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

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.

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.