C-star-algebra

C^*-algebras are studied in functional analysis and are used in some formulations of quantum mechanics. A C^*-algebra A is a Banach algebra over the field of complex numbers, together with a map ^* : A -> A called involution which has the following properties:

• (x + y)^* = x^* + y^* for all x, y in A
• (λ x)^* = λ^* x^* for every λ in C and every x in A; here, λ^* stands for the complex conjugation of λ.
• (xy)^* = y^* x^* for all x, y in A
• (x^*)^* = x for all x in A
• ||x x^*|| = ||x||² for all x in A.

If the last property is omitted, we speak of a B^*-algebra.

*-Homormorphisms and *-Isomorphisms

A map f : A -> B between B^*-algebras A and B is called a *-homomorphism if

• f is C-linear
• f(xy) = f(x)f(y) for x and y in A
• f(x^*) = f(x)^* for x in A

Such a map f is automatically continuous. If f is bijective, then its inverse is also a *-homorphism and f is called a *-isomorphism and A and B are called *-isomorphic. In that case, A and B are for all practical purposes identical; they only differ in the notation of their elements.

Examples of C^*-algebras

The algebra of n-by-n matrices over C becomes a C^*-algebra if we use the matrix norm ||.||₂ arising as the operator norm[?] from the Euclidean norm on Cⁿ. The involution is given by the conjugate transpose.

The motivating example of a C^*-algebra is the algebra of continuous linear operators defined on a complex Hilbert space H; here x^* denotes the adjoint operator[?] of the operator x : H -> H. In fact, every C^*-algebra is *-isomorphic to a closed subalgebra of such an operator algebra for a suitable Hilbert space H; this is the content of the Gelfand-Naimark theorem[?].

An example of a commutative C^*-algebra is the algebra C(X) of all complex-valued continuous functions defined on a compact Hausdorff space X. Here the norm of a function is the supremum of its absolute value, and the star operation is complex conjugation. Every commutative C^*-algebra with unit element is *-isomorphic to such an algebra C(X) using the Gelfand representation[?].

If one starts with a locally compact Hausdorff space X and considers the complex-valued continuous functions on X that vanish at infinity (defined in the article on local compactness), then these form a commutative C^*-algebra C₀(X); if X is not compact, then C₀(X) does not have a unit element. Again, the Gelfand representation[?] shows that every commutative C^*-algebra is *-isomorphic to an algebra of the form C₀(X).

C^*-algebras and quantum field theory

In quantum field theory, one typically describes a physical system with a C^*-algebra A with unit element; the self-adjoint elements of A (elements x with x^* = x) are thought of as the observables, the measurable quantities, of the system. A state of the system is defined as a positive functional on A (a C-linear map φ : A -> C with φ(u u^*) > 0 for all u∈A) such that φ(1) = 1. The expected value of the observable x, if the system is in state φ, is then φ(x).