C star state GNS construction, purity and Polish pure-state spaces

Define $L_{\varphi }=\{a\in A:\varphi (a^{\ast }a)=0\}$ and on $A\mathrm{/}{L}_{\varphi }$ set $⟨[a],[b]⟩=\varphi (b^{\ast }a)$, linear in the first variable. Positivity and Cauchy-Schwarz from [F2] make this a well-defined positive-definite inner product after quotienting by its null space; complete it to a Hilbert space $H_{\varphi }$ using [F3].

For $c,a\in A$, positivity of $\mathrm{\parallel }c{\mathrm{\parallel }}^{2}1-c^{\ast }c$ and conjugation order in [F1] give $\varphi ((ca{)}^{\ast }(ca))\le \mathrm{\parallel }c{\mathrm{\parallel }}^2\varphi (a^{\ast }a)$. Thus ${\pi }_{\varphi }(c)[a]=[ca]$ is well-defined and bounded with norm at most $\mathrm{\parallel }c\mathrm{\parallel }$; left multiplication gives ${\pi }_{\varphi }(cd)={\pi }_{\varphi }(c){\pi }_{\varphi }(d)$, and $⟨{\pi }_{\varphi }(c)[a],[b]⟩=\varphi (b^{\ast }ca)=⟨[a],{\pi }_{\varphi }(c^{\ast })[b]⟩$ gives ${\pi }_{\varphi }(c{)}^{\ast }={\pi }_{\varphi }(c^{\ast })$ by [F3].

If $A$ is unital, take ${\xi }_{\varphi }=[1]$; it is a unit cyclic vector, ${\pi }_{\varphi }(1)=I$ makes the representation nondegenerate, and it yields the stated vector functional. If $A$ is nonunital, fix a positive contractive approximate unit $(e_{\lambda })$. For every positive bounded functional $\theta$ on $A$, the norm formula [F2] and $e_{\lambda }b{e}_{\lambda }\le e_{\lambda }^2$ for $0\le b\le 1$ give $\theta (e_{\lambda }^2)\to \mathrm{\parallel }\theta \mathrm{\parallel }$: for each $\epsilon >0$ choose such a $b$ with $\theta (b)>\mathrm{\parallel }\theta \mathrm{\parallel }-\epsilon$, use $e_{\lambda }b{e}_{\lambda }\to b$, and let $\epsilon \downarrow 0$; also $0\le e_{\lambda }^2\le e_{\lambda }\le 1$ gives $\theta (e_{\lambda })\to \mathrm{\parallel }\theta \mathrm{\parallel }$. In particular $\varphi (e_{\lambda }),\varphi (e_{\lambda }^2)\to 1$. Define $\tilde{\varphi }(a+z1)=\varphi (a)+z$ on the minimal unitization. For $x=a+z1\ge 0$, each compression $e_{\lambda }x{e}_{\lambda }=e_{\lambda }a{e}_{\lambda }+z{e}_{\lambda }^2$ lies in $A^{+}$, and $\varphi (e_{\lambda }a{e}_{\lambda })+z\varphi (e_{\lambda }^2)\to \varphi (a)+z$; hence $\tilde{\varphi }$ is positive. As $\tilde{\varphi }(1)=1$, the positive-functional norm formula makes $\mathrm{\parallel }\tilde{\varphi }\mathrm{\parallel }=1$, so it is a state. In its GNS construction, the map $[a{]}_{\varphi }\mapsto [a{]}_{\tilde{\varphi }}$ is an isometry from $A\mathrm{/}{L}_{\varphi }$ because the inner products agree. Moreover $\mathrm{\parallel }[1{]}_{\tilde{\varphi }}-[e_{\lambda }{]}_{\tilde{\varphi }}{\mathrm{\parallel }}^2=1-2\varphi (e_{\lambda })+\varphi (e_{\lambda }^2)\to 0$, so $[1{]}_{\tilde{\varphi }}$ lies in the closure of the image of $A\mathrm{/}{L}_{\varphi }$; then $[a+z1{]}_{\tilde{\varphi }}={\lim }_{\lambda }[a+z{e}_{\lambda }{]}_{\tilde{\varphi }}$, and the image of $A\mathrm{/}{L}_{\varphi }$ is dense in the unitized GNS space. Thus this space is precisely the completion $H_{\varphi }$ from steps 1.1-1.2, with the restricted representation agreeing with ${\pi }_{\varphi }$. Its vector ${\xi }_{\varphi }=[1{]}_{\tilde{\varphi }}$ is unit and cyclic for $A$, and $\varphi (a)=⟨{\pi }_{\varphi }(a){\xi }_{\varphi },{\xi }_{\varphi }⟩$. Finally, ${\pi }_{\varphi }(e_{\lambda }){\pi }_{\varphi }(a){\xi }_{\varphi }=[e_{\lambda }a{]}_{\varphi }\to [a{]}_{\varphi }={\pi }_{\varphi }(a){\xi }_{\varphi }$ on the dense cyclic span; since $\mathrm{\parallel }{\pi }_{\varphi }(e_{\lambda })\mathrm{\parallel }\le 1$, this convergence extends to every vector in $H_{\varphi }$, proving nondegeneracy.

