Type III Von Neumann Algebra

Type III Von Neumann algebras are special algebras considered in the mathematical theory of Von Neumann algebras. It is the third of three types of type classification of Von Neumann algebras . These can be constructed from Type II Von Neumann algebras according to a theorem by M. Takesaki .

Definitions

A projection in a Von Neumann algebra is a self-adjoint idempotent element , that is, it holds . Such a projection is called finite if it follows from and always . A Von Neumann algebra is called of type III if it does not have any finite projections other than 0. ${\displaystyle A}$ ${\displaystyle e}$ ${\displaystyle e=e^{*}=e^{2}}$ ${\displaystyle e=v^{*}v}$ ${\displaystyle vv^{*}\leq e}$ ${\displaystyle vv^{*}=e}$

Examples

In the article on W * -dynamic systems , a construction is described that leads to type III Von Neumann algebras. The Connes classification of type III factors described below provides further examples.

Theorem of Takesaki

Takesaki's theorem reduces the Type III Von Neumann algebras to Type II _∞ algebras:

For every type III Von Neumann algebra there is a W * -dynamic system , where a type II _∞ -algebra is such that . ${\displaystyle A}$ ${\displaystyle (A_{1},\mathbb {R} ,\alpha )}$ ${\displaystyle A_{1}}$ ${\displaystyle A\cong A_{1}\ltimes _{\alpha }\mathbb {R} }$

To do this, one uses the W * -dynamic system , which results from the Tomita-Takesaki theory , and forms the Type II _∞ algebra . Then follows with the dual W * -dynamic system ${\displaystyle (A,\mathbb {R} ,\sigma )}$ ${\displaystyle A_{1}:=A\ltimes _{\sigma }\mathbb {R} }$ ${\displaystyle (A_{1},\mathbb {R} ,{\hat {\sigma }})}$

${\displaystyle A_{1}\ltimes _{\hat {\sigma }}\mathbb {R} =(A\ltimes _{\sigma }\mathbb {R} )\ltimes _{\hat {\sigma }}\mathbb {R} \cong A\otimes L(L^{2}(\mathbb {R} ))}$     because of duality

${\displaystyle \cong A}$, since it is a Type III Von Neumann algebra. ${\displaystyle A}$

Connes Classification of Type III Factors

For a type III factor, that is to say a type III Von Neumann algebra with a center , we construct an isomorphism invariant number , which then leads to the concept of the type III _λ factor. ${\displaystyle \mathbb {C} \cdot 1}$ ${\displaystyle \lambda \in [0,1]}$

Let be a normal state on the Von Neumann algebra . Then there is a smallest projection with . Then there is a Von Neumann algebra and the constraint of is a faithful, normal state to which the Tomita-Takesaki theory can be applied, that is, there is a modular operator . Since this is a positive operator, its spectrum is in . One defines ${\displaystyle \varphi }$ ${\displaystyle A}$ ${\displaystyle p\in A}$ ${\displaystyle \varphi (p)=1}$ ${\displaystyle pAp}$ ${\displaystyle \varphi }$ ${\displaystyle \Delta _{\varphi }}$ ${\displaystyle \sigma (\Delta _{\varphi })}$ ${\displaystyle \mathbb {R} _{0}^{+}}$

${\displaystyle S(A):=\bigcap \{\sigma (\Delta _{\varphi });\,\varphi {\text{ normaler Zustand auf }}A\}\subset \mathbb {R} _{0}^{+}}$.

