Is there a well-defined notion of connection on a measurable bundle of Hilbert spaces?



Connection on a Hilbert bundle
This is an answer to the refined question formulated in the comments:
the base space is the unitary dual of a Lie group $G$.


The definition can be carried out in the setting of stacks in groupoids (or simplicial sets) on the site of cartesian spaces ($\def\R{{\bf R}} \R^n$ with smooth maps, for all $n≥0$).


Specifically,
we define a stack $R\_G$ that gives the unitary dual of a Lie group $G$
and a stack $B\_∇$ that gives bundles of Hilbert spaces with connectiom.
Then a morphism $R\_G→B\_∇$ is precisely a bundle of Hilbert spaces
with connection over $R\_G$.


To define the stack $R\_G$ that gives the unitary dual of a Lie group,
assign to a cartesian space $T$ the following groupoid $R\_G(T)$.


Objects of $R\_G(T)$ are given by a Hilbert space $H$ together with a smooth $T$-indexed
family of irreducible unitary representations of $G$ on $H$.
This is simply a smooth map $\def\Hom{\mathop{\rm Hom}} f\colon T→\Hom(G,U(H))$ (landing in irreps), where $U(H)$ is equipped with the ultraweak topology and $\Hom(G,U(H))$ is the space of continuous group homomorphisms equipped with the compact-open topology. Smooth means that the adjoint map $T⨯G→U(H)$ composed with the inclusion $U(H)→B(H)$ is smooth as a map from a finite-dimensional smooth manifold to a topological vector space $B(H)$.


Morphisms of $R\_G(T)$ from $(H,f)$ to $(H',f')$ are given by a smooth map $h\colon T→U(H,H')$ that intertwines the action of $G$.
Here $U(H,H')$ is the topological space of unitary isomorphisms $H→H'$ equipped with the ultraweak topology.


Next, we define stacks $B$ and $B\_∇$ of bundles of Hilbert spaces,
(equipped with connection in the case of $B\_∇$) as follows.
Given a cartesian space $T$, we define a groupoid $B\_∇(T)$ as follows.


Objects of $B\_∇(T)$ are pairs $(H,∇)$, where $H$ is a Hilbert space
and $∇\colon T→Hom(T,I(H))$ is a smooth map,
where $I(H)$ denotes the space of unbounded skew-adjoint operators on $H$
(equivalently, one-parameter unitary groups on $H$)
and $\Hom(T,I(H))$ is the space of linear maps $T→I(H)$.


Morphisms of $B\_∇(T)$ from $(H,∇)$ to $(H',∇')$ are smooth maps $p\colon T→U(H,H')$
such that $∇'=p^\*\theta+\mathop{\rm Ad}\_{p^{-1}}∇$, where $\theta$ is the Maurer–Cartan form and Ad denotes the adjoint action.


The stack $B$ is defined analogously, but dropping $∇$ and the condition on $p$.
We have a canonical forgetful map $B\_∇→B$.


The obvious forgetful map $W\colon R\_G→B$ defines the canonical bundle $W$
over $R\_G$.
In particular, the fiber of $W$ of a point of $R\_G$ given
by an irreducible representation $ρ$ of $G$ is simply $ρ$ itself.


Now, a connection on $W$ is a lift of the map $W\colon R\_G→B$
through the forgetful map $B\_∇→B$.


