# Representation fields and quasi-state duality

A C*-algebra can be recovered from its representations on one sufficiently large Hilbert space. Bounded operator fields that respect intertwiners recover the bidual; continuity distinguishes the original algebra. The argument below constructs the needed local realizations of states using finite Gram matrices. No separability assumption is made.

## Representations and their essential spaces

Let \(A\) be a C*-algebra and let \(H\) be an infinite-dimensional Hilbert space whose Hilbert dimension \(\kappa\) is at least the dimension of every cyclic representation of \(A\). Inner products are linear in the first variable. Put

\[
\mathcal R=\operatorname{Rep}(A,H)
 =\{\pi:A\longrightarrow B(H):\pi\text{ is a *-homomorphism}\}.
\]

We include degenerate representations. For \(\pi\in\mathcal R\), let \(p_\pi\) be the projection onto

\[
H_\pi=\overline{\pi(A)H}.
\]

The representation is zero on \(H_\pi^\perp\) and nondegenerate on \(H_\pi\). Thus this is also the space of nondegenerate representations on closed subspaces of \(H\), extended by zero. The zero representation corresponds to the zero subspace.

An **admissible field** is a family \(F=(F_\pi)_{\pi\in\mathcal R}\) such that

\[
\begin{gathered}
F_\pi\in p_\pi B(H)p_\pi,\qquad
\|F\|_\infty=\sup_{\pi\in\mathcal R}\|F_\pi\|<\infty,\\
T\pi(a)=\rho(a)T\quad(a\in A)
\quad\Longrightarrow\quad
TF_\pi=F_\rho T
\end{gathered}
\tag{1.1}
\]

for every \(\pi,\rho\in\mathcal R\) and every \(T\in B(H)\). In particular \(F_0=0\). Equivalently one can specify operators on the variable spaces \(H_\pi\) and use all bounded intertwiners between those spaces. Indeed an intertwiner preserves essential spaces, as does its adjoint; its other block acts between the annihilated complements and contributes zero to both sides of (1.1).

Intertwining the inclusions of orthogonal reducing summands shows that (1.1) includes compatibility with orthogonal direct sums whenever their essential space embeds in \(H\). Conversely, direct-sum compatibility alone is not our definition: all bounded intertwiners are required.

The earlier programme proofs used here are the GNS construction and uniqueness, Theorems 5.3–5.5, the universal bidual construction and normal extension of a representation, Lemma 2.1, continuous functional calculus and its norm continuity, and continuous affine functions on the quasi-state space, Theorem 3.1. We use their actual conclusions: \(A^{**}\) is the universal von Neumann algebra with predual \(A^*\); each \(\pi\) has a normal extension \(\bar\pi\), with \(\bar\pi(1)=p_\pi\); and continuous real affine functions on \(Q(A)\) vanishing at zero are evaluations at self-adjoint elements of \(A\).

## The pointwise operator topologies agree

Give \(\mathcal R\) the pointwise weak operator topology: a net \(\pi_i\) converges to \(\pi\) when \(\langle\pi_i(a)\xi,\eta\rangle\to\langle\pi(a)\xi,\eta\rangle\) for every \(a\in A\) and \(\xi,\eta\in H\).

For clarity, the sigma-strong* topology on \(B(H)\) is generated by the seminorms

\[
q_{\boldsymbol\xi}(T)^2
 =\sum_{n=1}^\infty
   \bigl(\|T\xi_n\|^2+\|T^*\xi_n\|^2\bigr),
\qquad \sum_n\|\xi_n\|^2<\infty.
\tag{2.1}
\]

Pointwise sigma-strong* convergence of representations requires convergence in these seminorms separately for every \(a\).

**Theorem 2.1.** Pointwise weak operator, pointwise strong operator, pointwise strong*, and pointwise sigma-strong* topologies coincide on \(\mathcal R\).

