Original text: CC0 1.0. Prerequisite proofs and component terms.

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.

Editable source · Proof dependencies and component terms