W * -dynamic system

W * -dynamic systems are examined in the mathematical sub-area of functional analysis. It is a construction with which one obtains a new Von Neumann algebra from a Von Neumann algebra and a locally compact group that operates in a certain way on the Von Neumann algebra.

definition

A W * -dynamic system is a triple consisting of a Von Neumann algebra over a Hilbert space , a locally compact group and a homomorphism of into the group of * - automorphisms of , which is strongly continuous at points , i.e. all mappings are norm-continuous are. ${\displaystyle (A,G,\alpha )}$ ${\displaystyle A}$ ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle \alpha :G\rightarrow \mathrm {Aut} (A),s\mapsto \alpha _{s}}$ ${\displaystyle G}$ ${\displaystyle A}$ ${\displaystyle G\rightarrow H,s\mapsto \alpha _{s}(a)\xi ,\,a\in A,\xi \in H}$

You can replace the strong operator topology with the weak or ultra- weak operator topology and get the same term.

Construction of the cross product

For a W * -dynamic system we construct a Von Neumann algebra as follows . Here we reproduce the construction presented in. First we describe the Hilbert space on which the new Von Neumann algebra is supposed to operate. ${\displaystyle (A,G,\alpha )}$ ${\displaystyle A\ltimes _{\alpha }G}$

${\displaystyle A}$operate on the Hilbert space and let L ² (G) be the Hilbert space of the square-integrable functions with respect to the hair measure . The Tensor product of Hilbert spaces can with the space of measurable functions with identification. The mapping that assigns the function to an elementary tensor can become a unitary operator ${\displaystyle H}$ ${\displaystyle H\otimes L^{2}(G)}$ ${\displaystyle L^{2}(G,H)}$ ${\displaystyle \xi :G\rightarrow H}$ ${\displaystyle \textstyle \int _{G}\|\xi (s)\|^{2}\mathrm {d} s<\infty }$ ${\displaystyle \xi _{0}\otimes h}$ ${\displaystyle s\mapsto h(s)\xi _{0}}$

${\displaystyle H\otimes L^{2}(G)\rightarrow L^{2}(G,H)}$

be continued.

Now to the operators of the Von Neumann algebra to be defined. Since the space of continuous functions with compact support is close to , it is sufficient to state the effect of the operators on . For each we define the operator on through ${\displaystyle C_{c}(G,H)}$ ${\displaystyle G\rightarrow H}$ ${\displaystyle L^{2}(G,H)}$ ${\displaystyle C_{c}(G,H)}$ ${\displaystyle x\in A}$ ${\displaystyle \pi (x)}$ ${\displaystyle L^{2}(G,H)}$

${\displaystyle (\pi (x)f)(s):=\alpha _{s^{-1}}(x)f(s),\quad f\in C_{c}(G,H),s\in G}$

and for each the operator on through ${\displaystyle t\in G}$ ${\displaystyle \lambda (t)}$ ${\displaystyle L^{2}(G,H)}$

${\displaystyle (\lambda (t)f)(s):=f(t^{-1}s),\quad f\in C_{c}(G,H),s\in G}$.

Then there is a Hilbert space representation of and a group representation of on the Hilbert space and it applies ${\displaystyle \pi }$ ${\displaystyle A}$ ${\displaystyle \lambda }$ ${\displaystyle G}$ ${\displaystyle L^{2}(G,H)}$

${\displaystyle \lambda (t)\pi (x)\lambda (t)^{*}:=\pi (\alpha _{t}(x))}$for everyone . ${\displaystyle x\in A,t\in G}$

Therefore the linear hull of the operators is a partial algebra of , with respect to the involution, closed , of the restricted, linear operators on , whose weak closure is a Von Neumann algebra. This is called the Von Neumann Algebra of the W * -dynamic system or the cross product of and (fortune ) and is denoted by. Alternative names are , or . ${\displaystyle \pi (x)\lambda (t)}$ ${\displaystyle L(L^{2}(G,H))}$ ${\displaystyle L^{2}(G,H)}$ ${\displaystyle (A,G,\alpha )}$ ${\displaystyle A}$ ${\displaystyle G}$ ${\displaystyle \alpha }$ ${\displaystyle A\ltimes _{\alpha }G}$ ${\displaystyle A\otimes _{\alpha }G}$ ${\displaystyle A\times _{\alpha }G}$ ${\displaystyle W^{*}(A,G,\alpha )}$