**Proof.** Suppose \(\pi_i\to\pi\) pointwise weakly. Multiplicativity gives, for fixed \(a,\xi\),

\[
\begin{aligned}
\|\pi_i(a)\xi-\pi(a)\xi\|^2
={}&\langle\pi_i(a^*a)\xi,\xi\rangle\\
&-2\operatorname{Re}\langle\pi_i(a)\xi,\pi(a)\xi\rangle
 +\|\pi(a)\xi\|^2\longrightarrow0.
\end{aligned}
\tag{2.2}
\]

Applying this also to \(a^*\) proves pointwise strong* convergence. The bound
\(\|\pi_i(a)-\pi(a)\|\le2\|a\|\) makes the tail of (2.1), for \(T=\pi_i(a)-\pi(a)\), at most

\[
8\|a\|^2\sum_{n>N}\|\xi_n\|^2,
\]

uniformly in \(i\). Choose \(N\) to make this small, and then use convergence of the finite sum. This proves sigma-strong* convergence for nets as well as sequences. The reverse implications follow from the one-vector seminorms and Cauchy–Schwarz. Equality of convergent nets gives equality of the topologies. Indeed, if a neighborhood for one topology contained no neighborhood for the other, choose a point outside it from each neighborhood of the latter, ordered by reverse inclusion. That net converges only in the latter topology, a contradiction. \(\square\)

The calculation depends on the representations preserving both multiplication and adjoints. It is not an assertion that weak and strong topologies agree on an arbitrary bounded set of operators.

## Bounded fields recover the bidual

**Lemma 3.1 — cyclic decomposition.** A nondegenerate representation is an orthogonal direct sum of cyclic reducing subrepresentations.

**Proof.** For a nonzero vector \(\eta\), the closed space \(\overline{\pi(A)\eta}\) reduces the representation. A contractive approximate identity converges strongly to the identity, so it contains \(\eta\) and has \(\eta\) as a cyclic vector. Choose a maximal orthogonal family of such nonzero reducing subspaces by Zorn's lemma. The representation on their orthogonal complement remains nondegenerate, since the approximate identity still converges to the identity there. If that complement were nonzero, a vector in it would give a further cyclic subspace. Thus the family exhausts the Hilbert space. \(\square\)

**Theorem 3.2 — algebraic reconstruction.** Evaluation gives an isometric *-algebra bijection

\[
A^{**}\longrightarrow\{\text{admissible fields}\},
\qquad x\longmapsto(\bar\pi(x))_{\pi\in\mathcal R}.
\tag{3.1}
\]


**Proof.** For \(x\in A^{**}\), normal extension gives
\(\bar\pi(x)=p_\pi\bar\pi(x)p_\pi\) and \(\|\bar\pi(x)\|\le\|x\|\). If \(T\) intertwines \(\pi\) and \(\rho\), the normal maps
\(x\mapsto T\bar\pi(x)\) and \(x\mapsto\bar\rho(x)T\) agree on the weak* dense copy of \(A\); hence they agree everywhere. Evaluation therefore gives an admissible field.

For the converse, take the direct sum of the GNS representations of all states,

\[
\pi_u=\bigoplus_{\varphi\in S(A)}\pi_\varphi
\quad\text{on}\quad
H_u=\bigoplus_{\varphi\in S(A)}H_\varphi.
\]

Each \(H_\varphi\) embeds isometrically into \(H\), although \(H_u\) itself need not. For an isometry \(J_\varphi:H_\varphi\to H\), put

\[
\rho_\varphi(a)=J_\varphi\pi_\varphi(a)J_\varphi^*,
\qquad
X_\varphi=J_\varphi^*F_{\rho_\varphi}J_\varphi.
\]

If another embedding is used, the partial isometry between the two images intertwines their representations. Equation (1.1) therefore gives the same \(X_\varphi\). More generally, if \(S:H_\psi\to H_\varphi\) intertwines their cyclic representations, applying (1.1) to \(J_\varphi S J_\psi^*\) gives

\[
SX_\psi=X_\varphi S.
\tag{3.2}
\]

The operator \(X=\bigoplus_\varphi X_\varphi\) is bounded by \(\|F\|_\infty\). For \(B\in\pi_u(A)'\), every matrix entry \(B_{\varphi\psi}\) is such an intertwiner. Equation (3.2) says that every matrix entry of \(XB-BX\) vanishes. A bounded operator on a Hilbert direct sum is determined by these entries: test finite-support vectors and use their density. Hence

\[
X\in\pi_u(A)''.
\]

The universal bidual identification supplies a unique \(x\in A^{**}\) with \(\bar\pi_u(x)=X\).

