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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
  • 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 uA) such that φ(1) = 1. The expected value of the observable x, if the system is in state φ, is then φ(x).

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