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'noncommutative geometry'
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| Title of object: |
noncommutative geometry |
| Canonical Name: |
NoncommutativeGeometry |
| Type: |
Topic |
| Created on: |
2008-07-18 04:26:41 |
| Modified on: |
2008-09-18 16:09:28 |
| Classification: |
msc:81T75 |
| Keywords: |
C*-algebras, Quantum Gravity Theories, `deformation-quantization' of (commutative) spaces |
| Defines: |
`Geometry' of quantum spaces in terms of non-commutative algebras of functions and quantum operators, or `spectral (quantum) triples' |
| Synonyms: |
noncommutative geometry=Non-Commutative Geometry
noncommutative geometry=Non-Abelian Geometry
noncommutative geometry=Non-Abelian Topology
noncommutative geometry=Noncommutative Topology |
Revision comment (for changes between this and next version):
\textbf{Note:}
The Royal Swedish Academy of Sciences has awarded the 2001 Crafoord Prize in mathematics
to Professor Alain Connes of the Institut des Hautes Études Scientifiques (IHES) and the
Collège de France, Paris, ``for his penetrating work on the theory of... (quantum)... operator algebras
and for having been a founder of \emph{noncommutative geometry}".
\PMlinkexternal{Crafoord Prize in 2001 in Noncommutative Geometry and Quantum Operator Algebras}{http://www.ams.org/notices/200105/comm-crafoord.pdf} |
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Content:
Noncommutative `geometry' utilizes nonabelian methods for quantization of spaces through `deformation' to non-commutative 'spaces' (in fact {\em non-commutative} algebraic structures, or algebras of functions).
\emph{An alternative meaning is often given to Noncommutative Geometry (viz . A Connes et al.)}:
i.e., as a non-commutative `geometric' approach-- \emph{in the relativistic sense}-- to Quantum Gravity.\\
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.)
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