Let $A$ be a von Neumann algebra and let $H$ be a (separable) Hilbert space.
It is known (see e.g., Section IV, Thm. 5.5 of Takesaki I) that there exists a Hilbert space $K$ such that $A \subset \mathbb{B}(K)$ such that any normal $*$-homomorphism $\varphi : A \to \mathbb{B}(H)$ can be written as
$$ \varphi(a) = v a v^*,$$
where $v: K \to H$ is a partial isometry with $vv^* = \mathrm{id}_K$ and $v^*v \in A^\prime \subseteq \mathbb{B}(K)$.

We are led to consider the sets $\mathrm{Hom}(A, \mathrm{B}(K))$ with its u-topology (defined by seminorms $\varphi \mapsto \|\omega \circ \varphi\|_{A_*}$, where $\omega \in \mathrm{B}(H)_*$ is an element of the predual of $\mathrm{B}(H)$) and 
$$V(A, H) := \{v \in \mathrm{B}(H, K) \mid vv^* = \mathrm{id}_K, v^*v \in A^\prime\}$$
with the strong operator topology.
By the result stated above, the map $V(A, H) \to \mathrm{Hom}(A, \mathrm{B}(K))$ is surjective.

**Q: Is this map a Serre fibration?**

If not, what are the problems here, and can we assume something on $A$ or change the topologies somehow to ensure this?