C-star-algebra
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
- (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
- The C* identity:
C* algebras are also * algebras.
If the last property is omitted, we speak of a B*-algebra.
By the Gelfand-Naimark theorem, C*-algebras are (up to isomorphism) precisely those algebras of bounded operators on Hilbert spaces that are closed in the norm topology and under taking adjoints, with the involution map being taking the adjoint.
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2 Examples of C*-algebras 3 von Neumann algebras 4 C*-algebras and quantum field theory |
A map f : A → B between B*-algebras A and B is called a *-homomorphism if
*-Homormorphisms and *-Isomorphisms
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. The structure of a C*-algebra forces any *-homomorphism to be contractive; and a homomorphism is injective if and only if it is isometric.
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).
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).Examples of C*-algebras
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 branch of a Mathematics in itself, separate from C*-algebras. C*-algebras and quantum field theory
See also algebra, associative algebra, * algebra, B* algebra.