\displaystyle 1={\mbox{id}}_{H}\in A}$ ${\displaystyle 1\in A}$

Comparability rate

Any two projections need not be comparable. However, the Von Neumann algebra can be broken down into a direct sum of three Von Neumann algebras, so that there is comparability in every summand. The following sentence applies:

Theorem of comparability : Let projections in Von Neumann algebra be . Then there are clearly certain, pairwise orthogonal, central projections with , so that the following applies: ${\displaystyle e,f\in A}$ ${\displaystyle A}$ ${\displaystyle p_{0},p_{1},p_{2}\in A}$ ${\displaystyle p_{0}+p_{1}+p_{2}=1}$

• ${\displaystyle p_{0}e\,\sim \,p_{0}f}$.
• If a central projection is included , then applies . ${\displaystyle r\in A}$ ${\displaystyle \{0\}\not =r(H)\subset p_{1}(H)}$ ${\displaystyle re\prec rf}$
• If a central projection is included , then applies . ${\displaystyle r\in A}$ ${\displaystyle \{0\}\not =r(H)\subset p_{2}(H)}$ ${\displaystyle rf\prec re}$

The abbreviation stands for " , and " and two projections are called orthogonal (to each other) if their product is 0. ${\displaystyle \prec }$ ${\displaystyle \preceq }$ ${\displaystyle \not \sim }$

Type I, Type II, Type III

→ Main article : Type I Von Neumann algebra

• A Von Neumann algebra is the type I (read: type one) when an Abelian projection with there. ${\displaystyle A}$ ${\displaystyle e\in A}$ ${\displaystyle C_{e}=1}$
• ${\displaystyle A}$is called more precisely of type I _n , where if is of type I and is the sum of pairwise equivalent Abelian projections. ${\displaystyle n\in \mathbb {N} \cup \{\infty \}}$ ${\displaystyle A}$ ${\displaystyle 1\in A}$ ${\displaystyle n}$

→ Main article : Type II Von Neumann algebra

• A Von Neumann algebra is called of type II if it does not have any Abelian projections other than 0, but has a finite projection with . ${\displaystyle A\not =\{0\}}$ ${\displaystyle e\in A}$ ${\displaystyle C_{e}=1}$
• A Von Neumann algebra of type II is called of type II ₁ if is a finite projection. ${\displaystyle A}$ ${\displaystyle 1\in A}$
• A Von Neumann algebra of type II is called of type II _∞ if there is a truly infinite projection. ${\displaystyle A}$ ${\displaystyle 1\in A}$

→ Main article : Type III Von Neumann algebra

• A Von Neumann algebra is called type III if it does not have any finite projections other than 0. ${\displaystyle A\not =\{0\}}$

The conditions for the above type classification are laid out in such a way that a Von Neumann algebra can at most be of one type, but there are Von Neumann algebras that are of no type in the above sense. The following theorem shows that any Von Neumann algebra can be uniquely decomposed into a direct sum, so that all summands have a type:

Theorem of the type decomposition: Let be a Von Neumann algebra. Then there are unique, mutually orthogonal central projections , , and , such that with Total 1: ${\displaystyle A}$ ${\displaystyle p_{n},n\in \mathbb {N} \cup \{\infty \}}$ ${\displaystyle p_{c_{1}}}$ ${\displaystyle p_{c_{\infty }}}$ ${\displaystyle q_{\infty }}$

• ${\displaystyle Ap_{n}\,=\,p_{n}Ap_{n}}$is of type I _n or 0.
• ${\displaystyle Ap_{c_{1}}=p_{c_{1}}Ap_{c_{1}}}$is of type II ₁ or 0.
• ${\displaystyle Ap_{c_{\infty }}=p_{c_{\infty }}Ap_{c_{\infty }}}$is of type II _∞ or 0.
• ${\displaystyle Aq_{\infty }=q_{\infty }Aq_{\infty }}$is of type III or 0.

