Hilbert module over $C_0(\Lambda)$ as space of continuous sections of HIlbert bundle

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 $H_{\lambda}$ is isomorphic to $H(\lambda)$. Does anyone know some way to prove this or some reference with a similar proof? I've encounterd problems because $\overline{K_\lambda}$ is not necessarily the kernel of the evaluation map from $\Gamma_0^0$ to $\mathbb{C}$. There are several references pointing to properties similiar to this one, in fact in Mathematical Quantization by Nik Weaver there is a similar result regarding $C(\Lambda)$ modules in the case $\Lambda$ is a compact manifold but there is no proof. I've already checked the references provided by Weaver and didn't find a proof.

Answer 1

The reference I cited in my book is Fell and Doran, Representations of ${}^*$-Algebras, Locally Compact Groups, and Banach ${}^*$-Algebraic Bundles, vol. 1 (1988). Did you check there? I don't have a copy handy but I remember this treatment being very complete.

If that doesn't work, you could look at my paper with Chris Phillips, Modules with norms which take values in a C${}^*$-algebra. Theorem 8 gives a more general result about Banach modules/Banach bundles. The full proofs are not given here either, but there are specific references to Takahashi's dissertation, Fields of Hilbert Modules (Tulane, 1971) which should fill all the gaps.