Now take any \(\pi\in\mathcal R\). Decompose its essential space into cyclic subspaces by Lemma 3.1. A unit cyclic vector in each subspace determines a state, and GNS uniqueness identifies that subrepresentation with its state GNS representation. The inclusion into \(H\), composed with this identification, is a bounded intertwiner. Equation (1.1) forces \(F_\pi\) on that subspace to be the same operator as \(\bar\pi(x)\). Both vanish on \(H_\pi^\perp\), so they agree on all of \(H\).

Finally,

\[
\|x\|=\|X\|=\sup_\varphi\|X_\varphi\|
\le\|F\|_\infty\le\|x\|.
\]

Uniqueness follows from the faithful universal realization. Products and adjoints are preserved by every \(\bar\pi\), proving the *-algebra assertion. If \(A=0\), its only representation and only admissible field are zero, and the conclusion holds directly. \(\square\)

## Quasi-states and adjoining a new identity

Write

\[
Q(A)=\{\varphi\in A^*_+:\|\varphi\|\le1\}
\]

with its weak* topology. We will use a **forced unitization** \(A^+=A\oplus\mathbb C1\), even when \(A\) already has an identity. Its product is

\[
(a+\lambda1)(b+\mu1)=ab+\lambda b+\mu a+\lambda\mu1.
\]

It is a C*-algebra: using a faithful representation \(\tau\) of \(A\), realize it as
\((\tau(a)+\lambda I)\oplus\lambda\). This is a closed *-subalgebra of \(B(K)\oplus\mathbb C\): convergence controls \(\lambda\), and then controls \(\tau(a)\), whose range is norm closed. The scalar coordinate distinguishes the new identity from any old one.

**Lemma 4.1.** Restriction is a homeomorphism

\[
S(A^+)\longrightarrow Q(A).
\tag{4.1}
\]

Its inverse sends \(\varphi\) to
\(\varphi^+(a+\lambda1)=\varphi(a)+\lambda\).
Every cyclic representation of \(A^+\) has dimension at most \(\kappa\).

**Proof.** Put \(t=\|\varphi\|\). In its GNS representation, \(\varphi\) has a cyclic vector \(\eta\) of squared norm \(t\). Extend that representation to \(A^+\) by \(a+\lambda1\mapsto\pi_\varphi(a)+\lambda I\), and add the scalar quotient representation with vector \(\sqrt{1-t}\). The resulting vector has norm one and its vector state is exactly \(\varphi^+\). This includes \(t=0\), when the GNS summand is zero. Thus the displayed extension is positive and unital. Restriction of any state is positive and contractive, and its value at the new identity fixes the extension uniquely. Evaluation at \(a+\lambda1\) is evaluation at \(a\) plus a constant; therefore both restriction and its inverse are weak* continuous.

For the dimension assertion, let \((\sigma,L,\zeta)\) be cyclic for \(A^+\), with \(\|\zeta\|=1\), and let \(q\) be the essential projection of \(\sigma|_A\). The space \(qL\) is cyclic for \(A\) with vector \(q\zeta\): the closure of \(\sigma(A)\zeta\) equals \(qL\), using the approximate identity and cyclicity for \(A^+\). On \((1-q)L\), \(A\) acts by zero and \(A^+\) acts by scalars. Cyclicity makes that complement at most one-dimensional. Hence
\(\dim L\le\kappa+1=\kappa\), because \(\kappa\) is infinite. \(\square\)

## Matching finitely many vectors

**Lemma 5.1 — a small rotation.** For unit vectors \(\alpha,\beta\) in a complex Hilbert space there is a unitary \(U\) with

\[
U\alpha=\beta,\qquad \|U-I\|=\|\alpha-\beta\|.
\tag{5.1}
\]


**Proof.** If \(\beta=r\alpha\), multiply the line \(\mathbb C\alpha\) by \(r\) and leave its orthogonal complement fixed. Otherwise write
\(\beta=r\alpha+s\gamma\), where \(\gamma\perp\alpha\) is a unit vector, \(s>0\), and \(|r|^2+s^2=1\). On the ordered orthonormal pair \((\alpha,\gamma)\), take

\[
U=\begin{pmatrix}r&-s\\s&\overline r\end{pmatrix},
\]

and use the identity on the orthogonal complement. The two columns are orthonormal. Moreover \(U+U^*=2\operatorname{Re}(r)I\) on this plane, so
\((U-I)^*(U-I)=(2-2\operatorname{Re}r)I\). This is precisely \(\|\alpha-\beta\|^2I\). \(\square\)

