3
$\begingroup$

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 $<\xi,a\eta> = \int a(\xi|\eta)|D|^{-p}$

$\endgroup$

1 Answer 1

3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.