In which sense the GNS-construction is a functor?

Question

I asked this at mathstackexchange a week ago, without success.

I think the Gelfand–Naimark–Segal 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.

Answer 1

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 $\tilde{\varphi}$.

Answer 2

Although this response is a bit late, perhaps this perspective may help nonetheless. It only addresses the question about functoriality of the GNS construction.

The GNS construction is not quite a functor, though as Nik Weaver's response indicates, it has a certain functoriality associated with it. The GNS construction, is, however, (almost) a natural transformation that satisfies a universal property, which is indicated in Westerbaan's response. Let me make this more precise.

Background: There are two relevant functors on the category of $C^*$-algebras. One is the functor that associates the set of all states to every $C^*$-algebra and the function that pulls back states to states to every $C^*$-algebra $*$-homomorphism. This is the ``states'' functor $\mathcal{S}:\mathbf{C}^*\text{-}\mathbf{Alg}^{\mathrm{op}}\to\mathbf{Set}.$ There is another functor that sends a $C^*$-algebra to its category of representations and a $*$-homomorphism to a functor between categories of representations $\mathbf{Rep}:\mathbf{C}^*\text{-}\mathbf{Alg}^{\mathrm{op}}\to\mathbf{Cat}.$ A representation is not enough to produce a state, so we add the additional data in the category of representations to include a normalized vector. Morphisms in this category are then taken to be intertwining isometries preserving unit vectors. Thus, we write $\mathbf{Rep}^{\bullet}:\mathbf{C}^*\text{-}\mathbf{Alg}^{\mathrm{op}}\to\mathbf{Cat}$ for this new functor. The codomains of $\mathcal{S}$ and $\mathbf{Rep}^{\bullet}$ are not the same so it does not make sense to compare them. By viewing every set as a discrete category, we can take the codomain of $\mathcal{S}$ to be $\mathbf{Cat}$ as well. To distinguish $\mathcal{S}$ from this functor, let's call the latter $\mathbf{States}.$

Statement: The GNS construction furnishes a pseudo-natural transformation $\mathbf{GNS}^{\bullet}:\mathbf{States}\Rightarrow\mathbf{Rep}^{\bullet}.$

It is not a strict natural transformation. Technically, it is a 1-morphism in the 2-category of functors from $\mathbf{C}^*\text{-}\mathbf{Alg}^{\mathrm{op}}$ to $\mathbf{Cat}.$ The objects here are functors, 1-morphisms are pseudo-natural transformations (natural transformations that satisfy naturality up to a higher coherence), and 2-morphisms are modifications.

Universal property: The GNS construction is actually part of an adjunction in this 2-category mentioned in the previous paragraph. It is left adjoint to the natural transformation (which is an honest natural transformation) that takes a representation with a unit vector and simply pulls back the canonical state associated to this unit vector to the underlying $C^*$-algebra.

[There is supposed to be an image here---sorry, not enough reputation points---please see the introduction in the paper below]

More details can be found in https://arxiv.org/abs/1609.08975 "From observables and states to Hilbert space and back: a 2-categorical adjunction," which was coincidentally posted not long after your question, though I was unaware of this until now.

Answer 3

Expanding on Nik Weaver's answer: one can also make Stinespring's dilation into a functor and even get an adjunction (but maybe not the one you were after.) One can go even further and also give an adjunction for Paschke's GNS.

Minimal Stinespring dilation as an adjunction

Chris Heunen proposed the following construction. Let $C_1$ denote the category with as objects normal completely positive linear maps of the form $\varphi\colon A \to B(H)$ for a von Neumann algebra $A$ and Hilbert space $H$. An arrow between two objects $\varphi_1\colon A_1 \to B(H_1)$ and $\varphi_2\colon A_2 \to B(H_2)$ is a pair of maps: a normal $*$-homomorphism $m\colon A_1 \to A_2$ and an operator $T\colon H_1 \to H_2$ such that $\varphi_1 = \textrm{ad}_T \circ \varphi_2 \circ m$. Let $C_2$ denote the subcategory of $C_1$ restricting to those objects that are normal *-homomorphisms. Let $U$ denote the inclusion functor from $C_2$ to $C_1$. Then $U$ has a left-adjoint $S$ which sends $\varphi$ to its minimal Stinespring representation. The unit of the adjunction is given by the pair $(\textrm{id}, \textrm{ad}_V)$ where $\textrm{ad}_V$ is the right-hand map in the minimal Stinespring dilation. This is straight-forward to show with the UMP definition of adjoint functors once we know the following Proposition:

Proposition 13 of [1]. Let $(K, \pi, V)$ and $(K', \pi', V')$ be two normal Stinespring dilations for the same map $\textrm{ad}_V \circ \pi = \textrm{ad}_{V'} \circ \pi'$. If $(K, \pi, V)$ is minimal, then there is a unique $S\colon K \to K'$ with $SV=V'$ and $\pi = \textrm{ad}_S \circ \pi'$.

In the case both dilations are minimal, this result is well-known. We couldn't find this generalisation in the literature and so published a proof in [1]. It's not very long, but requires a trick.

Paschke's GNS as an adjunction

Now consider the category $C_3$ with as objects completely positive normal linear maps between arbitrary von Neumann algebra's. An arrow between $\varphi_1 \colon A_1 \to B_1$ and $\varphi_2 \colon A_2 \to B_2$ is given by a pair of a normal $*$-homomorphism $m\colon A_1 \to A_2$ and any completely positive normal linear map $h\colon B_2 \to B_1$ such that $h \circ \varphi_2 \circ m = \varphi_1$. Let $C_4$ denote the subcategory of $C_3$ restricting to those objects that are $*$-homomorphisms. The inclusion functor $U\colon C_4 \to C_3$ has a left-adjoint $P$, which sends a map $\varphi\colon A \to B$ to its Paschke GNS representation $A \to B^a(A \otimes_\varphi B)$. This Paschke GNS representation has a similar universal property as Stinespring's dilation with which you can show this fact. This universal property is the main topic of [1].

[1] Westerbaan & Westerbaan, Paschke Dilations http://arxiv.org/pdf/1603.04353v1.pdf