Let $B$ be a unital C*-algebra. Its normalized state space $S(B)$ is a weak-star closed subset of the dual unit ball: positivity and $\omega (1)=1$ are pointwise closed, and positive unital functionals have norm one by Cauchy-Schwarz and $x^{\ast }x\le \mathrm{\parallel }x{\mathrm{\parallel }}^{2}1$. Banach-Alaoglu and AC make $S(B)$ compact. If $B$ is separable, choose a countable norm-dense family $(b_n)$ in $B$; the metric $d(\omega ,\rho )={\sum }_{n\ge 1}{2}^{-n}\min (1,\mathrm{\mid }\omega (b_n)-\rho (b_n)\mathrm{\mid })$ induces the weak-star topology on $S(B)$, since all states have norm one and approximation by the $b_n$ controls evaluation on every element of $B$. Hence $S(B)$ is compact metrizable; this metric is complete because every Cauchy sequence has a convergent subsequence by compactness and therefore converges to the same limit.

If $(\rho ,K,\eta )$ is another cyclic nondegenerate representation with unit vector state $\varphi$, the assignment ${\pi }_{\varphi }(a){\xi }_{\varphi }\mapsto \rho (a)\eta$ preserves inner products because both give $\varphi (b^{\ast }a)$ on cyclic vectors. It extends uniquely to a unitary intertwiner on the dense cyclic spans. In the nonunital case, for any nondegenerate representation $\sigma$, an approximate unit satisfies $\sigma (e_{\lambda })\to I$ strongly: this holds on the dense span $\sigma (A)K$ since $e_{\lambda }a\to a$, and then on all vectors by $\mathrm{\parallel }\sigma (e_{\lambda })\mathrm{\parallel }\le 1$. Applying this to ${\pi }_{\varphi }$ and $\rho$ gives $U{\xi }_{\varphi }=\eta$. Thus the triple is unique up to exactly one unitary carrying cyclic vector to cyclic vector.

If $A$ is separable, choose a countable norm-dense subset $D\subseteq A$. Contractivity of ${\pi }_{\varphi }$ makes $\{{\pi }_{\varphi }(a){\xi }_{\varphi }:a\in D\}$ dense in ${\pi }_{\varphi }(A){\xi }_{\varphi }$, whose span is dense in $H_{\varphi }$ by step 1.3; its countable rational-complex span is a countable dense subset of $H_{\varphi }$.

Let $\psi$ be a bounded positive functional with $0\le \psi \le \varphi$. On cyclic vectors define $B_{\psi }({\pi }_{\varphi }(a){\xi }_{\varphi },{\pi }_{\varphi }(b){\xi }_{\varphi })=\psi (b^{\ast }a)$. Cauchy-Schwarz and domination give $\mathrm{\mid }{B}_{\psi }(v,w){\mathrm{\mid }}^2\le \psi (a^{\ast }a)\psi (b^{\ast }b)\le \mathrm{\parallel }v{\mathrm{\parallel }}^2\mathrm{\parallel }w{\mathrm{\parallel }}^2$, so this is a well-defined bounded positive sesquilinear form on the dense cyclic span and extends to $H_{\varphi }$.

