I asked this at [mathstackexchange](https://math.stackexchange.com/questions/1886719/in-which-sense-the-gns-construction-is-a-functor) a week ago, without success. 


I think the [Gelfand–Naimark–Segal construction](https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark%E2%80%93Segal_construction) must be a functor in some sense, but I can't find an explicit statement anywhere. Can anybody enlighten me?


For example, is the following hypothesis true?



> 
> Let $\varphi:A\to B$ be an involutive homomorphism of C\*-algebras, and let $f$ be a state on $B$. Consider the corresponding state $f\circ\varphi$ on $A$ and the GNS-constructions $\pi\_{f\circ\varphi}:A\to {\mathcal B}(H\_{f\circ\varphi})$ and $\pi\_f:B\to {\mathcal B}(H\_f)$. Let $\widetilde{A}$ and $\widetilde{B}$ be the von Neumann algebras in ${\mathcal B}(H\_{f\circ\varphi})$ and ${\mathcal B}(H\_f)$ generated by $\pi\_{f\circ\varphi}(A)$ and $\pi\_f(B)$ respectively, and let us consider $\pi\_{f\circ\varphi}$ and $\pi\_f$ as homomorphisms with ranges in $\widetilde{A}$ and $\widetilde{B}$. Is there a homomorphism $\widetilde{\varphi}:\widetilde{A}\to\widetilde{B}$ such that 
>  $$
> \pi\_f\circ\varphi=\widetilde{\varphi}\circ\pi\_{f\circ\varphi} \,\,\,\,\,\text{?}
> $$ 
> 
> 
> 


I believe, it is not essential here that $A$ and $B$ are C\*-algebras, they can just be topological algebras with involution is some sense, and the states can be defined as continuous positive functionals $f:A\to{\Bbb C}$ such that the map $x\mapsto f(x^\*\cdot x)$ is also continuous.


I need this for my current work, if anybody could help, I would appreciate this very much.



In which sense the GNS-construction is a functor?
If you want $\tilde{\varphi}$ to be normal then this is false. But first let me point out that there is a sense in which the GNS construction is a functor. Note that $\varphi$ induces an isometric embedding of $H\_{f\circ\varphi}$ into $H\_f$ (the respective GNS Hilbert spaces). So define a morphism between two representations $\pi: A \to B(H)$ and $\rho: B \to B(K)$ to be a $\*$-homomorphism $\varphi: A \to B$ together with an isometric embedding $V: H \to K$ such that $\pi = V^\*(\rho\circ\varphi)V$. Then the GNS construction is functorial.


Here is a counterexample to the hypothesis you give, assuming normality of $\tilde{\varphi}$. Let $A$ be the continuous functions on $[0,2]$ which are constant on $[1,2]$, acting by multiplication on $L^2[0,2]$. Let $B = B(L^2([0,2]))$. Let $\varphi: A \to B$ be the inclusion and let $f$ be the vector state given by the unit vector $1\_{[0,1]}$. Then $H\_{f\circ\varphi} = L^2[0,1]$ and $\pi\_{f\circ\varphi}$ is the restriction of the given representation to $H\_{f\circ\phi}$, whereas $H\_f = L^2[0,2]$ and $\pi\_f$ is the identity representation of $B$. So $\tilde{A} = L^\infty[0,1]$ and $\tilde{B} = B$, and if there were a normal homomorphism $\tilde{\varphi}$ of the desired type then it would restrict to $\varphi$ on $A$. But the functions $f\_n(t) = \begin{cases}\cos(2\pi nt)&0 \leq t \leq 1\cr 1&1 \leq t \leq 2\end{cases}$ in $A$ converge weak\* to $0$ in $\tilde{A}$ but they converge weak\* to $1\_{[1,2]}$ in $\tilde{B}$, contradicting normality of $\tilde{\varphi}$.