**Lemma 5.2 — finite Gram matching.** Let \(K\) be an infinite-dimensional Hilbert space of dimension \(\kappa\). Fix vectors \(v_0,\ldots,v_m\in K\) with \(\|v_0\|=1\). For every \(\varepsilon>0\), sufficiently close Gram matrices have the following consequence. If \(w_0,\ldots,w_m\) belong to a Hilbert space \(L\) of dimension at most \(\kappa\), \(\|w_0\|=1\), and

\[
G_v=(\langle v_k,v_j\rangle)_{j,k=0}^m,
\qquad G_w=(\langle w_k,w_j\rangle)_{j,k=0}^m
\]

are sufficiently close in norm, then an isometry \(V:L\to K\) exists with

\[
Vw_0=v_0,\qquad \|Vw_j-v_j\|<\varepsilon\quad(0\le j\le m).
\tag{5.2}
\]

No linear independence or constant-rank assumption is needed.

**Proof.** Define \(T_v:\mathbb C^{m+1}\to K\) by \(T_ve_j=v_j\); then \(T_v^*T_v=G_v\). The rule \(W(G_v^{1/2}u)=T_vu\) is a well-defined isometry on the range of \(G_v^{1/2}\), since both squared norms equal \(\langle G_vu,u\rangle\). Thus \(T_v=W G_v^{1/2}\). Extend \(W\) from its initial space to an isometry \(J:\mathbb C^{m+1}\to K\) by completing finite orthonormal families. Thus \(JG_v^{1/2}e_j=v_j\).

Put \(z_j=JG_w^{1/2}e_j\). Their Gram matrix is exactly \(G_w\), so the linear rule \(w_j\mapsto z_j\) is a well-defined isometry on their span: the squared norm of every linear combination is given by the same Gram matrix. Extend this map to an isometry \(V_0:L\to K\). Such an extension exists by choosing orthonormal bases of the complements: the prescribed subspace is finite-dimensional, so its complement in \(K\) still has dimension \(\kappa\).

Let \(\delta=\|G_w^{1/2}-G_v^{1/2}\|\). Then \(\|z_j-v_j\|\le\delta\). Both \(z_0\) and \(v_0\) are unit vectors. Lemma 5.1 supplies a unitary \(U\) with \(Uz_0=v_0\) and \(\|U-I\|\le\delta\). For \(V=UV_0\),

\[
\|Vw_j-v_j\|
\le\delta\|w_j\|+\delta.
\tag{5.3}
\]

When \(G_w\to G_v\), its diagonal entries bound the finitely many \(\|w_j\|\), and \(\delta\to0\). To see the last fact without assuming invertibility, bound both spectra in a common interval \([0,M]\), approximate \(\sqrt t\) uniformly there by a polynomial, and use norm continuity of that polynomial in the matrix entries. Functional calculus bounds the two uniform approximation errors. Equation (5.3) proves the assertion. \(\square\)

The same lemma can impose a finite-dimensional forbidden subspace \(Z\): if every \(v_j\perp Z\), apply it with target \(Z^\perp\). Its dimension remains \(\kappa\).

## The fixed-vector map is a quotient

Fix a unit vector \(\xi\in H\), and define

\[
\theta:\mathcal R\longrightarrow Q(A),\qquad
\theta(\pi)(a)=\langle\pi(a)\xi,\xi\rangle.
\tag{6.1}
\]


**Theorem 6.1.** The map \(\theta\) is continuous, onto, and a quotient map. Thus a function on \(Q(A)\) is continuous whenever its composite with \(\theta\) is continuous.

**Proof.** Continuity follows from the definition of the pointwise weak operator topology. Given \(\varphi\in Q(A)\), form the cyclic GNS representation of \(\varphi^+\) on \(A^+\). By Lemma 4.1 its Hilbert space embeds in \(H\). Choose an embedding taking its unit cyclic vector to \(\xi\), extend the representation by zero on the orthogonal complement, and restrict it to \(A\). The resulting \(\pi\) satisfies \(\theta(\pi)=\varphi\), proving surjectivity.

