I've been reading this paper by Rieffel and I'm a bit confused about the some of the claims, found in Section 1 (Modules over operator algebras). Let $B$ be a $C^\ast$-algebra and let $n(B)$ be its enveloping $W^\ast$-algebra. Let $\mathrm{Hermod}$ be the category of Hermitian modules over a $C^\ast$-algebra and $\mathrm{Normod}$ be the category of normal modules over a $W^\ast$-algebra. Then the paper claims that $$\mathrm{Hermod}\text{-}B \cong \mathrm{Normod}\text{-}n(B).$$
However, the comment which confuses me is "there are many $W^\ast$-algebras whose category of normal modules is not isomorphic to the category of Hermitian modules over any $C^\ast$-algebra". My understanding was that if $B$ is a $W^\ast$-algebra, it is in particular a $C^\ast$-algebra, and we have that $n(B) \cong B$. Therefore, by the claim above, we would have $$ \mathrm{Normod}\text{-}B \cong \mathrm{Normod}\text{-}n(B) \cong \mathrm{Hermod}\text{-}B. $$ That is, the category of normal modules over a $W^\ast$-algebra is always isomorphic to the category of Hermitian modules over itself, as a $C^\ast$-algebra. What have I missed?
The enveloping von Neumann algebra $\tilde A$ of a $C^\ast$-algebra $A$ has the property that every non-degenerate representation of $A$ lifts to a normal unital representation of $\tilde A$. Since von Neumann algebras usually have lots of non-normal (unital) representations, they do not coincide with their enveloping von Neumann algebra.
Another way of looking at this is that the enveloping von Neumann algebra is canonically isomorphic to $A^{\ast\ast}$ with the Arens product, and the inclusion $A\to A^{\ast\ast}$ is only surjective for finite-dimensional von Neumann algebras.
To give you concrete examples, it is known that if $M$ is a $\sigma$-finite von Neumann algebra (i.e. it admits a faithful normal state) that is the enveloping von Neumann algebra of a $C^\ast$-algebra, then $M$ is atomic. This means that plenty of von Neumann algebras are not isomorphic to enveloping von Neumann algebras, for example all II$_1$ factors.