States of a concretely represented C star algebra are weak star limits of finite sums of vector states

$u_{\lambda }\to I$ in the strong operator topology. For $b\in B$ one has $\mathrm{\parallel }{u}_{\lambda }b-b\mathrm{\parallel }\to 0$ by [F1], hence $u_{\lambda }(b\eta )\to b\eta$ for every $\eta \in K$; the vectors $b\eta$ span a dense subspace because $B$ acts nondegenerately, and $\mathrm{\parallel }{u}_{\lambda }\mathrm{\parallel }\le 1$ uniformly, so $u_{\lambda }\xi \to \xi$ for every $\xi \in K$.

If $K=0$, then $B=0$ has no state and the statement is vacuous. For $K\mathrm{\ne }0$ and self-adjoint $S\in \mathcal{B}(K)$, ${\sup }_{\mathrm{\parallel }\eta \mathrm{\parallel }=1}⟨S\eta ,\eta ⟩=\max \sigma (S)$, and this equals the supremum of $\mathrm{\mid }⟨S\eta ,\eta ⟩\mathrm{\mid }$ over unit vectors when $S\ge 0$. For $c\ge \mathrm{\parallel }S\mathrm{\parallel }$ the operator $S+cI$ is positive, so [F2] gives $\mathrm{\parallel }S+cI\mathrm{\parallel }={\sup }_{\mathrm{\parallel }\eta \mathrm{\parallel }=1}⟨(S+cI)\eta ,\eta ⟩={\sup }_{\mathrm{\parallel }\eta \mathrm{\parallel }=1}⟨S\eta ,\eta ⟩+c$; since $\mathrm{\parallel }S+cI\mathrm{\parallel }=c+\max \sigma (S)$ for the self-adjoint operator $S+cI$, the claim follows, using the norm and spectral image formulas of the calculus in $\mathcal{B}(K)$ from [F1].

For $h=h^{\ast }\in B$ the spectrum computed in $B$ (in $B$ itself if unital, in its minimal unitization otherwise) equals the operator spectrum ${\sigma }_{\mathcal{B}(K)}(h)$, hence $\max {\sigma }_B(h)=\max {\sigma }_{\mathcal{B}(K)}(h)$. If $B$ is unital then nondegeneracy forces its unit to be $I$: the unit is a projection $p$ with $BK\subseteq pK$, so $pK$ is dense and closed, hence $p=I$. If $B$ is nonunital, $B+\mathbb{C}I$ is closed: $d=\mathrm{dist}(I,B)>0$, and $\mathrm{\mid }{z}_n-z_m\mathrm{\mid }d\le \mathrm{\parallel }(b_n+z_{n}I)-(b_m+z_{m}I)\mathrm{\parallel }$ makes both terms of every Cauchy sequence converge separately. Thus it is a unital C*-subalgebra of $\mathcal{B}(K)$ with identity $I$ containing $B$, and its norm on the algebraic unitization restricts to the given norm on $B$, so by the uniqueness clause of the minimal unitization it is the minimal unitization of $B$. In both cases [F3] gives the claim.

Let $C$ be the weak-* closed convex hull of $V$ and suppose $\omega \mathrm{\notin }C$. A finite-evaluation weak-* neighbourhood of $\omega$ is disjoint from $C$. Thus, for some $b_1,\dots ,b_n\in B$, the map $L(\psi )=(\mathrm{Re}\psi (b_i),\mathrm{Im}\psi (b_i){)}_{i=1}^n$ into ${\mathbb{R}}^{2n}$ sends $\omega$ outside $\overline{L(C)}$. Applying finite-dimensional closed-convex separation [F4] to this nonempty closed convex set gives a real linear combination of the coordinates that is strictly larger at $\omega$ than its supremum over $L(C)$. Such a combination is $\mathrm{Re}\psi (h_0)$ for some $h_0\in B$. Write $h_0=h+ik$ with $h,k$ self-adjoint. On $V$, $\mathrm{Re}\psi (h_0)=\psi (h)$ because both quadratic forms at $h,k$ are real; the identity extends by linearity and weak-* continuity to $C$. For $\omega$ the same identity follows from [F3]. Therefore $\omega (h)>{\sup }_{\psi \in C}\psi (h)={\sup }_{\mathrm{\parallel }\eta \mathrm{\parallel }=1}⟨h\eta ,\eta ⟩$, the equality holding because evaluation is continuous and linear on the closed convex hull.

Each $\eta$ with $\mathrm{\parallel }\eta \mathrm{\parallel }=1$ defines a state ${\omega }_{\eta }(b):=⟨b\eta ,\eta ⟩$ of $B$: positivity is ${\omega }_{\eta }(b^{\ast }b)=\mathrm{\parallel }b\eta {\mathrm{\parallel }}^2\ge 0$, and the norm is one because ${\omega }_{\eta }(u_{\lambda })=⟨u_{\lambda }\eta ,\eta ⟩\to \mathrm{\parallel }\eta {\mathrm{\parallel }}^2=1$ by step 1.1 and $\mathrm{\parallel }{u}_{\lambda }\mathrm{\parallel }\le 1$, so $\mathrm{\parallel }{\omega }_{\eta }\mathrm{\parallel }\ge 1$ while $\mathrm{\mid }{\omega }_{\eta }(b)\mathrm{\mid }\le \mathrm{\parallel }b\mathrm{\parallel }$ gives $\mathrm{\parallel }{\omega }_{\eta }\mathrm{\parallel }\le 1$. Hence $V\subseteq$ the state space of $B$.

No state lies outside $C$: if $\omega \mathrm{\notin }C$, step 1.4 gives $h=h^{\ast }$ with $\omega (h)>{\sup }_{\mathrm{\parallel }\eta \mathrm{\parallel }=1}⟨h\eta ,\eta ⟩=\max {\sigma }_{\mathcal{B}(K)}(h)=\max {\sigma }_B(h)$ by steps 1.2 and 1.3, contradicting the state spectral bound $\omega (h)\le \max {\sigma }_B(h)$ of [F3]. Therefore every state of $B$ belongs to the weak-* closed convex hull of $V$.

Let $\omega$ be a state of $B$, so $\omega \in C$ by step 2.2. By definition of the weak-* closure of the convex hull, every basic weak-* neighbourhood of $\omega$ meets the convex hull of $V$; a basic neighbourhood is given by finitely many $b_1,\dots ,b_n$ and $ϵ>0$, and an element of the convex hull is a finite convex combination ${\sum }_j{\lambda }_j{\omega }_{{\eta }_j}$ with unit vectors ${\eta }_j$. This is exactly the displayed approximation, so the lemma follows.

The Axiom of Choice is used for the geometric separation of step 1.4 and is inherited from the approximate-unit and spectral suppliers of [F1]–[F3]; the remaining estimates use no further choice (The Axiom of Choice).