We prove a local assertion at this particular preimage. Write \(K\subseteq H\) for the embedded cyclic GNS space and \(\sigma:A^+\to B(K)\) for its unital representation, with cyclic vector \(\xi\). Thus \(\pi(a)=\sigma(a)\oplus0\) on \(K\oplus K^\perp\). Every neighborhood of \(\pi\) contains one specified by finitely many inequalities

\[
\|(\rho(a_i)-\pi(a_i))\zeta_j\|<\varepsilon
\quad(1\le i\le r,\ 1\le j\le s),
\tag{6.2}
\]

by Theorem 2.1. We show that the image of this neighborhood contains a weak* neighborhood of \(\varphi\).

Decompose \(\zeta_j=\eta_j+z_j\), with \(\eta_j\in K\) and \(z_j\in K^\perp\), and put \(Z=\operatorname{span}\{z_1,\ldots,z_s\}\). Cyclicity supplies \(b_j\in A^+\) with

\[
\|\eta_j-\sigma(b_j)\xi\|<d,
\tag{6.3}
\]

where \(d>0\) will be chosen below. Make the finite list

\[
(d_0,\ldots,d_m)=(1,\ b_j,\ a_i b_j\text{ for all }i,j),
\]

allowing repetitions, and set \(v_k=\sigma(d_k)\xi\). All these vectors lie in \(Z^\perp\), and \(v_0=\xi\).

For \(\varphi'\in Q(A)\), let \((\sigma',L',\xi')\) be the cyclic GNS representation of \((\varphi')^+\), and put \(w_k=\sigma'(d_k)\xi'\). The Gram entries are

\[
\langle w_k,w_l\rangle
 =(\varphi')^+(d_l^*d_k),
\qquad
\langle v_k,v_l\rangle
 =\varphi^+(d_l^*d_k).
\tag{6.4}
\]

These are finitely many weak* continuous functions of \(\varphi'\). Consequently a weak* neighborhood \(W\) of \(\varphi\) makes the two Gram matrices as close as required by Lemma 5.2. Since \(\dim L'\le\kappa\), that lemma, with target \(Z^\perp\), gives an isometry \(V:L'\to Z^\perp\) such that

\[
V\xi'=\xi,\qquad
\|V\sigma'(d_k)\xi'-\sigma(d_k)\xi\|<t
\tag{6.5}
\]

for all \(k\), where \(t>0\) is prescribed.

Define \(\rho(a)=V\sigma'(a)V^*\), extended by zero off \(VL'\). This is a representation of \(A\), it annihilates \(Z\), and \(\theta(\rho)=\varphi'\). Write \(v_{b_j}=\sigma(b_j)\xi\), \(w_{b_j}=\sigma'(b_j)\xi'\), and likewise for \(a_i b_j\). Since
\(\rho(a_i)Vw_{b_j}=Vw_{a_i b_j}\), (6.3)–(6.5) give

\[
\begin{aligned}
\|(\rho(a_i)-\pi(a_i))\zeta_j\|
={}&\|(\rho(a_i)-\pi(a_i))\eta_j\|\\
\le{}&2\|a_i\|d+\|a_i\|t+t.
\end{aligned}
\tag{6.6}
\]

For example, choose \(M=\max(1,\|a_1\|,\ldots,\|a_r\|)\), then \(d<\varepsilon/(4M)\) and \(t<\varepsilon/(2(M+1))\). The inequalities (6.2) follow. Thus every \(\varphi'\in W\) is the image of a representation in the specified neighborhood.

Finally, if \(U\subseteq Q(A)\) has open inverse image, take any \(\varphi\in U\) and the cyclic preimage just constructed. A neighborhood of that preimage lies in \(\theta^{-1}(U)\). The local assertion puts a neighborhood of \(\varphi\) inside \(U\). Hence \(U\) is open, which proves the quotient property. This only uses suitable cyclic preimages, and does not assert that \(\theta\) is open at every representation. \(\square\)

## Continuity recovers the original algebra

**Theorem 7.1 — continuous reconstruction.** An admissible field \(F\) is continuous from \(\mathcal R\) to \(B(H)\) with its weak operator topology if and only if there is an \(a\in A\) with

\[
F_\pi=\pi(a)\quad(\pi\in\mathcal R).
\tag{7.1}
\]

For these fields, continuity into the sigma-strong* topology also holds. The element \(a\) is unique and \(\|a\|=\|F\|_\infty\).

