Are Almost Commutative Geometries a sort of Kaluza Klein limit, or no?

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SUMMARY

The discussion centers on the relationship between almost commutative geometries and Kaluza Klein theories, specifically questioning whether Alain Connes' approach can be viewed as a limit of Kaluza Klein theories. The participant reflects on the Morita equivalence of the algebra ##A## in the structure ##C(M)\otimes A## and its connection to the isometry group of compact spaces. However, they express difficulty in substantiating this claim, noting that the standard techniques do not yield a finite matrix algebra or the expected limit spaces through volume contraction.

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  • Understanding of almost commutative geometries
  • Familiarity with Kaluza Klein theories
  • Knowledge of Morita equivalence in algebra
  • Concepts of isometry groups and compact spaces
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The discussion is beneficial for mathematicians, theoretical physicists, and researchers interested in the intersections of geometry, algebra, and physics, particularly those exploring noncommutative geometry and Kaluza Klein theories.

arivero
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I have always taken for granted that the extra algebra ##A## in an almost commutative geometry ##C(M)\otimes A## was not only Morita equivalent to a point, but shaped in a way that its inner automorphism group remembered the isometry group of a compact space that had been contracted to a point. Now while revisiting the theme, I can not find any way to prove or argue for this; the point of Connes being a "zero dim Kaluza Klein" seems just a intuition from the eighties. Straightforward collapse via the equivalence groupoid does not produce a finite matrix algebra, and limit spaces via contraction of the volume are not in the usual toolbox.

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So, what do you think? Is it possible to substantiate the claim that Connes method is a limit of Kaluza Klein theories?
 
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