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Noncommutative geometry
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==Invariants for noncommutative spaces== Some of the motivating questions of the theory are concerned with extending known [[topological invariant]]s to formal duals of noncommutative (operator) algebras and other replacements and candidates for noncommutative spaces. One of the main starting points of [[Alain Connes]]' direction in noncommutative geometry is his discovery of a new homology theory associated to noncommutative associative algebras and noncommutative operator algebras, namely the [[cyclic homology]] and its relations to the [[algebraic K-theory]] (primarily via Connes–Chern character map). The theory of [[characteristic classes]] of smooth manifolds has been extended to spectral triples, employing the tools of operator [[K-theory]] and [[cyclic cohomology]]. Several generalizations of now-classical [[index theorem]]s allow for effective extraction of numerical invariants from spectral triples. The fundamental characteristic class in cyclic cohomology, the [[JLO cocycle]], generalizes the classical [[Chern character]].
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