C * -dynamic system

C * -dynamic systems are examined in the mathematical sub-area of functional analysis. It is about a construction with which one obtains a new C * -algebra from a C * -algebra and a locally compact group that operates in a certain way on the C * -algebra . This construction generalizes the classical dynamic systems , in which the group of integers operates on a compact Hausdorff space . The prototype of a C * -dynamic system is the irrational rotational algebra .

definition

A C * -dynamic system is a triple consisting of a C * -algebra , a locally compact group and a homomorphism of in the group of * - automorphisms of , so that all mappings are continuous. (Morphisms on C * -algebras are always understood to be those that also receive the involution; one only writes , but * -automorphisms are meant.) ${\displaystyle (A,G,\alpha )}$ ${\displaystyle A}$ ${\displaystyle G}$ ${\displaystyle \alpha \colon G\rightarrow \mathrm {Aut} (A),s\mapsto \alpha _{s}}$ ${\displaystyle G}$ ${\displaystyle A}$ ${\displaystyle G\rightarrow A,s\mapsto \alpha _{s}(a),\,a\in A}$ ${\displaystyle \mathrm {Aut} (A)}$

The simplest and most important case for many applications is . Since the group is discrete , the continuity condition does not apply. Furthermore it is already determined by. A C * -dynamic system with a group is nothing more than a C * -algebra with an excellent automorphism. ${\displaystyle G=\mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \alpha }$ ${\displaystyle \alpha _{1}\in \mathrm {Aut} (A)}$ ${\displaystyle \mathbb {Z} }$

Covariant representations

As is well known, one can represent both C * -algebras and locally compact groups on Hilbert spaces . If a C * -dynamic system and a Hilbert space representation of and a unitary representation of on the same Hilbert space, then the pair is called a covariant representation, if ${\displaystyle (A,G,\alpha )}$ ${\displaystyle \pi \colon A\rightarrow B(H)}$ ${\displaystyle A}$ ${\displaystyle u\colon G\rightarrow B(H),s\mapsto u_{s}}$ ${\displaystyle G}$ ${\displaystyle (\pi ,u)}$

${\displaystyle \pi (\alpha _{s}(a))=u_{s}\pi (a)u_{s}^{*}}$for everyone and . ${\displaystyle a\in A}$ ${\displaystyle s\in G}$

By means of a covariant representation is therefore by -switched group operation of on by unitary operators shown. ${\displaystyle \alpha }$ ${\displaystyle G}$ ${\displaystyle A}$

The cross product

If a C * -dynamic system, one defines on the space of continuous functions with compact support for and : ${\displaystyle (A,G,\alpha )}$ ${\displaystyle K(A,G,\alpha )}$ ${\displaystyle G\rightarrow A}$ ${\displaystyle x,y\in K(A,G,\alpha )}$ ${\displaystyle \beta \in \mathbb {C} }$

• ${\displaystyle (\beta x)(t):=\beta x(t)}$
• ${\displaystyle (x+y)(t):=x(t)+y(t)}$
• ${\displaystyle (x\star y)(t):=\int _{G}x(s)\alpha _{s}(y(s^{-1}t))\,\mathrm {d} \mu (s)}$
• ${\displaystyle (x^{*})(t):=\Delta (t)^{-1}\alpha _{t}(x(t^{-1})^{*})}$
• ${\displaystyle \|x\|_{1}:=\int _{G}\|x(s)\|\,\mathrm {d} \mu (s)}$

Here , a left- haired measure of and the modular function of . One calculates that these definitions result in a normalized algebra with isometric involution . The product that depends on is called the cross product . The completion is then a Banach - * - algebra , which is denoted by. ${\displaystyle t\in G}$ ${\displaystyle \mu }$ ${\displaystyle G}$ ${\displaystyle \Delta }$ ${\displaystyle G}$ ${\displaystyle K(A,G,\alpha )}$ ${\displaystyle \alpha }$ ${\displaystyle \star }$ ${\displaystyle L^{1}(A,G,\alpha )}$

If there is a covariant representation of the C * -dynamic system on a Hilbert space , then becomes through ${\displaystyle (\pi ,u)}$ ${\displaystyle (A,G,\alpha )}$ ${\displaystyle H}$

${\displaystyle (\pi \times u)(x):=\int _{G}\pi (x(t))u_{t}\,\mathrm {d} \mu (t),\quad x\in L^{1}(A,G,\alpha )}$

a non-degenerate Hilbert space representation of defined. Conversely, if a non-degenerate Hilbert space representation is given, then there is exactly one covariant representation of the C * -dynamic system, so that the given * -representation results from the above formula. Knowledge of all covariant representations of the C * -dynamic system therefore corresponds to knowledge of all non-degenerate * -representations of the associated -algebra. ${\displaystyle L^{1}(A,G,\alpha )}$ ${\displaystyle L^{1}(A,G,\alpha )}$ ${\displaystyle L^{1}}$

The enveloping C * -algebra of is denoted by or and is called the cross product of the C * -dynamic system. The covariant representations of a C * -dynamic system thus lead to non-degenerate Hilbert space representations of and vice versa. ${\displaystyle L^{1}(A,G,\alpha )}$ ${\displaystyle C^{*}(A,G,\alpha )}$ ${\displaystyle A\ltimes _{\alpha }G}$ ${\displaystyle A\ltimes _{\alpha }G}$

