nLab functorial aspects of the GNS representation


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This is a small subcollection of my (Tom Mainiero‘s) personal notes on some functorial aspects of the Gelfand-Naimark-Segal construction (GNS construction) that have been sitting around for a while. Please feel free to contribute! The ideas here underlie some of the constructs of Homtools: specifically, one can think of a “ltipartite’‘ state as a presheaf valued in the category of states (defined below) over a collection of subsystems. The GNS representation can then be thought of as a functor that takes us into a “representation-theoretic” category where we can, e.g. compute an associated (co)homology using Čech methods.

Baby’s first GNS functor

For the purposes of this section, RR is a fixed a unital ring and QQ is a unital RR-algebra. Although RR will not appear explicitly in any of the discussion in this section (so that QQ could effectively just be thought of as a unital ring), we keep it around as the case where QQ is a finite-dimensional (semisimple) \mathbb{C}-algebra is most relevant to later discussions involving the GNS construction.


Let QMod {}_{Q}\mathbf{Mod}^{\bullet} denote the category with:

  • objects given by pointed left Q Q -modules: pairs (M,m)(M,m) of a left QQ-module MM with an element mMm \in M;

  • morphisms given by point preserving intertwiners.

QMod {}_{Q}\mathbf{Mod}^{\odot} is the full subcategory of QMod {}_{Q}\mathbf{Mod}^{\bullet} whose objects are cyclic pointed modules: (M,m)(M,m) such that Qm=MQ m = M.


For those that find the notion of a pointed module somewhat awkward, note that: QMod {}_{Q}\mathbf{Mod}^{\bullet} is equivalent to the coslice category QQ/ QMod{}_{Q}{Q}/{}_{Q}\mathbf{Mod}, where QMod{}_{Q}\mathbf{Mod} is the category of left Q Q -modules.


QMod {}_{Q}\mathbf{Mod}^{\odot} is a reflective subcategory of QMod {}_{Q}\mathbf{Mod}^{\bullet}: the right adjoint of the inclusion functor

ι: QMod QMod \iota \colon {}_{Q}\mathbf{Mod}^{\odot} \longrightarrow {}_{Q}\mathbf{Mod}^{\bullet}

is given by the “cyclification” functor:

Cyc: QMod QMod \mathsf{Cyc} \colon {}_{Q}\mathbf{Mod}^{\bullet} \longrightarrow {}_{Q}\mathbf{Mod}^{\odot}

that acts on objects by assigning to any pointed module (M,m)(M,m) the cyclic pointed submodule generated by the orbit of mm, i.e. Cyc:(M,m)(Qm,m)\mathsf{Cyc} \colon (M,m) \to (Q \cdot m, m) and acts on morphisms by restriction to cyclic submodules.


For any left QQ-module MM, we denote the annihilator of a point mMm \in M as:

Ann Q(m){qQ:qm=0}. \mathrm{Ann}_{Q}(m) \coloneqq \{ q \in Q : q \cdot m = 0 \}.

The following lemma, which will become useful later, describes the structure of the hom-sets QMod ((M,m),(N,n)){}_{Q}\mathbf{Mod}^{\bullet}((M,m), (N,n)) when (M,m)(M,m) is a cyclic pointed module.


Let (M,m)(M,m) and (N,n)(N,n) be pointed QQ-modules and suppose (M,m)(M,m) is cyclic. Then there exists a morphism from (M,m)(M,m) to (N,n)(N,n) if and only if Ann Q(m)Ann Q(n)\mathrm{Ann}_{Q}(m) \subseteq \mathrm{Ann}_{Q}(n). Moreover, this morphism is unique when it exists.


This follows quickly from the fact that a morphism f:(M,m)(N,n)f: (M,m) \rightarrow (N,n) must satisfy f(qm)=qnf(q m) =q n for all qQq \in Q.

There is an immediate corollary of this Lemma: QMod {}_{Q}\mathbf{Mod}^{\odot} is equivalent to the poset of left-ideals of QQ (thought of as a category). This equivalence of categories is actually part of the restriction of a larger adjunction. We now spell out the definitions behind this claim.


L(Q)\mathbf{L}(Q) is the thin category given by the poset of left ideals of QQ:

  • objects are left ideals of QQ, and

  • there is a unique morphism I!JI \overset{!}{\to} J if and only if IJI \subseteq J.

The process of taking annihilators of distinguished points of left QQ-modules defines a functor:

Ann Q: QMod L(Q). \mathsf{Ann}_{Q} \colon {}_{Q}\mathbf{Mod}^{\bullet} \longrightarrow \mathbf{L}(Q).

Explicitly: it acts on objects by:

Ann Q:(M,m)Ann Q(m), \mathsf{Ann}_{Q} \colon (M,m) \mapsto \mathrm{Ann}_{Q}(m),

and its action on morphisms is deduced from the following straightforward remark.


If f:(M,m)(N,n)f \colon (M,m) \to (N,n) is a morphism of pointed modules, then qn=qf(m)=f(qm)q \cdot n = q \cdot f(m) = f(q \cdot m); so we must have Ann Q(m)Ann Q(n)\mathrm{Ann}_{Q}(m) \subseteq \mathrm{Ann}_{Q}(n).

Thus, for f:(M,m)(N,n)f \colon (M,m) \to (N,n) a Mod Q \mathbf{Mod}^{\bullet}_{Q}-morphism, we can define Ann Q(f):Ann Q(m)Ann Q(n)\mathsf{Ann}_{Q}(f) \colon \mathsf{Ann}_{Q}(m) \to \mathsf{Ann}_{Q}(n) as the unique L(Q)\mathbf{L}(Q)-morphism Ann Q(m)!Ann Q(n)\mathsf{Ann}_{Q}(m) \overset{!}{\to} \mathsf{Ann}_{Q}(n) given by the inclusion of ideals Ann Q(m)Ann Q(n)\mathrm{Ann}_{Q}(m) \subseteq \mathrm{Ann}_{Q}(n).

Alternatively, given any left-ideal II of some RR-algebra QQ, one can produce an QQ-module Q/IQ/I. This module has a distinguished point [1 Q]1 Q+I[1_{Q}] \coloneqq 1_{Q} + I, where 1 Q1_{Q} is the identity of QQ. Moreover, Q[1 Q]=Q/IQ \cdot [1_{Q}] = Q/I; so, so the pointed module (Q/I,[1])(Q/I, [1]) is cyclic. This process defines a functor:

Quot:L(Q) QMod \mathsf{Quot} \colon \mathbf{L}(Q) \longrightarrow {}_{Q}\mathbf{Mod}^{\odot}

whose action on objects is as described above:

Quot:I(Q/I,[1 Q]) \mathsf{Quot} \colon I \mapsto (Q/I, [1_{Q}])

and action on morphisms is given by:

Quot:(I!J)τ J I \mathsf{Quot} \colon (I \overset{!}{\to} J) \mapsto \tau^{I}_{J}

where τ J I\tau^{I}_{J} is the tautological map of pointed modules acting by:

τ J I:q+Iq+J \tau^{I}_{J}: q + I \mapsto q + J

for any qQq \in Q; this is well-defined as JIJ \subseteq I.


There is an adjunction:

restricting to an equivalence

where Quot^ Q=ιQuot Q\widehat{\mathsf{Quot}}_{Q} = \iota \circ \mathsf{Quot}_{Q} for ι\iota the inclusion functor of cyclic pointed modules into pointed modules.


We will define a unit

η:id L(Q)Ann QQuot \eta \colon \mathrm{id}_{\mathbf{L}(Q)} \longrightarrow \mathsf{Ann}_{Q} \circ \mathsf{Quot}

and counit

ϵ:QuotAnn Qid QMod . \epsilon \colon \mathsf{Quot} \circ \mathsf{Ann}_{Q} \longrightarrow \mathrm{id}_{{}_{Q}\mathbf{Mod}^{\bullet}}.

satisfying the triangle identities. Note that η\eta is given by assigning the identity morphism to each object of L(Q)\mathbf{L}(Q): indeed, for any left-ideal we have Ann QQuot(I)=Ann Q[(Q/I,[1])]=I\mathsf{Ann}_{Q} \circ \mathsf{Quot}(I) = \mathsf{Ann}_{Q} \left[(Q/I,[1]) \right] = I. The counit ϵ\epsilon is a bit more interesting: let M=(M,m)\mathsf{M} = (M,m) be a pointed-module and define (K,k)QuotAnn Q(M)=(Q/Ann Q(m),[1])(K,k) \coloneqq \mathsf{Quot} \circ \mathsf{Ann}_{Q}(\mathsf{M}) = (Q/\mathrm{Ann}_{Q}(m),[1]). We have Ann Q(k)=Ann Q(m)\mathrm{Ann}_{Q}(k) = \mathrm{Ann}_{Q}(m); so by Lemma there is a unique morphism (K,k)(M,m)(K,k) \rightarrow (M,m); this defines the morphism ϵ (M,m)\epsilon_{(M,m)}. The triangle identities are trivial.

Connection to the GNS construction

The above observations are simple shadows of the functorality of the GNS construction for C*-algebras and W*-algebras. The opposite of the category of ideals L(Q)\mathbf{L}(Q) can be thought of as a stand-in for the category of states (to every C *C^\ast-algebraic state ρ:E\rho \colon E \to \mathbb{C} on a C *C^{\ast}-algebra EE we can associate the left ideal given by the “left kernel” {eE:ρ(e *e)=0}\{e \in E: \rho(e^{*} e) = 0 \}). The quotient functor Quot\mathsf{Quot} is a toy-model of the GNS construction.


In the following the letters E,FE, F and GG will denote C *C^\ast-algebras that are not necessarily W *\W^\ast-algebras and A,B,CA, B, C will denote W *W^\ast-algebras.

The term (normal) state will be taken to mean (normal) positive linear functional Note that this is at odds with some definitions in the physics literature, where “state” typically means “normalized” positive linear functional (if f(1 E)=1f(1_{E}) = 1 for f:Ef: E \to \mathbb{C} a positive linear functional). Using the definition of “state” here, the zero map 0:E0: E \to \mathbb{C} is a perfectly valid state.

The GNS construction as a functor on a fixed C*^\ast-algebra

With an appropriate notion of a “category of states”, one can explore functorial properties of the GNS construction as in Arthur Parzygnat’s work (see Parzygnat2016 and Parzygnat2018).

The construction here is from a slightly different perspective, using a mildly richer category of states based on the natural poset structure on the collection of (normal) states on a fixed W *W^\ast or C *C^\ast-algebra. The posetal relation is intimately related to the notion of a Radon-Nikodym derivative.

The category of (normal) states over a fixed (W*^\ast) C*^\ast-algebra.

Begin by recalling a well-known posetal relation on the set of states State(E)\text{State(E)} on a C *C^\ast-algebra EE:


For any f,gState(E)f, g \in \text{State}(E), define the relation \leq by fgf \leq g if fgf - g is a state, where - denotes pointwise subtraction; equivalently: fgf \leq g if f(e *e)g(e *e)f(e^\ast e) \leq g(e^\ast e) for all eEe \in E

It is a straightforward exercise to show this relation is posetal.

The story for normal states in the W *W^\ast-algebra world is nearly identical: let NState(A)\text{NState(A)} denote the set of normal states on a W *W^\ast-algebra AA (a subset of State(A)\text{State}(A)); for any f,gNState(A)f, g \in \text{NState(A)} we can define a relation norm\leq_{\text{norm}} by: f normgf \leq_{\text{norm}} g if fgf - g is a normal state. Because the space of linear functionals splits as an orthogonal sum into normal and singular parts, it follows immediately that f normgf \leq_{\text{norm}} g for normal linear functionals ff and gg if and only if fgf \leq g. As a result we can simply write \leq when discussing states on C *C^\ast-algebras or normal states on W *W^\ast-algebras without any issues if we forget down from the category of W *W^\ast-algebras to the category of C *C^\ast-algebras.


We can easily define weaker versions of the relation \leq that may also be of interest: for ff and gg positive linear functionals on a C *C^\ast algebra EE, we can define relations \ll and \lesssim in the following manner:

  • fgf \ll g if there is an inclusion of ker L(g)ker L(f)\ker^{L}(g) \subseteq \ker^{L}(f), where, for any positive linear functional hh on EE, the “left kernel” or “vanishing ideal” is defined by:

    ker L(h){eE:h(e *e)=0}. \ker^{L}(h) \coloneqq \{e \in E: h(e^\ast e ) = 0 \}.
  • fgf \lesssim g if there exists a C0C \geq 0 such that f(e *e)Cg(e *e)f(e^\ast e) \leq C g(e^\ast e) for every eEe \in E.

We have the following immediate properties:

  • fgfgfgf \leq g \Rightarrow f \lesssim g \Rightarrow f \ll g.

  • \leq is a partial order as it is antisymmetric: fgf \leq g and gfg \leq f if and only if f=gf = g.

However, the weaker preorders \ll and \lesssim do not satisfy antisymmetry: e.g. 2ff2 f \lesssim f and f2ff \lesssim 2f.

Moreover, while \lesssim is only a mild weakening of \leq, the weakest preorder \ll is vastly different. In particular, we will show there is a unique Radon-Nikodym derivative associated to every pair of functionals related by the preorders \leq and \lesssim; the result can be weakened for \ll, but the Radon-Nikodym derivative will be an unbounded (densely defined) operator. See Gudder.


Let EE be a C *C^\ast-algebra. The category State(E)\mathbf{State}(E) is the opposite of the thin category determined by the preorder \leq, i.e. it is the category whose objects are all states (positive linear functionals) on EE, and whose morphism sets are given by

State(E)(ω,ν)={{}, νω , otherwise \mathbf{State}(E)(\omega, \nu) = \left\{ \begin{array}{ll} \{\star\}, & \nu \leq \omega\\ \emptyset, & \text{otherwise} \end{array} \right.

where {}\{\star\} denotes the one point set. Similarly if AA is a W *W^\ast-algebra, let NState(A)\mathbf{NState}(A) be defined as the full subcategory of State(A)\mathbf{State}(A) whose objects are normal states.


For EE a C *C^\ast-algebra, the category Rep (E)\mathbf{Rep}^{\bullet}(E) is the category with:

  • objects given by pointed *\ast-representations of EE: triples (,v,π:EB)(\mathcal{H}, v, \pi: E \to \mathrm{B} \mathcal{H}) of a Hilbert space \mathcal{H} a vector vv \in \mathcal{H} and a *\ast-representation π\pi;

  • morphisms given by contractive linear point-preserving intertwiners: a morphism from ( 1,v 1,π 1)(\mathcal{H}_{1}, v_{1}, \pi_1) to ( 2,v 2,π 2)(\mathcal{H}_2, v_2, \pi_2) is a linear map f: 1 2f: \mathcal{H}_{1} \to \mathcal{H}_{2} such that f1\|f\| \leq 1 with respect to the operator norm, fv 1=v 2f v_{1} = v_{2}, and

    fπ 1(e)=π 2(e)f f \circ \pi_{1}(e) = \pi_{2}(e) \circ f

    for all eEe \in E.

Rep (E)\mathbf{Rep}^{\odot}(E) is the full subcategory of Rep (E)\mathbf{Rep}^{\bullet}(E) whose objects are cyclic representations: (,v,π:EB)(\mathcal{H}, v, \pi: E \to \mathrm{B} \mathcal{H}) such that the subspace Ev={ev:eE}E v = \{e v : e \in E \} is dense in \mathcal{H}.

We also define normal versions of the above.


Similarly, for AA a W *W^\ast-algebra, the category NRep (A)\mathbf{NRep}^{\bullet}(A) is the category whose objects are pointed normal *\ast-representations of AA, and morphisms are contractive linear intertwiners (equivalently the full subcategory of Rep (A)\mathbf{Rep}^{\bullet}(A) generated by objects whose underlying representation is normal). NRep (A)\mathbf{NRep}^{\odot}(A) is the full subcategory of NRep (A)\mathbf{NRep}^{\bullet}(A) generated by cyclic representations.

Maybe add a bit here about how NRep and Rep interact.

The following are a few straightforward observations about the subcategories (N)Rep ()\mathbf{(N)Rep}^{\odot}(\ldots) generated by cyclic objects.


The cyclic subcategories are thin: there is at most one morphism between two objects, every morphism is uniquely determined by where distinguished point is sent.


Just as the full subcategory of cyclic pointed QQ-modules (for QQ an RR-algebra) is reflective inside of the category of pointed AA-modules (c.f.\ Remark ), the subcategory Rep (E)\mathbf{Rep}^{\odot}(E) (resp. NRep (A)\mathbf{NRep^{\odot}}(A)) is a reflective subcategories of Rep (A)\mathbf{Rep}^{\bullet}(A) (resp. NRep (A)\mathbf{NRep^{\bullet}}(A)). The right adjoint of the inclusion functor is given on objects by assigning to any pointed representation the completion of the orbit of the distinguished point; it is defined on morphisms by restriction.

The observations in the above remark are shadows of adjunctions (EE a C *C^{\ast}-algebra and AA a W *W^{\ast}-algebra)

which restricts to an equivalence of categories:

The analogous statements hold for normal states/pointed representations. Our next step is to unravel the functors in the above diagrams, both which should be clear to those familiar with the GNS construction


The functor:

Fnl:Rep (E)State(E) \mathsf{Fnl} \colon \mathbf{Rep}^{\bullet}(E) \longrightarrow \mathbf{State}(E)

acts on objects by:

Fnl:(,v,π:EB)(ev,π(e)v ), \mathsf{Fnl} \colon (\mathcal{H}, v, \pi: E \rightarrow \operatorname{B} \mathcal{H}) \mapsto (e \mapsto \langle v, \pi(e) v \rangle^{\mathcal{H}} ),

and takes a morphism fRep (E)[( 1,v 1,π 1),( 2,v 2,π 2)]f \in \mathbf{Rep}^{\bullet}(E)[(\mathcal{H}_{1}, v_{1}, \pi_{1}), (\mathcal{H}_{2}, v_{2}, \pi_{2})] to the unique morphism in State(E)\mathbf{State}(E) defined by the relation:

v 2,π 2(e *e)v 2=fv 1,π 2(e *e)fv 1=v 1,f *fπ 1(e *e)v 1v 1,π 1(e *e)v 1, \langle v_{2}, \pi_{2}( e^{*} e ) v_{2} \rangle = \langle f v_{1}, \pi_{2}( e^{*} e ) f v_{1} \rangle = \langle v_{1}, f^{*} f \pi_{1}( e^{*} e ) v_{1} \rangle \leq \langle v_{1}, \pi_{1}( e^{*} e) v_{1} \rangle,

for all eEe \in E.

Before defining the functor L 2\mathsf{L^2}, we note that to any positive linear functional ω:E\omega \colon E \to \mathbb{C}, there is an associated left ideal ker Lω\ker^{L} \omega. Thus, repeating the observations of Section, there is an associated pointed (algebraic) left EE-module E/ker LωE/\ker^{L} \omega


The functor L 2:State(E)Rep (E)\mathsf{L^2} \colon \mathbf{State}(E) \rightarrow \mathbf{Rep}^{\bullet}(E) acts on objects by taking a positive linear functional ω\omega to its corresponding GNS representation, and acts on a morphism νω\nu \to \omega corresponding the relation ων\omega \leq \nu by assigning the quotient map q¯(ω,ν):L 2νL 2ω\overline{q}(\omega, \nu) \colon \mathsf{L^{2}} \nu \to \mathsf{L^{2}} \omega


Last revised on June 6, 2023 at 08:16:45. See the history of this page for a list of all contributions to it.