In Connes' approach to non-commutative geometry, the notion of a spectral triple is said to generalize compact Riemannian manifolds to the non-commutative setting. Motivating classical examples include the operator $d + d^*$ on Riemannian manifold, where $d$ is the exterior derivative and $d^*$ is its adjoint with respect to the metric. Another example is the operator $\overline{\partial} + \overline{\partial}^*$ on a compact Hermitian manifold.

According to Wikipedia, a spectral triple is regular if

Let $\delta(T)$ denote the commutator of $|D|$ with an operator $T$ on $H$. A spectral triple is said to be regular when the elements in $A$ and the operators of the form $[a, D]$ for $a \in A$ are in the domain of the iterates $\delta^n$ of $\delta$.

Is it easy to see that the two commutative examples given above are regular?


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