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.