If one observes the isomorphism given above , one can show that is contained in the tensor product . ${\displaystyle H\otimes L^{2}(G)\cong L^{2}(G,H)}$ ${\displaystyle A\ltimes _{\alpha }G}$ ${\displaystyle A{\overline {\otimes }}L(L^{2}(G))}$

duality

Be a commutative, locally compact group. Then there is the dual group of continuous group homomorphisms . With the topology of compact convergence, this is again a commutative, locally compact group. For such a group homomorphism we define the unitary operator auf by the formula ${\displaystyle G}$ ${\displaystyle {\hat {G}}}$ ${\displaystyle G\rightarrow \{z\in \mathbb {C} ;\,|z|=1\}}$ ${\displaystyle \chi \in {\hat {G}}}$ ${\displaystyle v_{\chi }}$ ${\displaystyle L^{2}(G)}$

${\displaystyle (v_{\chi }f)(s):={\overline {\chi (s)}}f(s),\quad f\in C_{c}(G),s\in G}$.

Then there is a unitary operator over and one can show that it holds, that is, that through ${\displaystyle 1_{H}\otimes v_{\chi }}$ ${\displaystyle H\otimes L^{2}(G)}$ ${\displaystyle (1\otimes v_{\chi })\,A\ltimes _{\alpha }G\,(1\otimes v_{\chi })^{*}=A\ltimes _{\alpha }G}$

${\displaystyle {\hat {\alpha }}_{\chi }(y):=(1\otimes v_{\chi })y(1\otimes v_{\chi })^{*}}$

an automorphism is defined which makes a W * -dynamic system. So you can build the cross product and show that it is isomorphic to . ${\displaystyle A\ltimes _{\alpha }G}$ ${\displaystyle (A\ltimes _{\alpha }G,{\hat {G}},{\hat {\alpha }})}$ ${\displaystyle (A\ltimes _{\alpha }G)\ltimes _{\hat {\alpha }}{\hat {G}}}$ ${\displaystyle A{\overline {\otimes }}L(L^{2}(G))}$

Applications

Construction of factors

Let it be a Borel space that is Borel isomorphic to [0,1] and a σ-finite measure on without atoms, that is, it is for everyone . We consider injective group homomorphisms ${\displaystyle T}$ ${\displaystyle \mu }$ ${\displaystyle T}$ ${\displaystyle \mu (\{t\})=0}$ ${\displaystyle t\in T}$

${\displaystyle \alpha ':G\rightarrow \mathrm {Iso} (T,\mu )}$

a discrete group in the group of Borel isomorphisms , so that the following applies: ${\displaystyle G}$ ${\displaystyle (T,\mu )}$

• From follows for everyone too . ${\displaystyle \mu (N)=0}$ ${\displaystyle g\in G}$ ${\displaystyle \mu (\alpha '_{g}(N))=0}$
• ${\displaystyle G}$operate freely , that is, for everyone different from the neutral element . ${\displaystyle (T,\mu )}$ ${\displaystyle \mu (\{t\in T;\,\alpha '_{g}(t)=t\})=0}$ ${\displaystyle g\in G}$
• ${\displaystyle G}$operate ergodically , that is, is with for a different from the neutral element , so is or . ${\displaystyle (T,\mu )}$ ${\displaystyle N\subset T}$ ${\displaystyle \mu (\alpha '_{g}(N)\setminus N)=0}$ ${\displaystyle g\in G}$ ${\displaystyle \mu (N)=0}$ ${\displaystyle \mu (T\setminus N)=0}$

A group homomorphism is obtained from ${\displaystyle \alpha ':G\rightarrow \mathrm {Iso} (T,\mu )}$

${\displaystyle \alpha :G\rightarrow \mathrm {Aut} (L^{\infty }(T,\mu )),\quad (\alpha _{g}(f))(t):=f(\alpha '_{g^{-1}}(t))}$

into the automorphism group of the Abelian Von Neumann algebra and one obtains a W * -dynamic system . Hence one can form the cross product . For this applies: ${\displaystyle L^{\infty }(T,\mu )}$ ${\displaystyle (L^{\infty }(T,\mu ),G,\alpha )}$ ${\displaystyle L^{\infty }(T,\mu )\ltimes _{\alpha }G}$