If every positive contraction in ${\pi }_{\varphi }(A{)}^{\mathrm{\prime }}$ is scalar, shifting and rescaling any self-adjoint member shows it is scalar, and taking real and imaginary parts gives ${\pi }_{\varphi }(A{)}^{\mathrm{\prime }}=\mathbb{C}I$. A closed invariant subspace $M$ for a *-representation is reducing: if $v\in M^{\perp }$, $w\in M$, and $a\in A$, then $⟨{\pi }_{\varphi }(a)v,w⟩=⟨v,{\pi }_{\varphi }(a^{\ast })w⟩=0$. By the orthogonal-decomposition theorem [F7], whose projection existence uses Countable Choice, the projection onto $M$ exists; reduction makes it commute with every ${\pi }_{\varphi }(a)$, so scalarity forces that projection to be $0$ or $I$ and the representation is irreducible. Conversely, extend ${\pi }_{\varphi }$ to a unital representation ${\tilde{\pi }}_{\varphi }$ of $B=A$ if unital and $B=A^{+}$ otherwise. The unitary group $U(B)$ with its norm topology is a topological group: multiplication is norm-continuous by submultiplicativity and inversion is $u\mapsto u^{\ast }$, which is isometric on unitaries. Contractivity of ${\tilde{\pi }}_{\varphi }$ gives $\mathrm{\parallel }{\tilde{\pi }}_{\varphi }(u)\eta -{\tilde{\pi }}_{\varphi }(v)\eta \mathrm{\parallel }\le \mathrm{\parallel }u-v\mathrm{\parallel }\mathrm{\parallel }\eta \mathrm{\parallel }$, so it is a strongly continuous unitary representation. Every self-adjoint contraction $h\in B$ is $(u+u^{\ast })\mathrm{/}2$ for $u=h+i(1-h^2{)}^{1\mathrm{/}2}$, which is unitary since $h$ commutes with its positive square root; scaling self-adjoint elements and decomposing arbitrary elements into real and imaginary parts shows the unitaries linearly span $B$. Thus the commutant of ${\tilde{\pi }}_{\varphi }(U(B))$ equals ${\pi }_{\varphi }(A{)}^{\mathrm{\prime }}$. Any closed subspace invariant under all these unitary images is invariant under their linear span ${\tilde{\pi }}_{\varphi }(B)$, hence under ${\pi }_{\varphi }(A)$; therefore irreducibility of ${\pi }_{\varphi }$ makes this unitary representation irreducible. Schur's lemma [F7] makes its commutant scalar. Hence ${\pi }_{\varphi }$ is irreducible exactly when its commutant is scalar.

Suppose $B$ is separable and fix a compatible metric $d$ on $S(B)$. For each $n\ge 1$, the set of pairs $({\omega }_0,{\omega }_1)$ with $d({\omega }_0,{\omega }_1)\ge 1\mathrm{/}n$ is compact; the midpoint map is weak-star continuous because each evaluation of its value is the average of the two evaluations, so its image is compact and consists of nonextreme states. Conversely, if $\omega =t\alpha +(1-t)\beta$ with distinct states and $0<t<1$, choosing $0<\delta <\min (t,1-t)$ makes $\omega$ the midpoint of the distinct states $(t+\delta )\alpha +(1-t-\delta )\beta$ and $(t-\delta )\alpha +(1-t+\delta )\beta$. Thus the nonextreme states are exactly a countable union of compact sets, so the pure states form a $G_{\delta }$ subset of $S(B)$.

Let $a\in A$ be positive and nonzero, and put $B=A$ when unital and $B=A^{+}$ otherwise. The character of $C^{\ast }(1,a)$ at the maximal spectral value of $a$ is a state taking value $\mathrm{\parallel }a\mathrm{\parallel }$ at $a$, since positive calculus gives $\max \sigma (a)=\mathrm{\parallel }a\mathrm{\parallel }$. Extend it to a norm-one functional $F$ on $B$ by complex Hahn-Banach; $F(1)=1$. For self-adjoint $c$, $\mathrm{\mid }1+itF(c)\mathrm{\mid }\le \mathrm{\parallel }1+itc\mathrm{\parallel }$ for every real $t$, and $\mathrm{\parallel }1+itc{\mathrm{\parallel }}^2=\mathrm{\parallel }1+t^2{c}^2\mathrm{\parallel }\le 1+t^2\mathrm{\parallel }c{\mathrm{\parallel }}^2$, so letting $t$ approach zero from both signs shows $F(c)$ is real. If $0\le c\le 1$, then $\mathrm{\mid }1-F(c)\mathrm{\mid }=\mathrm{\mid }F(1-c)\mathrm{\mid }\le \mathrm{\parallel }1-c\mathrm{\parallel }\le 1$, hence $F(c)\ge 0$; scaling proves positivity. Since $F$ is positive and $F(1)=1$, the norm formula [F2] gives $\mathrm{\parallel }F\mathrm{\parallel }=1$, so it is a state norming $a$. The norm-attaining states form a nonempty compact face of $S(B)$ by step 1.4: every state has value at most $\mathrm{\parallel }a\mathrm{\parallel }$, so a convex combination reaches $\mathrm{\parallel }a\mathrm{\parallel }$ only when each endpoint does. Krein-Milman [F8] gives an extreme point of that face, hence a pure state $\omega$ of $B$ still satisfying $\omega (a)=\mathrm{\parallel }a\mathrm{\parallel }$.

