$\mathrm{W}^*$-algebraic Integration Theory

Given a pair of $\mathrm{W}^*$-algebras $(\mathcal{M}_\mathcal{S},\mathcal{M}_\mathcal{R})$ with $(\mathcal{M}_\mathcal{S})_*$ separable, a measurable space $(\Sigma, \mathcal{F})$ and a POVM $\mathsf{E}: \mathcal{F} \to \mathcal{E}(\mathcal{M}_\mathcal{R})$, the integral of a function $f: \Sigma \to \mathcal{M}_\mathcal{S}$ 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 $B_b(\Sigma,\mathcal{F},\mathcal{M}_\mathcal{S})$ of uniformly bounded ultraweakly measurable functions is the universal domain of integration; once $\mathsf{E}$ is fixed it refines to the quotient $L^\infty_\mathsf{E}(\Sigma,\mathcal{M}_\mathcal{S}) = B_b(\Sigma,\mathcal{F},\mathcal{M}_\mathcal{S})/\mathcal{N}_\mathsf{E}$ by $\mathsf{E}$-null functions.

When $(\mathcal{M}_\mathcal{R})_*$ is also separable, $L^\infty_\mathsf{E}(\Sigma,\mathcal{M}_\mathcal{S}) \cong \mathcal{M}_\mathcal{S} \bar{\otimes} L^\infty_\mathsf{E}(\Sigma)$ 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 $\boldsymbol{1}_{\mathcal{M}_\mathcal{S}} \hat{\otimes} \Phi_\mathsf{E}$ where $\Phi_\mathsf{E}: L^\infty_\mathsf{E}(\Sigma) \to \mathcal{M}_\mathcal{R}$ 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.