CAlgebren

CAlgebren, or C-algebras, are algebras over the field of complex numbers. In its simplest form, a C-algebra is a complex vector space equipped with a bilinear multiplication that makes it an associative algebra, possibly with a unit. These structures can be finite-dimensional or infinite-dimensional and serve as a common framework across algebra and analysis. Finite-dimensional C-algebras decompose into direct sums of matrix algebras, and every simple finite-dimensional complex algebra is isomorphic to a full matrix algebra Mn(C).

Typical examples include the complex numbers C (as a one–dimensional algebra), the algebra of n-by-n complex matrices Mn(C), the polynomial algebra C[x], and the algebra C(X) of complex-valued continuous functions on a topological space X (all with pointwise operations). Direct sums, tensor products, and subalgebras of such algebras provide a wide range of C-algebras.

A prominent subclass is the C*-algebras. A C*-algebra is a complex algebra equipped with an involution * and a norm such that the algebra is complete with respect to the norm, the norm satisfies ||ab|| ≤ ||a||·||b||, and the C*-identity ||a* a|| = ||a||^2 holds. Bounded operators on a Hilbert space, B(H), form a canonical example. Commutative C*-algebras are classified by locally compact spaces via Gelfand duality, linking algebraic and topological structures.

In applications, C-algebras provide a unifying language for representation theory, functional analysis, and aspects of noncommutative geometry, where the emphasis often shifts to extra structures such as norms, involutions, or topologies.