## OA-MOD-SC-02. Positive observations can be implemented by vector sums **Lemma.** If \(A\subseteq B(L)\) is a concrete unital von Neumann algebra and \(\omega\in A_*^+\), there are vectors \(v_j\in L\), indexed by positive integers, such that \[ \sum_j\|v_j\|^2=\omega(1),\qquad \omega(a)=\sum_j\langle av_j,v_j\rangle \quad(a\in A). \tag{SC.3} \] The series is absolutely convergent. Every component functional \(\omega_{v_j}\) is positive normal and satisfies \(\omega_{v_j}\le\omega\). No separability hypothesis on \(L\) is needed. **Proof.** CP-08 gives square-summable vectors \(\zeta=(\zeta_j)\in\ell^2(\mathbb N,L)\) such that \[ \omega(a)\le\vartheta(a):= \sum_j\langle a\zeta_j,\zeta_j\rangle \qquad(a\ge0). \] Let \(\rho(a)=\operatorname{diag}(a,a,\ldots)\), a bounded *-representation on \(\ell^2(\mathbb N,L)\), and put \(K=\overline{\rho(A)\zeta}\). The subspace is invariant under \(\rho(A)\) and its adjoints, so it reduces this representation. Also \(\zeta\in K\), because \(A\) is unital. On the dense subspace \(\rho(A)\zeta\subseteq K\), define \[ B(\rho(a)\zeta,\rho(b)\zeta)=\omega(b^*a). \] The inequality \(\omega(a^*a)\le\vartheta(a^*a)=\|\rho(a)\zeta\|^2\), together with the positive-functional Cauchy–Schwarz inequality, proves that this rule is well-defined and \[ |B(\rho(a)\zeta,\rho(b)\zeta)| \le\|\rho(a)\zeta\|\,\|\rho(b)\zeta\|. \] It extends to a positive bounded sesquilinear form on \(K\). The bounded-form theorem gives a unique positive contraction \(h\in B(K)\) with \(B(u,v)=\langle hu,v\rangle\). For \(c\in A\), the equality \[ B(\rho(c)\rho(a)\zeta,\rho(b)\zeta) =B(\rho(a)\zeta,\rho(c^*)\rho(b)\zeta) \] holds because both sides equal \(\omega(b^*ca)\). Extension from the dense cyclic subspace shows that \(h\) commutes with \(\rho(c)|_K\). Its square root commutes as well. Set \(v=h^{1/2}\zeta\in K\), and write its components as \(v_j\in L\). For every \(a\in A\), \[ \omega(a)=B(\rho(a)\zeta,\zeta) =\langle\rho(a)v,v\rangle =\sum_j\langle av_j,v_j\rangle. \] Evaluation at \(1\) gives the asserted sum of squared norms. Absolute convergence follows from \(|\langle av_j,v_j\rangle|\le\|a\|\|v_j\|^2\). For positive \(a\), each nonnegative summand is at most the sum, proving \(\omega_{v_j}\le\omega\). Single-vector functionals are ultraweakly continuous by CP. If \(\omega=0\), the construction may be replaced by the zero sequence. ∎ The use of a countable series concerns one predual functional. The family of all normal functionals can be arbitrarily large. We have not chosen a countable family which detects the entire algebra.