fa.functional analysis - Schwartz kernel theorem for A-linear operators

Let $X,Y subset mathbb{R}^n$ be open subsets. Denote by $C^infty(X)$ the smooth functions on $X$, let $mathcal{E}'(Y)$ be its dual space considered as a space of distributions. Let $L(C^infty(X), mathcal{E}'(Y))$ be the continuous linear operators. This space itself is equipped with the topology of bounded convergence (see for example Operator topology).

An instance of the Schwartz kernel theorem states that

$L(C^infty(X), mathcal{E}'(Y)) simeq mathcal{E}'(X times Y)$,

which morally says that I can write every continuous linear operator $T colon C^infty(X) to mathcal{E}'(Y)$ as an integral operator with a distributional kernel $k(y,x) in mathcal{E}'(X times Y)$ as

$
(Tu)(y) = int k(y,x)u(x) dx
$

(where the integral denotes the pairing of functions with distributions).

Now let $E$ and $F$ be Hilbert $A$-modules for a $C^*$-algebra $A$. I can still consider smooth functions with values in the Banach space $E$. Also $F$-valued distributions still make sense. They can be defined by

$mathcal{E'}(Y,F) = L(Y,F) = mathcal{E}'(Y) hat{otimes} F$,

where the tensor product is the completion of the projective tensor product (well, since $mathcal{E}'(Y)$ is nuclear this doesn't really matter).

I was wondering if in this case there
is an analogue of the Schwartz kernel
theorem with $A$-linearity and
adjointability build into it, thus
stating something like

$L_A^*(C^infty(X,E),
> mathcal{E}'(Y,F)) simeq
> mathcal{E}'(X times Y,
> text{Hom}_A^*(E,F))$

Here the left hand side should denote
adjointable continuous $A$-linear
operators (part of the question is to
find the right notion of
adjointability, $A$-linearity should
be clear) and $text{Hom}_A^*(E,F)$
are the adjointable $A$-linear
operators from $E$ to $F$.

I know that there exists a Schwartz kernel theorem for vector-valued distributions (see e.g. theorem 1.8.9 in http://user.math.uzh.ch/amann/files/distributions.pdf). This looks like

$mathcal{E}'(X times Y,Z') simeq L(C^infty(X), mathcal{E}'(Y,Z'))$

for a Banach space $Z$ and I heard that Schwartz has proven similar theorems in much greater generality, but without considering Hilbert modules of course. The above theorem is close to what I want, if $text{Hom}_A^*(E,F)$ has a predual, then it implies

$mathcal{E}'(X times Y,text{Hom}_A^*(E,F)) simeq L(C^infty(X), mathcal{E}'(Y,text{Hom}_A^*(E,F)))$.