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
- (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.
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)
- 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]?.