In mathematics and mathematical physics, and in particular in functional analysis, by analogy with the Gelfand representation, which shows that commutative C* algebras are dual to locally compact Hausdorff spaces, noncommutative C* algebras are often now 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.

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