Concrete von Neumann algebra and faithful vector

Assume the Axiom of Choice. Let $H$ be a complex Hilbert space and $\mathcal{B}(H)$ the space of bounded linear operators on $H$ (The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators). For a set $\mathcal{S}\subseteq \mathcal{B}(H)$, write

${\mathcal{S}}^{\mathrm{\prime }}:=\{T\in \mathcal{B}(H):Ts=sT\text{ for every }s\in \mathcal{S}\},{\mathcal{S}}^{\mathrm{\prime }\mathrm{\prime }}:=({\mathcal{S}}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}\mathrm{.}$

A directed preorder is a nonempty set $I$ with a reflexive, transitive relation $\le$ such that every pair $i,j\in I$ has a common upper bound in $I$. A net in a set $X$ is a function from a directed preorder into $X$; its values are written $(x_i{)}_{i\in I}$.

The strong operator topology on $\mathcal{B}(H)$ is the topology of pointwise norm convergence: $T_{\alpha }\to T$ strongly exactly when $\mathrm{\parallel }(T_{\alpha }-T)\xi \mathrm{\parallel }\to 0$ for every $\xi \in H$. The weak operator topology is the topology of pointwise matrix-coefficient convergence: $T_{\alpha }\to T$ weakly exactly when $⟨T_{\alpha }\xi ,\eta ⟩\to ⟨T\xi ,\eta ⟩$ for every $\xi ,\eta \in H$. Use the linear-first inner-product convention.

A concrete von Neumann algebra on $H$ is a unital star subalgebra $M\subseteq \mathcal{B}(H)$ satisfying $M=M^{\mathrm{\prime }\mathrm{\prime }}$. Under Choice, the double commutant theorem makes this equivalent to being closed in the weak operator topology.

An operator $a\in \mathcal{B}(H)$ is positive when $a=a^{\ast }$ and $⟨a\xi ,\xi ⟩\ge 0$ for every $\xi \in H$, using the bounded adjoint convention of Hilbert-adjoint identities. Write $M_{+}$ for the positive operators in $M$ and $a\le b$ when $b-a$ is positive.

A vector $\mathrm{\Omega }\in H$ is cyclic for $M$ when $\overline{M\mathrm{\Omega }}=H$, and is separating for $M$ when $x\mathrm{\Omega }=0$ for $x\in M$ implies $x=0$. For a unit vector $\mathrm{\Omega }$, define its vector functional by ${\omega }_{\mathrm{\Omega }}(x)=⟨x\mathrm{\Omega },\mathrm{\Omega }⟩$. It is faithful when ${\omega }_{\mathrm{\Omega }}(x^{\ast }x)=0$ implies $x=0$; this is equivalent to $\mathrm{\Omega }$ being separating because ${\omega }_{\mathrm{\Omega }}(x^{\ast }x)=\mathrm{\parallel }x\mathrm{\Omega }{\mathrm{\parallel }}^2$.

For a bounded linear functional $\phi :M\to \mathbb{C}$, positive means $\phi (a)\in [0,+\mathrm{\infty })$ for every $a\in M_{+}$; in particular, each such value is real. A positive functional is normal when $\phi ({\sup }_i{x}_i)={\sup }_i\phi (x_i)$ for each bounded increasing net $(x_i{)}_{i\in I}$ in $M_{+}$ whose supremum is taken in $M$.

Von Neumann algebras and commutants

The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators

Hilbert-adjoint identities

Strong and weak operator topologies

The Axiom of Choice

AC implies DC implies countable choice

For a quasi-local state arising from a consistent family of finite local density matrices, assume Countable Choice together with the GNS supplier's C*-algebra completion and finite-matrix positive-square-root interfaces, exactly as stated in that supplier's Statement. Under these hypotheses it gives a cyclic representation. That construction does not assert that the cyclic vector is separating for the represented algebra, and it does not supply a separating vector for every state.

Cyclic representation of a quasi-local state

The definition $M=M^{\mathrm{\prime }\mathrm{\prime }}$ is the chosen concrete representation interface. The canonical imported definition starts with weak-operator closure; the cited double-commutant theorem proves the equivalence under the stated Choice hypothesis.