6
$\begingroup$

In noncommutative geometry, one is given a triple $(A,D,H)$, where $A$ is a commutative C* algebra, $H$ is a Hilbert space, and $D$ is an operator. There is a somewhat long list of conditions that this data has to satisfy.

There is a finite-dimensional compact manifold $M$ associated with this data. It is equal to the spectrum of $A$.

Question. How are coordinate charts on $M$ constructed? I.e. how is it locally identified with $\mathbb{R}^n$?

Improve this question
$\endgroup$
1
  • 2
    $\begingroup$ The spectrum of an abelian C${}^\ast$-algebra is not a compact manifold in general. This will only happen if $A = C(M)$ for some compact manifold $M$, in which case the coordinate charts are whatever were initially given on $M$. $\endgroup$
    – Nik Weaver
    Commented Feb 6 at 17:02

1 Answer 1

Reset to default
6
$\begingroup$

This is proved in

A. Rennie, J. C. Várilly: Reconstruction of manifolds in noncommutative geometry

for a special class of triples that the Authors call "spectral manifolds", see Definition 5.1.

For such a triple $(\mathcal{A}, \, \mathcal{H}, \, \mathcal{D})$ the spectrum $X:=\operatorname{sp}(\mathcal{A})$ is a smooth compact orientable manifold and $\mathcal{A}=C^{\infty}(X)$, see Theorems 7.20 and 7.26.

The (rather involved) construction of the coordinate charts is given in Sections 6-7 using a Lipschitz functional calculus developed in Section 5, see also the Introduction for a quick explanation of the strategy.

Improve this answer
$\endgroup$
4
  • 4
    $\begingroup$ Unfortunately, there were gaps in the preprint of Rennie and Várilly. The safe reference is A. Connes, On the spectral characterization of manifolds, J. Noncommut. Geom. 7 (2023), 1-8. In particular, the introduction sketches out the strategy for constructing coordinate charts. $\endgroup$
    – Branimir Ćaćić
    Commented Feb 7 at 21:07
  • $\begingroup$ The essential part in this paper is construction of manifolds $W_{jk}$ as in (6.2) and (7.1). They are constructed using operators $a_\alpha$. Do I understand it correctly that these operators are part of the {\it assumptions } we impose on the triple, i.e., they actually come as coefficients of the explicit Hochschild cycle? In other words, we have to assume something more than just existence of the operator $D$. In Connes' paper, I believe it corresponds to Lemma 4.5 $\endgroup$
    – 0x11111
    Commented Feb 11 at 21:54
  • 1
    $\begingroup$ In both cases, the operators whose joint spectra ultimately yield coordinate charts come from the Hochschild cycle implementing NC orientability. Again, I must stress that Rennie and Varilly's preprint requires stronger hypotheses than Connes's published paper and contains serious gaps, though it was a trailblazing and greatly admired attempt; Connes's paper, whose completion from much older unfinished work was arguably spurred by the appearance of Rennie and Várilly's preprint, is now the authoritative reference in the NCG literature. $\endgroup$
    – Branimir Ćaćić
    Commented Feb 11 at 22:11
  • 1
    $\begingroup$ [If I recall the relevant lore correctly, what Connes had been missing was the technical machinery concerning exponentiating relevant derivations and having "enough" such exponentiable derivations.] $\endgroup$
    – Branimir Ćaćić
    Commented Feb 11 at 22:14

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .