# In which sense the GNS-construction is a functor?

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.

## 3 Answers

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 $\tilde{\varphi}$.

• Nik, I did not understand, how $f$ acts on operators from $B(L_2[0,2])$. – Sergei Akbarov Aug 16 '16 at 15:23
• Sergei --- $f(x) = \langle x 1_{[0,1]}, 1_{[0,1]}\rangle$ for $x \in B(L^2[0,2])$. – Nik Weaver Aug 16 '16 at 15:24
• Nik, something is wrong: as far as I understand, $(f\circ\varphi)(a)=a(0)$ for $a\in A$, so $H_{f\circ\varphi}={\mathbb C}\ne L_2[0,1]$. – Sergei Akbarov Aug 16 '16 at 16:53
• You're right, there's a typo. I meant that functions in $A$ should be constant on $[1,2]$. – Nik Weaver Aug 16 '16 at 17:28
• Nik, I suddenly realized that this mapping $b\in B(K)\mapsto V^* b V\in B(H)$ is not a homomorphism of algebras (unless $V$ is an isomorphism of Hilbert spaces). Is that correct? – Sergei Akbarov Feb 24 '20 at 20:48

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.

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