GNS Hilbert reconstruction with invariant operator domain

Let $H$ be a complex Hilbert space with inner product linear in its first variable. An operator on $H$ is a linear map $T:D(T)\to H$ whose domain $D(T)$ is a linear subspace of $H$; it is densely defined when $D(T)$ is dense in $H$. It is bounded when some $C\ge 0$ satisfies $\mathrm{\parallel }Tx\mathrm{\parallel }\le C\mathrm{\parallel }x\mathrm{\parallel }$ for all $x\in D(T)$, and unbounded when no such $C$ exists. No boundedness of the operators ${\pi }_{\omega }(a)$ is assumed here.

For a complex vector space $E$, $\mathrm{End}(E)$ denotes the set of complex-linear maps from $E$ to itself. The product norm on $H\times H$ is $\mathrm{\parallel }(x,y){\mathrm{\parallel }}_{\times }:=(\mathrm{\parallel }x{\mathrm{\parallel }}^2+\mathrm{\parallel }y{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}$.

For a densely defined $T$, define its Hilbert-space adjoint by

$D(T^{\ast }):=\{y\in H:\text{there is }z\in H\text{ such that }(Tx,y)=(x,z)\text{ for every }x\in D(T)\},T^{\ast }y:=z\mathrm{.}$

The vector $z$ is unique because $D(T)$ is dense. The graph of $T$ is $G(T):=\{(x,Tx):x\in D(T)\}\subseteq H\times H$. The operator $T$ is symmetric when $D(T)\subseteq D(T^{\ast })$ and $T^{\ast }x=Tx$ for every $x\in D(T)$; it is self-adjoint when $D(T)=D(T^{\ast })$ and the same equality holds there. It is closable when the closure of $G(T)$ in the product norm on $H\times H$ is the graph of an operator, denoted $\bar{T}$. A symmetric operator is essentially self-adjoint when it is closable and $\bar{T}$ is self-adjoint.

Let $A$ be a complex vector space with a complex-bilinear associative product, unit $1_A$, and conjugate-linear involution satisfying $(ab{)}^{\ast }=b^{\ast }{a}^{\ast }$, $(a^{\ast }{)}^{\ast }=a$ and $1_A^{\ast }=1_A$. Let $\omega :A\to \mathbb{C}$ be complex-linear with $\omega (1_A)=1$ and real nonnegative $\omega (a^{\ast }a)$ for every $a\in A$. Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$: every countable family of nonempty sets has a choice function. Put $N_{\omega }:=\{a\in A:\omega (a^{\ast }a)=0\}$ and ${\mathcal{D}}_{\omega }:=A\mathrm{/}{N}_{\omega }$, with

$([a],[b]{)}_{\omega }:=\omega (b^{\ast }a),$

using the first-variable-linear convention. Let $H_{\omega }$ be the Hilbert completion of this inner-product space, and identify ${\mathcal{D}}_{\omega }$ with its canonical dense image in $H_{\omega }$. For each $a\in A$ define left multiplication on the common domain ${\mathcal{D}}_{\omega }$ by ${\pi }_{\omega }(a)[b]:=[ab]$. Then ${\pi }_{\omega }$ is a unital algebra representation of $A$ on the invariant dense domain ${\mathcal{D}}_{\omega }$, and

${\mathcal{D}}_{\omega }\subseteq D({\pi }_{\omega }(a{)}^{\ast }),{\pi }_{\omega }(a{)}^{\ast }{\mathrm{\mid }}_{{\mathcal{D}}_{\omega }}={\pi }_{\omega }(a^{\ast })\mathrm{.}$

Each ${\pi }_{\omega }(a)$ is closable, meaning that the closure of its graph in $H_{\omega }\times H_{\omega }$ is the graph of an operator. The vector ${\mathrm{\Omega }}_{\omega }:=[1_A]$ is a unit cyclic vector, ${\pi }_{\omega }(A){\mathrm{\Omega }}_{\omega }={\mathcal{D}}_{\omega }$, and $\omega (a)=({\pi }_{\omega }(a){\mathrm{\Omega }}_{\omega },{\mathrm{\Omega }}_{\omega })$. For uniqueness, let $K$ be a Hilbert space, ${\mathcal{D}}^{\mathrm{\prime }}\subseteq K$ a dense linear subspace invariant under every operator $\rho (a)$, and $\rho :A\to \mathrm{End}({\mathcal{D}}^{\mathrm{\prime }})$ a unital algebra representation satisfying $(\rho (a)x,y)=(x,\rho (a^{\ast })y)$ for all $a\in A$ and $x,y\in {\mathcal{D}}^{\mathrm{\prime }}$. If ${\mathrm{\Omega }}^{\mathrm{\prime }}\in {\mathcal{D}}^{\mathrm{\prime }}$, ${\mathcal{D}}^{\mathrm{\prime }}=\rho (A){\mathrm{\Omega }}^{\mathrm{\prime }}$, and $\omega (a)=(\rho (a){\mathrm{\Omega }}^{\mathrm{\prime }},{\mathrm{\Omega }}^{\mathrm{\prime }})$, then there is a unique unitary $V:H_{\omega }\to K$ with $V{\mathrm{\Omega }}_{\omega }={\mathrm{\Omega }}^{\mathrm{\prime }}$ and $V{\pi }_{\omega }(a)x=\rho (a)Vx$ for all $a\in A$ and $x\in {\mathcal{D}}_{\omega }$. This theorem supplies neither a C*-norm nor essential self-adjointness of ${\pi }_{\omega }(a)$ when $a=a^{\ast }$.