A map f : A > B between B^{*}algebras A and B is called a *homomorphism if
The algebra of nbyn matrices over C becomes a C^{*}algebra if we use the matrix norm ._{2} arising as the operator norm[?] from the Euclidean norm on C^{n}. 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 GelfandNaimark theorem[?].
An example of a commutative C^{*}algebra is the algebra C(X) of all complexvalued 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 complexvalued continuous functions on X that vanish at infinity (defined in the article on local compactness), then these form a commutative C^{*}algebra C_{0}(X); if X is not compact, then C_{0}(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 C_{0}(X).
In quantum field theory, one typically describes a physical system with a C^{*}algebra A with unit element; the selfadjoint 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 Clinear map φ : A > C with φ(u u^{*}) > 0 for all u∈A) 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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