C*-Algebras

C*-Algebras: Definitions and Examples

-Algebra & -Homomorphism

A -algebra is an algebra over equipped with an involution such that , , , for all and . A -homomorphism between -algebras is an algebra homomorphism also preserving the involution. A bijective -homomorphism is called a -isomorphism.

Banach -Algebra & C*-Algebra

A Banach -algebra is a Banach algebra that is also a -algebra, with the involution such that for all . If is unital, we also require . A C*-algebra is a Banach -algebra that satisfies the C*-identity:

Remark

If a C*-algebra has a unit , then . This follows from the C*-identity: . Since , this gives , and since , we must have .

e.g.

1. The complex numbers with the involution being conjugation is a C*-algebra.
2. For a locally compact Hausdorff space , the algebra of continuous complex-valued functions vanishing at infinity is a C*-algebra with pointwise operations and the involution . The norm is the supremum norm.
3. For a Hilbert space , the algebra of bounded linear operators on is a unital C*-algebra, where the involution is the operator adjoint.

-Subalgebra & C*-Subalgebra

Let be a -algebra. A -subalgebra of is a -subalgebra if it is closed under the involution, i.e., whenever . Similarly, a C*-subalgebra of a C*-algebra is a norm-closed -subalgebra of .

Special Elements

Let be a -algebra. An element is called:

• Self-adjoint (or Hermitian) if
• Normal if
• A projection if

If is unital, an element is called:

• Unitary if
• An isometry if

Fundamental Properties

Properties of Unital C*-Algebras

Let be a unital C*-algebra.

1. If is invertible, then is invertible and .
2. For any , the spectrum of its adjoint is given by .
3. Any can be uniquely written as , where and are self-adjoint.
4. If is self-adjoint (), then its spectrum is a subset of the real numbers , and .
5. If is unitary, then and its spectrum is a subset of the unit circle .

Uniqueness of the C*-Norm

There is at most one norm on a -algebra that makes it a C*-algebra.

Proof Suppose and are two norms on a -algebra that both satisfy the C*-identity. For any , the element is self-adjoint. For any self-adjoint element , its norm is equal to its spectral radius, . Since the spectral radius depends only on the algebraic structure, it is independent of the norm. Thus, for any : This implies .

Unitarization

If a C*-algebra does not have a unit, it’s possible to embed it into a larger, unital C*-algebra.

Unitarization

Let be a non-unital Banach -algebra. The unitarization of is the algebra , with operations:

• Addition:
• Multiplication:
• Involution:

is a unital *-algebra with unit .

The C*-Norm on the Unitization

Let be a non-unital C*-algebra. The norm on its unitization defined by makes a C*-algebra.

Proof This norm is the operator norm of the left multiplication operator acting on . It can be shown that this norm is complete and satisfies the C*-identity. The key step in proving the C*-identity is showing .

Homomorphisms

-Homomorphisms are Contractive

Any -homomorphism between C*-algebras is a contraction, i.e., .

Proof For any self-adjoint element , we have . This implies . For a C*-algebra, the norm of a self-adjoint element equals its spectral radius. So, . For a general element , we have: Thus, .

-Isomorphisms are Isometries

Any -isomorphism between C*-algebras is an isometry.

Characters on Abelian C*-Algebras

Any character (non-zero multiplicative linear functional) on an abelian C*-algebra is a -homomorphism.

Proof WLOG, assume is unital. Let be a self-adjoint element. One can show that must be a real number. For an arbitrary , write it as . Then: Since and are real, this is equal to .

Spectral Permanence

Let be a unital C*-algebra and let be a C*-subalgebra of containing the unit of . Then for any element , its spectrum in is the same as its spectrum in .

Proof The inclusion is always true. To prove the other direction, one shows that if is invertible in , it must also be invertible in . This is first proven for self-adjoint elements, leveraging the fact that for such elements , , which implies that is connected. Since the spectra must agree for self-adjoint elements, one can extend the argument to arbitrary elements by considering the self-adjoint element .