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 )
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]?.