C*-algebra

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C*-algebras (pronounced "C-star") are an important area of research in functional analysis, a branch of mathematics. The prototypical example of a C*-algebra is a complex algebra A of linear operators on a complex Hilbert space with two additional properties:

  • A is closed under the operation of taking adjoints of operators.

It is generally believed that 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.

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.

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