noncommutative geometry topic

Non-commutative Geometry and Non-Abelian Algebraic Topology

Noncommutative `geometry' utilizes non-Abelian (or nonabelian) methods for quantization of spaces through `deformation' to non-commutative 'spaces' (in fact non-commutative algebraic structures, or algebras of functions). therefore, it can be considered a subfield of non-Abelian algebraic topology (NAAT).

An alternative meaning is often given to Noncommutative Geometry (viz . A Connes et al.): that is, as a non-commutative `geometric' approach- in the relativistic sense- to quantum gravity. This approach is therefore relevant to Non-Abelian Quantum Algebraic Topology (NA-QAT, or NAQAT).

A specific example due to A. Connes is the convolution $ C^*$ -algebra of (discrete) groups; other examples are non-commutative $ C^*$ -algebras of operators defined on Hilbert spaces of quantum operators and states. (Please see also the other PM entries on $ C^*$ -algebra and noncommutative topology.)

Recent Developments in NCG



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