Noncommutative geometry

In analogy to the Gelfand-Naimark theorem, which states that commutative C* algebras are dual to locally compact Hausdorff spaces, noncommutative C* algebras are called noncommutative spaces.

Examples:

• The symplectic phase space of classical mechanics is deformed into a noncommutative phase space generated by the position and momentem operators.

Also, in analogy to the duality between affine schemes and polynomial algebras, we can also have noncommutative affine schemes.

For the duality between locally compact measure spaces and commutative von Neumann algebras, we call noncommutative von Neumann algebras noncommutative measure spaces.

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