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In Connes' noncommutative geometry, the starting point is a spectral triple $(A,D,H)$ where $A$ is a commutative C* algebra, e.g. as in Connes "ON THE SPECTRAL CHARACTERIZATION OF MANIFOLDS" , where 5 conditions are stated which the triple must satisfy.

In order to construct coordinate charts on the $X = Spec(A)$, Connes proceeds with analysis of derivations $\delta_j$ on A, which satisfy exponentiability conditions (Lemma 3.3 in the above paper).

Question: where do these derivations come from?

They do not seem to be part of the package of the 5 conditions.

Is my understanding correct that Connes uses the spectrum of these derivations to identify local charts on X with $\mathbb{R}^p$?

For user's convenience, here are the 5 conditions:

(1) The n-th characteristic value of the resolvent of D is $O(n^{-1/p}$.

(2) $[[D,a],b]=0$ $\forall a,b \in A$

(3) For any $a\in A$, both $a$ and $[D,a]$ belong to the domain of $\delta^m$ for any integer m and $\delta(T) = [|D|,T]$

(4) There exists a Hochschild cycle $c\in Z_p(A,A)$ such that $\pi(c)=1$ for p odd, and $\pi(c)=\gamma$ is $\mathbb{Z}/2$ grading for p even

(5) The A-module $H_\infty = \cap Dom(D^m)$ is finite and projective. It also has hermitian structure $(|)$ defined as $<\xi,a\eta> = \int a(\xi|\eta)|D|^{-p}$

0x11111
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1 Answers1

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If I'm not mistaken, the derivations in questions are those of the form of equation (22) in Lemma 4.3, i.e., derivations of the form $a \mapsto \mathrm{i}(\xi \vert [D,a]\xi)$ for $\xi \in \mathcal{H}^\infty$. The paper is a difficult one, but the introduction does provide a decent roadmap to understanding where and how the construction of the coordinate charts plays out in the body of the paper.

Branimir Ćaćić
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