It is a direct sum of Von Neumann algebras. ${\displaystyle A=Ap_{1}\oplus \ldots \oplus Ap_{\infty }\oplus Ap_{c_{1}}\oplus Ap_{c_{\infty }}\oplus Aq_{\infty }}$

Many of these projections can of course be 0, in which case they have no corresponding type component. is a Von Neumann algebra of type I _∞ . Type I Von Neumann algebras are sometimes called discrete because they are a direct sum ; the summation index runs through a discrete set. Examples of Von Neumann algebras of type II or III are more complex, they can be obtained, among other things, by suitable group constructions or as factors that are generated by representations of UHF algebras , especially the CAR algebra . In the article on W * -dynamic systems , a weight-theoretical construction of Type II and Type III Von Neumann algebras is presented. ${\displaystyle A}$ ${\displaystyle L(\ell ^{2})}$ ${\displaystyle A}$ ${\displaystyle A=Ap_{1}\oplus \ldots \oplus Ap_{\infty }}$

Von Neumann algebras of type II are also called continuous . Therefore, the terms and were chosen in the above sentence (c stands for continuous). Some authors also consider type III algebras to be continuous. Type III algebras are purely infinite. ${\displaystyle p_{c_{1}}}$ ${\displaystyle p_{c_{\infty }}}$

A Von Neumann algebra without a type III component (that is, in the above sentence) is called semiendlich . ${\displaystyle q_{\infty }=0}$

The article Tensor Product for Von Neumann Algebras explains how the type classification presented here behaves in the formation of tensor products.

Factors, dimension function

Since a factor does not contain any other central projections apart from 0 and 1, a factor always has exactly one well-defined type. Type III factors can be further classified; For each one can define factors according to the Connes classification type III _λ , which goes back to Alain Connes , which will not be discussed further here. There are factors for each type, even on separable Hilbert spaces. ${\displaystyle \lambda \in [0,1]}$

It immediately follows from the comparability principle that in one factor two projections are comparable. The minimal projections coincide with the Abelian projections. If one only considers factors on separable Hilbert spaces and the set of projections is in , then one can describe the types via the order structure of . The following sentence applies: ${\displaystyle \preceq }$ ${\displaystyle A}$ ${\displaystyle {\mathcal {P}}(A)}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {P}}(A)/\sim }$

Theorem (dimension function): If a factor is on a separable Hilbert space, there is a function with the following properties: ${\displaystyle A}$ ${\displaystyle D:{\mathcal {P}}(A)\rightarrow [0,\infty ]}$

• for true ${\displaystyle e,f\in {\mathcal {P}}(A)}$ ${\displaystyle e\sim f\Leftrightarrow D(e)=D(f)}$
• for true ${\displaystyle e,f\in {\mathcal {P}}(A)}$ ${\displaystyle e\preceq f\Leftrightarrow D(e)\leq D(f)}$
• The following applies to two mutually orthogonal projections . ${\displaystyle e,f\in {\mathcal {P}}(A)}$ ${\displaystyle D(e+f)\,=\,D(e)+D(f)}$
• For is: finite . ${\displaystyle e\in {\mathcal {P}}(A)}$ ${\displaystyle e}$ ${\displaystyle \Leftrightarrow D(e)<\infty }$

The function is uniquely determined except for a constant factor and is called the dimension function . Except for a scaling factor, the image is one of the following quantities: ${\displaystyle D}$ ${\displaystyle D({\mathcal {P}}(A))}$

• ${\displaystyle \{0,1,\ldots n\}}$for a ; is then of type I _n . ${\displaystyle n\in N}$ ${\displaystyle A}$
• ${\displaystyle \{0,1,\ldots \}\cup \{\infty \}}$; is then of type I _∞ . ${\displaystyle A}$
• ${\displaystyle [0,\,1]}$; is then of type II ₁ . ${\displaystyle A}$
• ${\displaystyle [0,\infty ]}$; is then of type II _∞ . ${\displaystyle A}$
• ${\displaystyle \{0,\infty \}}$; is then of type III . ${\displaystyle A}$

For the type I _∞ factor one obtains with the specified scaling for all . That explains the name dimension function . ${\displaystyle A=L(\ell ^{2})}$ ${\displaystyle D(e)={\mbox{dim}}\,e(\ell ^{2})}$ ${\displaystyle e\in {\mathcal {P}}(A)}$

Note that an orderly bijection induces. The type of a factor is therefore determined according to the above sentence by the order structure of . ${\displaystyle D}$ ${\displaystyle ({\mathcal {P}}(A)/\sim )\rightarrow D({\mathcal {P}}(A))}$ ${\displaystyle {\mathcal {P}}(A)/\sim }$

swell

• RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press Inc. 1986
• VS Sunder: An Invitation to Von Neumann Algebras (1987), ISBN 0387963561