Crossed Products of C*-Algebras

The Full Crossed Product Algebra

The Algebra of Test Functions

Let be a C*-dynamical system. We define the vector space of -valued functions on with finite support as: We equip with a convolution product and an involution, making it a -algebra.

• Convolution: For , the product is defined as:
• Involution: The involution is defined as:

An element can be formally written as a sum , where is the function that is at and elsewhere.

Integrated Form of a Covariant Representation

Given a covariant representation of on a Hilbert space , we can define a map by:

Proposition

The map is a -representation of the algebra .

The Full Crossed Product

We define the universal norm on as: where the supremum is taken over all covariant representations of . This norm is bounded by the -norm:

The (full) crossed product C*-algebra, denoted , is the completion of with respect to the universal norm .

Universal Property

Canonical Maps

There exist canonical maps from and into the (multiplier algebra of) .

1. The map defined by is an isometric *-homomorphism.
2. The map defined by is an injective group homomorphism into the group of unitaries.

Proof Sketch For the isometry of , we first note that for any covariant representation : This implies . To show the reverse inequality, one can construct a specific covariant representation using the universal representation of and the left regular representation of on . For this representation, one finds that . Therefore, . For the map :

• It is a homomorphism since and .
• To show injectivity, we can use the same covariant representation on . We have . Since the left regular representation is injective, must also be injective.

Universal Property of the Crossed Product

For every covariant representation of on , there exists a unique -representation of the crossed product, which we also denote by , from to such that:

• for all .
• for all .

Conversely, every non-degenerate representation of arises in this way from a covariant representation of .

The Reduced Crossed Product

Reduced vs. Full

We can also define a reduced norm by taking the supremum only over regular covariant representations: The completion of with respect to this norm is the reduced crossed product, denoted .

In general, there is a surjective *-homomorphism . However, this map may not be injective.

Examples

Trivial Group Action

If the group is trivial, , and the action is the identity, then .

Group C*-Algebras

Let the C*-algebra be the complex numbers, , and the action be the trivial action, . The crossed product is called the group C-algebra* of G, denoted .

• In this case, is just the space of complex-valued functions with finite support, .
• The convolution product becomes .
• The involution is .
• is the universal C*-algebra generated by a set of unitaries satisfying the group relations .

Abelian Groups

Consider the group C*-algebra where is an abelian group.

• The algebra is abelian.
• Since is an abelian, unital C*-algebra, the Gelfand-Naimark theorem states that there exists a compact Hausdorff space such that .

This space can be identified with the Pontryagin dual of G, denoted .

Character Group (Pontryagin Dual)

A character of an abelian group is a group homomorphism . The set of all characters is denoted by . When endowed with the topology of pointwise convergence, becomes a compact Hausdorff space.

Proposition

For an abelian group , we have the isomorphism .

Proof Sketch The proof involves showing a correspondence between the character space (maximal ideal space) of the C*-algebra and the character group .

• Any character gives rise to a multiplicative linear functional on by setting . This extends to a character on the full C*-algebra .
• Conversely, any character on gives a group character on G via .
• The topology of pointwise convergence on corresponds to the weak-* topology on the character space of .

Examples of Dual Groups:

• If , its characters are of the form for . Thus, . (In general, for any finite abelian group, ).
• If , any character is determined by . Thus, .

Free Groups

Consider the free group on two generators, .

• The group C*-algebra is the universal C*-algebra generated by two unitaries .
• This contrasts with , which is the universal C*-algebra generated by two commuting unitaries (since is abelian).

Proposition

The C*-algebra has a family of finite-dimensional representations such that their direct sum is an isometric representation.

Proof Sketch The proof relies on approximating the universal representation.

1. By the Gelfand-Naimark theorem, has a faithful (isometric) universal representation on a separable Hilbert space . Let and be the unitaries corresponding to the generators.
2. Let be an orthonormal basis for and let be the projection onto the span of .
3. Define compressed operators and . These are not necessarily unitary.
4. “Purify” these contractions to unitaries on a larger space. Define unitaries and on the space using and . For example:
5. This defines a sequence of finite-dimensional representations by setting and .
6. One can then show that for any element , the norm converges to . This implies that the direct sum representation is isometric: This isometry extends from to all of .