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

[arivero]

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.

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

 

[Roberto Pavani]

I surely miss something, but it seems to me that the answer should be no, at least for a non-trivial $K$. For a universe that is not flat and completely empty, I can't understand how the combinatorial complexity, the number of possible field/geometric configurations on a compact space $K$, can be encoded into a finite-dimensional matrix algebra.

It seems to me that the counting of degrees of freedom forbids it: as soon as I place particles at arbitrary positions, doesn't the number of configurations explode beyond what a finite algebra can hold?

If so, wouldn't the limit KK → almost-commutative geometry necessarily lose information?