• Is now -invariant, ie for all measurable subsets , it is a type II factor , namely a type II ₁ factor if , and otherwise a type II _∞ factor. ${\displaystyle \mu }$ ${\displaystyle G}$ ${\displaystyle \mu (\alpha '_{g}(N))=\mu (N)}$ ${\displaystyle N\subset T}$ ${\displaystyle L^{\infty }(T,\mu )\ltimes _{\alpha }G}$ ${\displaystyle \mu (T)<\infty }$

• If it is not -invariant, but is invariant with regard to a subgroup of , which also operates ergodically , then it is a type III factor. ${\displaystyle \mu }$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle (T,\mu )}$ ${\displaystyle L^{\infty }(T,\mu )\ltimes _{\alpha }G}$

The following specific examples can be given for this:

Concrete examples

(i) Let be the circular line with the hair measure . It be and ${\displaystyle T=\mathbb {T} :=\{z\in \mathbb {C} ;\,|z|=1\}=\{e^{2\pi it};t\in [0,1]\}}$ ${\displaystyle \mu }$ ${\displaystyle G=\mathbb {Q} /\mathbb {Z} }$

${\displaystyle \alpha ':G\rightarrow \mathrm {Iso} (\mathbb {T} ,\mu ),\quad \alpha '_{q+\mathbb {Z} }(e^{2\pi it}):=e^{2\pi i(t+q)}}$

This fulfills the requirements of the above sentence, and it follows that a Type II ₁ factor is. ${\displaystyle L^{\infty }(\mathbb {T} ,\mu )\ltimes _{\alpha }\mathbb {Q} /\mathbb {Z} }$

(ii) Be with the Lebesgue measure . ${\displaystyle T=\mathbb {R} }$ ${\displaystyle \lambda }$

${\displaystyle \alpha ':\mathbb {Q} \rightarrow \mathrm {Iso} (\mathbb {R} ,\lambda ),\quad \alpha '_{q}(t):=t+q}$

This fulfills the requirements of the above theorem, and it follows that there is a type II _∞ -factor. ${\displaystyle L^{\infty }(\mathbb {R} ,\lambda )\ltimes _{\alpha }\mathbb {Q} }$

(iii) Let be with the Lebesgue measure and be the multiplicative matrix group . For be . Then meets the requirements of the above sentence, and it follows that is a type III factor. ${\displaystyle T=\mathbb {R} }$ ${\displaystyle \lambda }$ ${\displaystyle G}$ ${\displaystyle G=\{{\begin{pmatrix}a&b\\0&1\end{pmatrix}};\,a,b\in \mathbb {Q} \}}$ ${\displaystyle g={\begin{pmatrix}a&b\\0&1\end{pmatrix}}}$ ${\displaystyle \alpha '_{g}(t):=at+b}$ ${\displaystyle \alpha ':G\rightarrow \mathrm {Iso} (\mathbb {R} ,\lambda )}$ ${\displaystyle L^{\infty }(\mathbb {R} ,\lambda )\ltimes _{\alpha }G}$

The modular group

For σ-finite Von Neumann algebras , the Tomita-Takesaki theory yields a W * -dynamic system for every true, normal state . The dependence on the state is described by a so-called Connes cocycle , from which it follows that the cross products of the W * -dynamic systems are isomorphic to different states. One can therefore speak of the cross product with the modular group. ${\displaystyle A}$ ${\displaystyle (A,\mathbb {R} ,\sigma )}$ ${\displaystyle A\ltimes _{\sigma }\mathbb {R} }$

Duality plays an important role in Takesaki's theorem on the structure of Type III Von Neumann algebras . ${\displaystyle (A\ltimes _{\sigma }\mathbb {R} )\ltimes _{\hat {\sigma }}\mathbb {R} \cong A\otimes L(L^{2}(\mathbb {R} ))}$

See also

• C * -dynamic system

Individual evidence

1. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , 7.4.2
2. ^ A van Daele: Continuous crossed products and type III von Neumann algebras , Cambridge University Press (1978), ISBN 0-521-21975-2
3. ^ A van Daele: Continuous crossed products and type III von Neumann algebras , Cambridge University Press (1978), ISBN 0-521-21975-2 , Theorem 4.11
4. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , November 7 , 2016
5. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-1239-3302-1 , Theorem 8.6.10