Titlebook: Elements of Noncommutative Geometry; José M. Gracia-Bondía,Joseph C. Várilly,Héctor Fig Textbook 2001 Springer Science+Business Media New

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D. S. Deenadayal,Vyshanavi BommakantiNoncommutative topology brings techniques of operator algebra to algebraic topology —and vice versa. It is relatively difficult to extend the standard homotopy and (co)homology functors. On the other hand, Atiyah’s .functor 12 generalizes very smoothly. We shall continue with that.

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https://doi.org/10.1007/978-3-0348-8992-6In this chapter, our goal is to construct geometries on commutative spaces. That is to say, given a noncommutative space admitting a differential calculus that is in fact commutative —in other words, an algebra A=.∞ (.) for some differential manifold .— we shall find the extra structures that give rise to geometries.

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Jean-Pierre Droz,Riccardo A. AudisioThis last part of the book sets out to explore several interfaces between noncommutative geometry (NCG) and physics. We have chosen to report on the avenues that look more promising, as the century draws to a close, rather than on the more established “applications”.

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The Noncommutative IntegralThe road to integral calculus on noncommutative manifolds passes through spectral analysis. In fact, inasmuch as it tries to discover the geometry of manifolds from the analysis of operators strategically associated to them, spectral analysis . noncommutative geometry ..

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Commutative GeometriesIn this chapter, our goal is to construct geometries on commutative spaces. That is to say, given a noncommutative space admitting a differential calculus that is in fact commutative —in other words, an algebra A=.∞ (.) for some differential manifold .— we shall find the extra structures that give rise to geometries.

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