Editing Noncommutative geometry (section)

==Noncommutative differentiable manifolds==

A smooth [[Riemannian manifold]] ''M'' is a [[topological space]] with a lot of extra structure. From its algebra of continuous functions ''C''(''M''), we only recover ''M'' topologically. The algebraic invariant that recovers the Riemannian structure is a [[spectral triple]]. It is constructed from a smooth vector bundle ''E'' over ''M'', e.g. the exterior algebra bundle. The Hilbert space ''L''<sup>2</sup>(''M'',&nbsp;''E'') of square integrable sections of ''E'' carries a representation of ''C''(''M)'' by multiplication operators, and we consider an unbounded operator ''D'' in ''L''<sup>2</sup>(''M'',&nbsp;''E'') with compact resolvent (e.g. the [[signature operator]]), such that the commutators [''D'',&nbsp;''f''] are bounded whenever ''f'' is smooth. A deep theorem<ref>{{cite journal |doi=10.4171/JNCG/108|title=On the spectral characterization of manifolds |year=2013 |last1=Connes |first1=Alain |journal=Journal of Noncommutative Geometry |volume=7 |pages=1–82 |s2cid=17287100|arxiv=0810.2088}}</ref> states that ''M'' as a Riemannian manifold can be recovered from this data.

This suggests that one might define a noncommutative Riemannian manifold as a [[spectral triple]] (''A'',&nbsp;''H'',&nbsp;''D''), consisting of a representation of a C*-algebra ''A'' on a Hilbert space ''H'', together with an unbounded operator ''D'' on ''H'', with compact resolvent, such that [''D'',&nbsp;''a''] is bounded for all ''a'' in some dense subalgebra of ''A''. Research in spectral triples is very active, and many examples of noncommutative manifolds have been constructed.