W*-algebraic Integration Theory

Abstract

Given a pair of ${\mathrm{W}}^{\ast }$-algebras $({\mathcal{M}}_{\mathcal{S}},{\mathcal{M}}_{\mathcal{R}})$ with $({\mathcal{M}}_{\mathcal{S}}{)}_{\ast }$ separable, a measurable space $(\hat{I}\pounds ,\mathcal{F})$ and a POVM $\mathsf{E}:\mathcal{F}\to \mathcal{E}({\mathcal{M}}_{\mathcal{R}})$, the integral of a function $f:\hat{I}\pounds \to {\mathcal{M}}_{\mathcal{S}}$ 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 $B_b(\hat{I}\pounds ,\mathcal{F},{\mathcal{M}}_{\mathcal{S}})$ of uniformly bounded ultraweakly measurable functions is the universal domain of integration; once $\mathsf{E}$ is fixed it refines to the quotient $L_{\mathsf{E}}^{\infty }(\hat{I}\pounds ,{\mathcal{M}}_{\mathcal{S}})=B_b(\hat{I}\pounds ,\mathcal{F},{\mathcal{M}}_{\mathcal{S}})/{\mathcal{N}}_{\mathsf{E}}$ by $\mathsf{E}$-null functions. When $({\mathcal{M}}_{\mathcal{R}}{)}_{\ast }$ is also separable, $L_{\mathsf{E}}^{\infty }(\hat{I}\pounds ,{\mathcal{M}}_{\mathcal{S}})\cong {\mathcal{M}}_{\mathcal{S}}\bar{\otimes }{L}_{\mathsf{E}}^{\infty }(\hat{I}\pounds )$ is a ${\mathrm{W}}^{\ast }$-algebra. The integration map is a faithful normal unital completely positive (CP) map, a $\ast$-homomorphism for PVMs and an isometry for localizable POVMs. It can be identified with the spatial tensor product $1_{{\mathcal{M}}_{\mathcal{S}}}\hat{\otimes }\hat{I}{¦}_{\mathsf{E}}$ where $\hat{I}{¦}_{\mathsf{E}}:L_{\mathsf{E}}^{\infty }(\hat{I}\pounds )\to {\mathcal{M}}_{\mathcal{R}}$ 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.