C*-algebra

C∗-algebras (pronounced 'C-star') are subjects of research in functional analysis, a branch of mathematics. A C*-algebra is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:Theorem. A finite-dimensional C*-algebra, A, is canonically isomorphic to a finite direct sumTheorem. If A is a C*-subalgebra of K(H), then there exists Hilbert spaces {Hi}i∈I such that C∗-algebras (pronounced 'C-star') are subjects of research in functional analysis, a branch of mathematics. A C*-algebra is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties: Another important class of non-Hilbert C*-algebra include the algebra of continuous functions C 0 ( X ) {displaystyle C_{0}(X)} . C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators. These papers considered a special class of C*-algebras which are now known as von Neumann algebras. Around 1943, the work of Israel Gelfand and Mark Naimark yielded an abstract characterisation of C*-algebras making no reference to operators on a Hilbert space. C*-algebras are now an important tool in the theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quantum mechanics. Another active area of research is the program to obtain classification, or to determine the extent of which classification is possible, for separable simple nuclear C*-algebras.