C*-algebra

C*-algebras are an important area of research in functional analysis. A C*-algebra can be defined concretely as a complex algebra A of linear operators on a complex Hilbert space with two additional properties:

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• A is a topologically closed set in the norm topology of operators.
• A is closed under the operation 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 in an extremely rudimentary form 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.

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Around 1943, the work of Israel Gelfand, Mark Naimark and Irving Segal yielded an abstract characterisation of C*-algebras making no reference to operators.

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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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C*-algebra: Abstract characterization ►



 

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