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.)
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.
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
for everyone and .
By means of a covariant representation is therefore by -switched group operation of on by unitary operators shown.
The cross product
If a C * -dynamic system, one defines on the space of continuous functions with compact support for and :
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.
If there is a covariant representation of the C * -dynamic system on a Hilbert space , then becomes through
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.
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.
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.
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 .
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:
- ,
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 .
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 .
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:
- If a C * -dynamic system with an indirect group , the left-regular representation is an isomorphism.
Especially for compact and for Abelian groups (important special case ) one does not have to differentiate between and , because these groups are indirect.
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 .
See also
Individual evidence
- ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , 7.4.1
- ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , 7.4.8
- ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , 7.6.1
- ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , Theorem 7.6.4
- ↑ 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
- ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , 7.6.5
- ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , 7.7.4
- ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , Theorem 7.7.7
- ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-821-80599-1 , Chapter VIII.3