Hilbert bundle and set of sections.

Question

Let $\Lambda$ be a manifold and $p:H\to\Lambda$ a continuous Hilbert bundle with $H(\lambda):=p^{-1}(\lambda)$. Suppose $\Gamma_0^0(\Lambda)$ is the space of continuous sections vanishing at infinity of $H$. I proved that $\Gamma_0^0(\Lambda)$ has the structure of a $C_0(\Lambda)$ module (with $C_0$ being the space of continuous functions vanishing at infinity). Define $H_{\lambda}=\Gamma_0^0(\Lambda)/\overline{K_{\lambda}}$ where $$K_{\lambda}=\text{span}\{f\varphi:\varphi\in \Gamma_0^0(\Lambda)\text{ and } f(\lambda)=0\}$$ I am interested in showing that $H_{\lambda}$ i isomorphic to $H(\lambda)$. Does anyone know some way to prove this?

Answer 1

In a local trivialization of $\Lambda$ around $\lambda$, let $U_n$ be the open ball centered at $\lambda$ with radius $1/n$, and use Uryhson to choose a continuous (bump) function $h_n$ on $\Lambda$ with $0≤h_n≤1$, vanishing off $U_n$ and with $h_n(\lambda)=1$. It is then easy to prove that for every section $\varphi$ vanishing at $\lambda$, one has that $\Vert h_n\varphi\Vert \to 0$.

Setting $f_n:= 1-h_n$ we deduce that $\|\varphi - f_n\varphi\|\to0$.

The given definition of $K_\lambda$ does not specify exactly what kind of functions $f$ are used there, but assuming bounded continuous functions are OK, then we are done because our $f_n$ above clearly satisfies these properties.

Should one prefer to define $K_\lambda$ as $$ K_{\lambda}=\text{span}\{f\varphi:\varphi\in \Gamma_0^0(\Lambda)\text{, } f\in C_0(\Lambda) \text{ and } f(\lambda)=0\} $$ (mind you that $f$ is now required to be in $C_0(\lambda)$) the above method will not work but it can be fixed as follows:

Using that $\varphi$ vanishes at $\infty$, one may show that there exists some $g$ in $C_0(\Lambda)$ such that $\varphi$ is near $g\varphi$. It will then follow that $\varphi$ is near $f_ng\varphi$, and bingo, $f_ng$ now lies in $C_0(\Lambda)$ and vanishes at $\lambda$, so $\varphi$ is seen to lie in the closure of the newly defined $K_\lambda$.