For each $v$, Riesz represents the bounded linear functional $w\mapsto \overline{B_{\psi }(v,w)}$ by a unique vector $Tv$ with $B_{\psi }(v,w)=⟨Tv,w⟩$. Uniqueness makes $T$ linear; polarization of the nonnegative quadratic form gives $T=T^{\ast }$, and $0\le B_{\psi }(v,v)\le \mathrm{\parallel }v{\mathrm{\parallel }}^2$ then gives $0\le T\le I$. For $v={\pi }_{\varphi }(a){\xi }_{\varphi }$, $w={\pi }_{\varphi }(b){\xi }_{\varphi }$, and $c\in A$, one has $⟨T{\pi }_{\varphi }(c)v,w⟩=\psi (b^{\ast }ca)=⟨{\pi }_{\varphi }(c)Tv,w⟩$; density implies $T\in {\pi }_{\varphi }(A{)}^{\mathrm{\prime }}$. Finally, $⟨T{\pi }_{\varphi }(a){\xi }_{\varphi },{\pi }_{\varphi }(e_{\lambda }){\xi }_{\varphi }⟩=\psi (e_{\lambda }a)\to \psi (a)$, while ${\pi }_{\varphi }(e_{\lambda }){\xi }_{\varphi }\to {\xi }_{\varphi }$ by step 1.3; hence $\psi (a)=⟨T{\pi }_{\varphi }(a){\xi }_{\varphi },{\xi }_{\varphi }⟩=⟨{\pi }_{\varphi }(a)T{\xi }_{\varphi },{\xi }_{\varphi }⟩$. In the unital case use $e=1$.

The $G_{\delta }$ theorem [F5] makes the pure-state subspace of separable unital $B$ completely metrizable by step 2.5. It is second countable as a subspace of $S(B)$, so [F5] also makes it separable and therefore Polish. For nonunital separable $A$, a countable dense subset of $A$ together with rational-complex multiples of the unit gives a countable dense subset of its minimal unitization $B=A^{+}$. By step 1.3, restriction identifies $S(A)$ with the states $\omega \in S(B)$ for which $\mathrm{\parallel }\omega {\mathrm{\mid }}_A\mathrm{\parallel }=1$, since any such restriction has the unique extension $a+z1\mapsto \omega (a)+z$. The augmentation $ϵ$ is pure: if $ϵ=t{\omega }_0+(1-t){\omega }_1$ with $0<t<1$, then for every $x\in A=\ker ϵ$, positivity gives $0=ϵ(x^{\ast }x)=t{\omega }_0(x^{\ast }x)+(1-t){\omega }_1(x^{\ast }x)$ and hence ${\omega }_j(x^{\ast }x)=0$; Cauchy-Schwarz [F2] implies ${\omega }_j(x)=0$, so ${\omega }_j(a+z1)=z=ϵ(a+z1)$. A pure state $\omega$ of $B$ other than $ϵ$ restricts to a state on $A$: if $s=\mathrm{\parallel }\omega {\mathrm{\mid }}_A\mathrm{\parallel }$ lay strictly between $0$ and $1$, then $\omega =s\widetilde{(\omega {\mathrm{\mid }}_A\mathrm{/}s)}+(1-s)ϵ$ would be a nontrivial convex decomposition, and $s=0$ would give $\omega =ϵ$. Conversely, if $\varphi$ is pure on $A$ and $\tilde{\varphi }=t{\omega }_0+(1-t){\omega }_1$ on $B$, the restrictions have norms at most one; the equality $1=\mathrm{\parallel }\varphi \mathrm{\parallel }\le t\mathrm{\parallel }{\omega }_0{\mathrm{\mid }}_A\mathrm{\parallel }+(1-t)\mathrm{\parallel }{\omega }_1{\mathrm{\mid }}_A\mathrm{\parallel }\le 1$ forces each restriction to be a state, and purity plus unique unitization extension forces ${\omega }_0={\omega }_1=\tilde{\varphi }$. Restriction and unique extension are weak-star continuous inverses because their evaluations are $\omega (a)$ and $\varphi (a)+z$, respectively. Thus restriction identifies the pure-state space of $A$ homeomorphically with $P(B)\setminus \{ϵ\}$. This set is $G_{\delta }$ in $S(B)$: $P(B)$ is $G_{\delta }$ by step 2.5 and the complement of the closed singleton $\{ϵ\}$ is open, hence $G_{\delta }$ in the metric space $S(B)$. The pure-state space of $A$ is therefore Polish in its weak-star topology.

