C*-Dynamical Systems with Integer Actions

Crossed Product C*-Algebras

C*-Dynamical System

A C*-dynamical system is a triple , where is a C*-algebra, is a locally compact group, and is a strongly continuous group homomorphism. In these notes, we focus on the case where .

Crossed Product C*-Algebra

Given a C*-dynamical system , the crossed product C-algebra* is the universal C*-algebra generated by a copy of and a unitary satisfying the covariance relation: The set of finite sums with forms a dense *-subalgebra.

The Canonical Expectation

The Gauge Action

For each , there exists a unique *-automorphism of determined by:

• for all
• This family of automorphisms is called the gauge action.

The Canonical Expectation

Let be a C*-dynamical system. The map defined by the integral over the gauge action is a faithful conditional expectation from onto .

Proof We verify the properties of a faithful conditional expectation. Let be an element in the dense subalgebra of “polynomials”.

• Projection onto A: We compute on the generators. For , . For the unitary , By linearity, for , we have . Since the image is in for the dense subalgebra, by continuity it holds for all of .
• Expectation Property: For and ,
• Positivity: If is positive, then is positive for all . The integral of a continuous path of positive elements is positive, so .
• Faithfulness: Suppose for . The coefficient of is . Thus, . Since is a *-automorphism, each term is positive. A sum of positive elements is zero if and only if each element is zero. Thus, , which implies , and hence for all . So . The property extends to the whole algebra by density.

The Rotation Algebra

Definition

Let . Consider the C*-dynamical system where the action is rotation: The corresponding crossed product C*-algebra is called the rotation algebra.

Generators and Relations

The algebra is generated by the unitary position operator , where . The action on this generator is Let be the unitary implementing the action in the crossed product. The covariance relation becomes . This is equivalent to the commutation relation: Thus, can be described as the universal C*-algebra generated by two unitaries satisfying this relation.

Simplicity of the Rotation Algebra

Let be the rotation algebra for .

1. If is rational, then is not simple.
2. If is irrational, then is simple (it contains no non-trivial closed two-sided ideals).

Proof (Irrational case) Let be irrational and let be a non-zero closed two-sided ideal in .

1. Find a non-zero element in . Let be a non-zero element. Then is a non-zero positive element in . Let be the canonical expectation from the theorem above. Since is faithful, is a non-zero positive element in . It is a standard result of crossed product theory that if is a non-zero ideal, then is also a non-zero ideal of . Let .

2. Show is -invariant. Let . Then and . For any , we have . Since and is a two-sided ideal, it follows that . As is an automorphism of , is also in . Therefore, .

3. Use minimality of the action. is a non-zero, closed, -invariant ideal of . By Gelfand theory, any closed ideal of is of the form for some closed subset . Since is -invariant, the set must be invariant under the rotation . Because is irrational, the orbit of any point in under this rotation is dense in . If were non-empty, it would have to contain a dense subset, and being closed, it would have to be all of . But if , then , which contradicts that is non-zero. Thus, the only possibility for an invariant closed subset is or . Since , we must have . This implies that is the entire algebra .

4. Conclude . We have shown that if is a non-zero ideal, it must contain all of . Since is an ideal and , it must also contain all elements of the form for and . The linear span of these elements is the dense subalgebra of polynomials. Since is closed and contains a dense subalgebra of , we must have .

Thus, has no non-trivial closed two-sided ideals and is simple.