**Proof.** Theorem 3.2 gives a unique \(x\in A^{**}\) with \(F_\pi=\bar\pi(x)\). On \(Q(A)\), put

\[
f_x(\varphi)=x(\varphi).
\]

This is a bounded complex affine function with \(f_x(0)=0\). For the fixed unit vector used in Theorem 6.1, normal extension of vector functionals gives

\[
f_x(\theta(\pi))=\langle\bar\pi(x)\xi,\xi\rangle
 =\langle F_\pi\xi,\xi\rangle.
\tag{7.2}
\]

Indeed the two normal functionals of \(x\) agree on \(A\), hence on its weak* dense bidual realization. If \(F\) is weakly continuous, the right side is continuous in \(\pi\). The quotient theorem makes \(f_x\) continuous on \(Q(A)\).

The continuous affine representation theorem from the earlier lesson now supplies \(a,b\in A_{\mathrm{sa}}\) such that

\[
\operatorname{Re}f_x(\varphi)=\varphi(a),\qquad
\operatorname{Im}f_x(\varphi)=\varphi(b)
\quad(\varphi\in Q(A)).
\]

Thus \(x=a+ib\) on all positive functionals, by scaling. Positive functionals span \(A^*\): decompose a functional into its hermitian real and imaginary parts, and apply the norm-additive Jordan decomposition to each. Therefore equality on positive functionals is equality in \(A^{**}\), so \(x\in A\).

Conversely evaluation at \(a\in A\) is continuous by the definition of the pointwise topology, and Theorem 2.1 makes it sigma-strong* continuous. The uniqueness and norm assertion follow from Theorem 3.2. \(\square\)

## Four examples

**Example 8.1 — scalar representations.** For \(A=\mathbb C\), a representation on \(H\) has the form \(\pi(\lambda)=\lambda P\), where \(P\) is an orthogonal projection. Theorems 3.2 and 7.1 say that its admissible fields are exactly \(F_P=\lambda P\). Weak convergence of projections to a projection is strong convergence: (2.2) with \(a=1\) proves this directly. The zero projection is included.

**Example 8.2 — a bounded field that is not continuous.** Let \(A=c_0(\mathbb N)\), fix a rank-one projection \(P\), and put \(\pi_n(a)=a_nP\). Then \(\pi_n\to0\) pointwise in operator norm. The identity of \(A^{**}\) gives the admissible field \(F_\pi=p_\pi\). Thus \(F_{\pi_n}=P\) while \(F_0=0\), so this field is not weakly continuous. This exhibits exactly why the bounded-field theorem reconstructs a larger algebra than the continuous-field theorem.

**Example 8.3 — Gram matrices can change rank.** With orthonormal \(e_0,e_1\), take \(v_0=w_0=e_0\), \(v_1=0\), and \(w_1=\delta e_1\). The Gram matrices differ by \(\operatorname{diag}(0,|\delta|^2)\), although one has rank one and the other rank two when \(\delta\ne0\). Lemma 5.2 allows this: it matches the unit vectors exactly and the second vectors within \(|\delta|\). Inverting a Gram matrix would exclude this necessary case.

**Example 8.4 — why the new identity matters.** For \(A=\mathbb C\), quasi-states are \(\varphi_t(\lambda)=t\lambda\), \(0\le t\le1\). The forced unitization is isomorphic to \(\mathbb C\oplus\mathbb C\) by \((a,\lambda)\mapsto(a+\lambda,\lambda)\). Under this identification, \(\varphi_t^+\) is the state with weights \(t\) and \(1-t\). Its state space is the whole interval \(Q(A)\). Simply keeping the old identity would retain only the state \(t=1\), and would lose the quotient construction at smaller norms.

For a freely accessible treatment of representation-space duality and quasi-state quotients, see S. A. Yuhjtman, [*The quasi-state space of a C*-algebra is a topological quotient of the representation space*](https://arxiv.org/abs/1304.4260). The finite Gram matching and local quotient arguments above supply the proofs used in this lesson.