C-star-algebra

C^*-algebras are one of the tools and objects of study 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 follow properties:

• (x + y)^* = x^* + y^* for all x, y in A

(the involution of the sum of x and y is equal to the sum of the involution of x with the involution of y)

• (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

(the involution of the product of x and y is equal to the product of the involution of x with the involution of y)

• (x^*)^* = x for all x in A

(the involution of the involution of x is equal to x)

• ||x x^*|| = ||x||² for all x in A.

(the norm of the product of x and the involution of x is equal to the norm of x squared )

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

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 (and x and y in B)

(the function applied to the product of x and y is equal to product of the function applied to x with the function applied to y )

• f(x^*) = f(x)^*

Such a map is automatically continuous. If it is bijective, then its inverse is also a *-homorphism and it 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.

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 subalgebra of such an operator algebra for a suitable Hilbert space; 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. Every commutative C^*-algebra with unit element is *-isomorphic to such an algebra C(X) using the [Gelfand representation]?.