- (
*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*.

Table of contents |

2 Examples of C ^{*}-algebras3 W* algebras 4 C ^{*}-algebras and quantum field theory |

A map *f* : *A* `->` *B* between B^{*}-algebras *A* and *B* is called a ***-homomorphism** if

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 **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 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 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 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 *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*).

See also algebra, associative algebra, * algebra, B* algebra.