Does Stinespring's dilation theorem hold if the algebra of interest is a topological $\ast$-algebra instead of the usual $C^{\ast}$-algebra?


I will now state the version of Stinespring's dilation theorem that I know, and am wondering what happens if I were to replace the $C^{\ast}$-algebra with a topological $\ast$-algebra.


**Theorem.** Let $\mathfrak{A}$ be a unital $C^{\ast}$-algebra, and let $\Phi : \mathfrak{A} \to \mathcal{B}(\mathcal{H})$ be a completely positive map. Then there exists a Hilbert space $\hat{\mathcal{H}}$, a unital $\ast$-homomorphism $\pi : \mathfrak{A} \to \mathcal{B}(\hat{\mathcal{H}})$ and a bounded operator $V : \mathcal{H} \to \hat{\mathcal{H}}$ with $||\Phi(1\_{\mathfrak{A}})||=||V||^{2}$ such that 
\begin{equation}
\Phi (a)= V^{\*} \pi (a) V, \; \; \; \; \; a \in \mathfrak{A}.
\end{equation}


**Notation.** Let $1\_{\mathfrak{A}}$ denote the unit in $\mathfrak{A}.$


References to literature on the subject would be greatly appreciated as well. 



Stinespring's dilation without $C^{\ast}$-algebras
The surprising fact is that the GNSStinespringKasparov theory is in fact completely algebraic, at least to a very large extend: the following results have been obtained by a PhD of mine but are, unfortunately, not published (yet?). 


So the setting is a \*-algebra over a ring $C$ which is of the form $C = R(i)$ with $i^2 = -1$ and $R$ being an ordered ring. In this framework one can define positive functionals to be linear functionals $\omega\colon A \longrightarrow C$ satisfying $\omega(a^\*a) \ge 0$. This gives also rise to the notion of a positive element by asking for $\omega(a) \ge 0$ for all positive $\omega$. One can then state what a positive linear map is (mapping positive elements to positive ones) and hence what $n$-positive and completely positive maps are. So far, the notion is entirely algebraic and, of course, reproduces the notions for a $C^\*$-algebra you know.


Then one can proceed by constructiong pre-Hilbert modules over \*-algebras. Without topology, we can of course not speak of completeness, but this "pre" is already interesting. The only catch is that one has to ask for completely positive inner products: for a Hilbert module over a $C^\*$-algebra, this is a consequence of positivity, but in this algebraic setting, there seems to be no way to prove that (hint very welcome!).


From there on, one can proceed with many things like a strong Morita theory for \*-algebras, etc. Among them is the algebraic part of the Stinespring construction: if you go through it, then we will see that most of it is algebraic and can be transfered to this setting.


The name of the PhD student is Florian Becher and you can find his thesis somewhere on the document server of Freiburg university.


On my homepage you find a lot of ref's on the algebraic \*-representation theory of \*-algebras, Morita theory and things, including a (not yet finished) lecture note on this.


