Kompleksalgebra

Kompleksalgebra is a mathematical structure consisting of a complex vector space equipped with a bilinear multiplication into itself, making it an algebra over the field of complex numbers. If the multiplication is associative, it is called an associative complex algebra; if it also has a multiplicative identity, it is a unital complex algebra. The scalar multiplication by complex numbers is compatible with the algebra multiplication, so (αa)b = α(ab) = a(αb) for all α in C and a,b in A. The center typically contains the image of C, and many authors view A as a C-algebra.

Common examples include the complex numbers C themselves, viewed as a complex algebra; the matrix algebra M_n(C); the polynomial algebra C[x] of polynomials with complex coefficients; the algebra C(X) of complex-valued functions on a set or topological space X with pointwise operations; and group algebras C[G] formed from a group G. Given a complex vector space V, the endomorphism algebra End_C(V) with composition of linear maps is also a complex algebra.

Complex algebras may be commutative or noncommutative, and they can be finite- or infinite-dimensional. They play a central role in representation theory, algebraic geometry, and functional analysis. When equipped with a norm making it a Banach space and with a compatible submultiplicative norm, a complex algebra becomes a complex Banach algebra; if it also carries an involution satisfying the C*-identity, it is a C*-algebra. These additional structures underpin much of modern analysis and operator theory.