Let be a compact smooth manifold. Show that any -algebra morphism is of the form for some , where .

Advertisements

Skip to content
# “Representability”

##
2 thoughts on ““Representability””

### Leave a Reply

Let be a compact smooth manifold. Show that any -algebra morphism is of the form for some , where .

Advertisements

%d bloggers like this:

Let be compact smooth manifolds. By Gelfand duality, an -algebras map is induced by the pullback of a map . In our case, and thus an -algebra morphism is the pullback of , which simply is evaluation at a point.

LikeLiked by 2 people

Necroposting. Let an -algebra morphism. Let’s establish in first place the following fact: there exist and an open neighbourhood such that whenever , is . Indeed, if it wasn’t the case, by compactness there should exist a finite cover of by proper open sets together with smooth functions such that and . By hypothesis, there is also a smooth function supported on with . Then, is identically but , a contradiction.

Now, the conclusion follows easily. Let and take a smooth function supported on with . By the choice of and the open set , . Since , this implies .

LikeLike