The basic idea is that the algebraic structure of a space is just as important as its geometric structure, and that by studying the algebra of functions on a space, we can learn a lot about its geometry.
The Gelfand-Naimark theorem is a fundamental result in functional analysis that establishes a correspondence between commutative c-star algebras and compact Hausdorff spaces. Specifically, the theorem states that every commutative C*-algebra is isomorphic to the algebra of continuous complex-valued functions on some compact Hausdorff space, and conversely, every such algebra of functions corresponds to a commutative C*-algebra.
This correspondence between algebra and geometry is a key insight of noncommutative geometry, and it motivates the study of noncommutative C*-algebras, whose "geometric counterpart" can be thought of as "noncommutative spaces". By studying the properties of these algebras, we can gain insights into the geometry of the noncommutative spaces they represent. In this way, noncommutative geometry provides a powerful tool for studying a wide range of mathematical and physical phenomena.
I think that in noncommutative space we can define a kind of measure (and I wonder: are these the spectral measures?). If we consider that in commutative spaces, points can be interpreted as measures (in particular as a probability measure), called Dirac measures, and that therefore other measures are something like "smeared-out points", then in noncommutative space we only have smeared-out points.
Introduction @bongaartsshort
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Author of the notes: Antonio J. Pan-Collantes
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