Conversely, for $T\in {\pi }_{\varphi }(A{)}^{\mathrm{\prime }}$ with $0\le T\le I$, set ${\psi }_T(a)=⟨{\pi }_{\varphi }(a)T{\xi }_{\varphi },{\xi }_{\varphi }⟩$. This is bounded since $\mathrm{\mid }{\psi }_T(a)\mathrm{\mid }\le \mathrm{\parallel }a\mathrm{\parallel }\mathrm{\parallel }T{\xi }_{\varphi }\mathrm{\parallel }\mathrm{\parallel }{\xi }_{\varphi }\mathrm{\parallel }$. Then ${\psi }_T(a^{\ast }a)=⟨T{\pi }_{\varphi }(a){\xi }_{\varphi },{\pi }_{\varphi }(a){\xi }_{\varphi }⟩\ge 0$, and $(\varphi -{\psi }_T)(a^{\ast }a)=⟨(I-T){\pi }_{\varphi }(a){\xi }_{\varphi },{\pi }_{\varphi }(a){\xi }_{\varphi }⟩\ge 0$. The same formula on pairs of cyclic vectors recovers $B_{\psi }$ from ${\psi }_T$, so the correspondence is injective; moreover ${\psi }_T\le {\psi }_S$ exactly when the quadratic form of $S-T$ is nonnegative on the dense cyclic span, equivalently $T\le S$. Thus it is an order isomorphism.

For positive $\psi \le \varphi$, put $\theta =\varphi -\psi$. Along the same approximate unit, step 1.3 gives $\mathrm{\parallel }\psi \mathrm{\parallel }={\lim }_{\lambda }\psi (e_{\lambda })$, $\mathrm{\parallel }\theta \mathrm{\parallel }={\lim }_{\lambda }\theta (e_{\lambda })$ and $\mathrm{\parallel }\varphi \mathrm{\parallel }={\lim }_{\lambda }\varphi (e_{\lambda })$, so $\mathrm{\parallel }\varphi \mathrm{\parallel }=\mathrm{\parallel }\psi \mathrm{\parallel }+\mathrm{\parallel }\theta \mathrm{\parallel }$ (in the unital case take $e=1$). If $\varphi$ is pure, the endpoints $\psi =0$ and $\psi =\varphi$ are scalar multiples of $\varphi$; otherwise both norms are positive and normalization gives $\varphi =\mathrm{\parallel }\psi \mathrm{\parallel }(\psi \mathrm{/}\mathrm{\parallel }\psi \mathrm{\parallel })+(1-\mathrm{\parallel }\psi \mathrm{\parallel })(\theta \mathrm{/}\mathrm{\parallel }\theta \mathrm{\parallel })$, so purity forces $\psi =\mathrm{\parallel }\psi \mathrm{\parallel }\varphi$. Conversely, if $\varphi =t{\varphi }_0+(1-t){\varphi }_1$ is a nontrivial convex decomposition into distinct states, then $t{\varphi }_0\le \varphi$ and this subfunctional cannot be proportional to $\varphi$ (proportionality would force ${\varphi }_0={\varphi }_1=\varphi$). By steps 2.3, 3.1, and 4.1, the order correspondence identifies scalarity of all dominated positive subfunctionals with scalarity of all positive contractions in ${\pi }_{\varphi }(A{)}^{\mathrm{\prime }}$. Combined with step 2.4, this is equivalent to irreducibility of the GNS representation.

If $A$ is nonunital, then $\omega (a)>0$ shows $\omega \mathrm{\ne }ϵ$, and step 3.2 makes its restriction a pure state of $A$; if $A$ is unital take $\varphi =\omega$. In either case $\varphi (a)=\mathrm{\parallel }a\mathrm{\parallel }$. For a nonzero closed two-sided ideal $J$, choose $x\in J\setminus \{0\}$. By [F2], $J$ is self-adjoint, so $a=x^{\ast }x\in J$; the C*-identity gives $a\mathrm{\ne }0$. The pure norming state from step 2.6 has $⟨{\pi }_{\varphi }(a){\xi }_{\varphi },{\xi }_{\varphi }⟩=\mathrm{\parallel }a\mathrm{\parallel }>0$, so ${\pi }_{\varphi }(a)\mathrm{\ne }0$. Its GNS representation is irreducible by steps 2.4 and 5.1, and its kernel therefore does not contain $J$.