One can show that 0 lies in if and only if is of type III, otherwise holds . For σ-finite factors there is exactly one of the following three cases: ${\displaystyle S(A)}$ ${\displaystyle A}$ ${\displaystyle S(A)=\{1\}}$

• ${\displaystyle S(A)\setminus \{0\}=\{1\}}$
• ${\displaystyle S(A)\setminus \{0\}=\{\lambda ^{n};\,n\in \mathbb {Z} \}}$ for a ${\displaystyle 0<\lambda <1}$
• ${\displaystyle S(A)\setminus \{0\}=\mathbb {R} ^{+}}$

In the first case one calls a type III ₀ factor, in the second case a type III _λ factor and in the third case a type III ₁ factor. This is Connes' classification of Type III factors. ${\displaystyle A}$

If different, a type III _λ -factor is not isomorphic to a type III _µ -factor, because the set is an isomorphism invariant. So there is a continuum of pairwise non-isomorphic Type III factors. ${\displaystyle \lambda ,\mu \in [0,1]}$ ${\displaystyle S(A)}$

Let us briefly discuss the existence of Type III _λ factors. To do this, we construct a state on the CAR algebra . You can recursively define states for one, where ${\displaystyle \varphi _{\mu }}$ ${\displaystyle \mu \in [0,1]}$ ${\displaystyle \varphi _{\mu }^{(n)}:M_{2^{n}}\rightarrow \mathbb {C} }$

• ${\displaystyle \varphi _{\mu }^{(0)}:M_{2^{0}}\cong \mathbb {C} \rightarrow \mathbb {C} }$ let the identical mapping be and
• ${\displaystyle \varphi _{\mu }^{(n)}(x)=\mu \varphi _{\mu }^{(n-1)}(x_{1,1})+(1-\mu )\varphi _{\mu }^{(n-1)}(x_{2,2})}$for each , where as -Matrix with elements from is written. ${\displaystyle n>0}$ ${\displaystyle x=(x_{i,j})}$ ${\displaystyle 2\times 2}$ ${\displaystyle M_{2^{n-1}}}$

Then the restriction from to is equal , because according to the embedding ${\displaystyle \varphi _{\mu }^{(n)}}$ ${\displaystyle M_{2^{n-1}}}$ ${\displaystyle \varphi _{\mu }^{(n-1)}}$

${\displaystyle M_{2^{n-1}}\ni x\mapsto {\begin{pmatrix}x&0\\0&x\end{pmatrix}}\in M_{2^{n}}}$

is

${\displaystyle (\varphi _{\mu }^{(n)}|_{M_{2^{n-1}}})(x)=\varphi _{\mu }^{(n)}({\begin{pmatrix}x&0\\0&x\end{pmatrix}})=\mu \varphi _{\mu }^{(n-1)}(x)+(1-\mu )\varphi _{\mu }^{(n-1)}(x)=\varphi _{\mu }^{(n-1)}(x)}$.

Hence there is a unique state on CAR algebra that is consistent with on all . A Hilbert space representation on a Hilbert space belongs to the state by means of a GNS construction . For the image is a C * -algebra, the closure of which in the weak operator topology is a factor of type III _λ , where . ${\displaystyle \varphi _{\mu }}$ ${\displaystyle M_{2^{n}}}$ ${\displaystyle \varphi _{\mu }^{(n)}}$ ${\displaystyle \varphi _{\mu }}$ ${\displaystyle \pi _{\mu }:A\rightarrow L(H_{\mu })}$ ${\displaystyle H_{\mu }}$ ${\displaystyle 0<\mu <{\frac {1}{2}}}$ ${\displaystyle \pi _{\mu }(A)\subset L(H_{\mu })}$ ${\displaystyle \textstyle \lambda ={\frac {\mu }{1-\mu }}}$

See also

• Type I Von Neumann algebra
• Type II Von Neumann algebra

Individual evidence

1. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , definition 6.5.1
2. ^ A van Daele: Continuous crossed products and type III von Neumann algebras , Cambridge University Press (1978), ISBN 0-521-21975-2 , Theorem II.4.8
3. ^ A van Daele: Continuous crossed products and type III von Neumann algebras , Cambridge University Press (1978), ISBN 0-521-21975-2 , Appendix C.
4. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , Theorem 8.15.5
5. ↑ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , Theorem 8.15.7 + 8.15.11
6. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , Theorem 8.15.13