If special , then every locally compact group operates trivially on , that is, for all , and the above construction yields the group C * -algebra . The construction of the cross product therefore generalizes the construction of the group C * algebra. ${\displaystyle A=\mathbb {C} }$ ${\displaystyle G}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \alpha _{s}=\mathrm {id} _{\mathbb {C} }}$ ${\displaystyle s\in G}$ ${\displaystyle C^{*}(G)}$

The reduced cross product

As in the case of group C * -algebras, one also considers left-regular representations for C * -dynamic systems , but here one obtains one for every given Hilbert space representation of . ${\displaystyle (A,G,\alpha )}$ ${\displaystyle A}$

If there is a Hilbert space representation of , one constructs a covariant representation on the Hilbert space of all measurable functions with the following formulas: ${\displaystyle \pi \colon A\rightarrow B(H)}$ ${\displaystyle A}$ ${\displaystyle ({\tilde {\pi }},\lambda )}$ ${\displaystyle L^{2}(G,H)}$ ${\displaystyle \xi \colon G\rightarrow H}$ ${\displaystyle \textstyle \int _{G}\|\xi (t)\|^{2}\,\mathrm {d} \mu (s)<\infty }$

• ${\displaystyle ({\tilde {\pi }}(a)\xi )(t)=\pi (\alpha _{t^{-1}}(a))(\xi (t))}$
• ${\displaystyle (\lambda _{s}\xi )(t)=\xi (s^{-1}t)}$,

where , and . One calculates that this actually defines a covariant representation. If the universal representation of is special , then the norm closure of in is called the reduced cross product of the C * -dynamic system; this is denoted by or . ${\displaystyle a\in A}$ ${\displaystyle s,t\in G}$ ${\displaystyle \xi \in L^{2}(G,H)}$ ${\displaystyle \pi _{u}\colon A\rightarrow H_{u}}$ ${\displaystyle A}$ ${\displaystyle ({\tilde {\pi _{u}}}\times \lambda )(L^{1}(A,G,\alpha ))}$ ${\displaystyle B(L^{2}(G,H_{u}))}$ ${\displaystyle C_{r}^{*}(A,G,\alpha )}$ ${\displaystyle A\ltimes _{\alpha r}G}$

If we look again at the special case with the trivial operation of the group , the construction of the reduced cross product yields exactly the reduced group C * algebra . ${\displaystyle A=\mathbb {C} }$ ${\displaystyle G}$

Since the covariant representation leads to a * representation of the cross product , a surjective homomorphism is obtained , which is also called the left-regular representation. As in the case of group C * algebras, the following theorem applies: ${\displaystyle ({\tilde {\pi }},\lambda )}$ ${\displaystyle A\ltimes _{\alpha }G}$ ${\displaystyle A\ltimes _{\alpha }G\rightarrow A\ltimes _{\alpha r}G}$

If a C * -dynamic system with an indirect group , the left-regular representation is an isomorphism. ${\displaystyle (A,G,\alpha )}$ ${\displaystyle G}$ ${\displaystyle A\ltimes _{\alpha }G\rightarrow A\ltimes _{\alpha r}G}$

Especially for compact and for Abelian groups (important special case ) one does not have to differentiate between and , because these groups are indirect. ${\displaystyle \mathbb {Z} }$ ${\displaystyle A\ltimes _{\alpha }G}$ ${\displaystyle A\ltimes _{\alpha r}G}$

Classic dynamic systems

Classical dynamic systems are group operations on a compact Hausdorff space . More precisely, a homeomorphism is given and this defines the group operation . also defines an automorphism on the C * algebra of continuous functions , the on maps. This results in a C * dynamic system , where . Relationships between the classical dynamic system and the C * -algebra can then be established. The prototype of this construction is the irrational rotational algebra . ${\displaystyle \mathbb {Z} }$ ${\displaystyle X}$ ${\displaystyle \sigma \colon X\rightarrow X}$ ${\displaystyle \mathbb {Z} \times X\rightarrow X,(n,x)\mapsto \sigma ^{n}(x)}$ ${\displaystyle \sigma }$ ${\displaystyle C(X)}$ ${\displaystyle X\rightarrow \mathbb {C} }$ ${\displaystyle f\in C(X)}$ ${\displaystyle f\circ \sigma ^{-1}}$ ${\displaystyle (C(X),\mathbb {Z} ,\alpha )}$ ${\displaystyle \alpha _{n}(f)=f\circ \sigma ^{-n}}$ ${\displaystyle (X,\sigma )}$ ${\displaystyle C(X)\ltimes _{\alpha }\mathbb {Z} }$

See also

• Takai duality
• W * -dynamic system

Individual evidence

1. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , 7.4.1
2. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , 7.4.8
3. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , 7.6.1
4. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , Theorem 7.6.4
5. ↑ Thomas Skill: Toeplitz quantization of symmetrical areas based on the C * -Duality , Teubner-Verlag (2011), ISBN 3-834-81541-1 , chap. 4.1: Group C * algebras and cross products of C * algebras
6. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , 7.6.5
7. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , 7.7.4
8. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , Theorem 7.7.7
9. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-821-80599-1 , Chapter VIII.3