C*algebras are studied in functional analysis. They are a theoretical tool in the theory of unitary representations of locally compact groups, and are also used in the socalled algebraic 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:


 for all x, y in A

 stands for the complex conjugation of λ.
 for all x, y in A

 = x for all x in A

  = x^{2} for all x in A.
A bounded linear map f : A → B between B*algebras A and B is called a *homomorphism if
 f(xy) = f(x)f(y) for x and y in A

 for x in A

 isomorphic.
Examples
Finite dimensional C*algebras
The algebra M_{n}(C) of nbyn matrices over C becomes a C*algebra if we consider matrices as operators on the Euclidean space C^{n} and use the operator norm . on matrices. The involution is given by the conjugate transpose. More generally, one can consider finite direct sums of matrix algebras.
Theorem. A finite dimensional C*algebra A is canonically isomorphic to a finite direct sum
 where min A is the set of minimal nonzero selfadjoint central projections of A. Each C*algebra A e is isomorphic (in a noncanonical way) to the full matrix algebra M_{dim(e)}(C). The finite family indexed on min A given by {dim(e)}_{e} is called the dimension vector of A. This vector uniquely determines the isomorphism class of a finite dimensional C*algebra.
C*algebras of operators
The prototypical example of a C*algebra is the algebra L(H) 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 A is *isomorphic to a normclosed adjoint closed subalgebra of L(H) for a suitable Hilbert space H; this is the content of the GelfandNaimark theorem.
Commutative C*algebras
Let X be a locally compact Hausdorff space. The space of C_{0}(X) of complexvalued continuous functions on X that vanish at infinity (defined in the article on local compactness) form a commutative C*algebra C_{0}(X) under pointwise multiplication and addition. The involution is pointwise conjugation. C_{0}(X) has a multiplicative unit element iff X is compact. As does any C*algebra, C_{0}(X) has an approximate identity. In the case of C_{0}(X) this is immediate: consider the directed set of compact subsets of X, and for each compact K let f_{K} be a function of compact support which is identically 1 on K. Such functions exist by the TietzeUrysohn theorem which applies to locally compact Hausdorff spaces. {f_{K}}_{K} is an approximate identity.
The Gelfand representation states that every commutative C*algebra is *isomorphic to an algebra of the form C_{0}(X).
The C*algebra of compact operators
Let H be a separable infinite dimensional Hilbert space. K(H) is the algebra of compact operators on H. It is a norm closed subalgebra of L(H). K(H) is also closed under involution; hence it is a C*algebra. Though K(H) does not have an identity element; an approximate identity for K(H) can be easily displayed. To be specific, let H = l^{2}; for each natural number n let H_{n} be the subspace of sequences of l^{2} which vanish for indices k ≥ n and let e_{n} be the orthogonal projection onto H_{n}. The sequence {e_{n}}_{n} is an approximate identity for K(H).
C*enveloping algebra
Given a B*algebra A with an approximate identity, there is (up to C*isomorphism) unique C*algebra E(A) and *morphism π from A into E(A) which is universal, that is every other B*morphism
π': A → B factors uniquely through π. E(A) is called the C*enveloping algebra of the B*algebra A.
Of particular importance are the enveloping algebras of the group algebras of locally compact groups. These C*algebras are used to provide a context for general harmonic analysis.
von Neumann algebras
von Neumann algebras, known as W* algebras before the 1960s, are a special kind of C*algebra. They are required to be closed in a topology which is weaker than the norm topology. Their study is a specialized area of functional analysis in itself, separate from C*algebras.
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 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).
Properties of C*algebras
C*algebras have a large number of good technical properties; some of these properties can be established by reduction to commutative C*algebras. In this case, we can use the fact that the structure of these is completely determined by the Gelfand isomorphism theorem or we can use the continuous functional calculus.

 algebras is contractive.

 algebras is isometric.
 algebra has an approximate identity.


 algebra in a unique way.

 y for some y. We can assume y itself is selfadjoint.
 algebra.
