W*-algebraic Integration Theory

Abstract

Given a pair of $\mathrm{W}^*$-algebras $(\mathcal{M}_\mathcal{S},\mathcal{M}_\mathcal{R})$ with $(\mathcal{M}_\mathcal{S})_*$ separable, a measurable space $(Σ, \mathcal{F})$ and a POVM $\mathsf{E}: \mathcal{F} \to \mathcal{E}(\mathcal{M}_\mathcal{R})$, the integral of a function $f: Σ\to \mathcal{M}_\mathcal{S}$ is defined as an element of the spatial tensor product $\int f \otimes d\mathsf{E} \in \mathcal{M}_\mathcal{S} \bar{\otimes} \mathcal{M}_\mathcal{R}$. The space $B_b(Σ,\mathcal{F},\mathcal{M}_\mathcal{S})$ of uniformly bounded ultraweakly measurable functions is the universal domain of integration; once $\mathsf{E}$ is fixed it refines to the quotient $L^\infty_\mathsf{E}(Σ,\mathcal{M}_\mathcal{S}) = B_b(Σ,\mathcal{F},\mathcal{M}_\mathcal{S})/\mathcal{N}_\mathsf{E}$ by $\mathsf{E}$-null functions. When $(\mathcal{M}_\mathcal{R})_*$ is also separable, $L^\infty_\mathsf{E}(Σ,\mathcal{M}_\mathcal{S}) \cong \mathcal{M}_\mathcal{S} \bar{\otimes} L^\infty_\mathsf{E}(Σ)$ is a $\mathrm{W}^*$-algebra. The integration map is a faithful normal unital completely positive (CP) map, a $*$-homomorphism for PVMs and an isometry for localizable POVMs. It can be identified with the spatial tensor product $\boldsymbol{1}_{\mathcal{M}_\mathcal{S}} \hat{\otimes} Φ_\mathsf{E}$ where $Φ_\mathsf{E}: L^\infty_\mathsf{E}(Σ) \to \mathcal{M}_\mathcal{R}$ is the faithful normal positive map corresponding to $\mathsf{E}$. Complete positivity of integration maps is derived from Stinespring factorization through Naimark dilation. We establish an operator-valued Leibniz rule and Fubini theorem.