Let $M$ be a smooth manifold.

I have been trying to figure out from the literature I know whether (any flavor of) pseudo-differential operators form a sheaf of algebras (w.r.t. the usual topology on $M$). Sadly the best results I could find showed at best that pseudo-differential form a sheaf of left $C^{\infty}$-modules without treating the question of whether the multiplication is well defined nevermind associative. So my question is rather simple:

Do pseudo-differential operators form a sheaf of (associative?) $\mathbb{C}$-algebras on $M$? If not, what fails?

I'm pretty sure that if one considers pseudo-differential operators modulo smothing operators than these do form a sheaf of algebras (However a precise reference here would be welcome). As for the entire space of pseudo-differential operators this seems rather non-trivial to me (I'm actually not entirely sure whether this should be true or not). I apologize if this is too elementary for this site.

actionon nice functions... and so on. $\endgroup$ – paul garrett Feb 19 '18 at 22:20