C*-Dynamical Systems

Group Actions and C*-Dynamical Systems

Throughout, we assume is a countable group equipped with the discrete topology.

Recall the definition of a group action:

A set equipped with an action of is called a -set.

Link to original

If is a topological space, we say that acts continuously if the action map is continuous.

e.g.

• Let and . Consider an initial value problem given by for a sufficiently nice function . For any , let be the unique solution with initial condition . We can define an action of on by .

  • .
  • . The solution starting at is just a time-shift of the original solution, i.e., . Thus, .

• Let be a locally compact topological space and let be a homeomorphism. The group acts continuously on via for and .

A group action on a space induces a natural action on the C*-algebra of continuous functions on .

Proposition

Let be a group acting continuously on a locally compact space . This induces a map given by This map is a group homomorphism.

C*-Dynamical System

A C*-dynamical system is a triple , where is a C*-algebra, is a group, and is a homomorphism. The homomorphism is often required to be continuous in an appropriate sense.

Proposition

If is a C*-dynamical system, then there exists a continuous action of on that induces .

Proof Sketch For each , is an automorphism of . By the Gelfand-Naimark theorem, there exists a unique homeomorphism such that for all . The homomorphism property implies , defining a group action .

Covariant Representations

Covariant Representation

Let be a C*-dynamical system. A covariant representation of on a Hilbert space is a pair where:

1. is a -representation of on .
2. is a unitary representation of on .
3. They satisfy the covariance condition: for all and .

e.g. Let be the dynamical system where acts on itself by left multiplication, and is the action on functions by left translation: . Define a representation of on by multiplication operators: Let be the left regular representation of on : Then is a covariant representation.

The Regular Representation Construction

Given any C*-dynamical system and a representation , one can construct a covariant representation on the larger Hilbert space . It consists of functions such that . The inner product is .

1. Unitary Representation : Define by This is a unitary representation of .
2. Algebra Representation : Define by

Theorem

The pair constructed above is a covariant representation of .

Proof We must check that . For any and : The calculation follows from the definitions of , , and the homomorphism property of . Thus, the covariance condition holds.