Regular weights on algebras of unbounded operators
We define weights on \(O^*\)-algebras and quasi-weights on \(O^*\)-algebras which is strictly weaker than the notion of weights. And we construct the GNS-representation from (quasi-) weight on the similar method to positive linear functionals. We define the notions of regularity and singularity for (quasi-) weights. A (quasi-) weight \(\varphi\) is said to be regular if \(\varphi= \sup_\alpha f_\alpha\) on the domain \({\mathcal P}({\mathfrak N}_\varphi)\) of \(\varphi\) for some net \(\{f_\alpha\}\) of positive linear functionals on an \(O^*\)-algebra \({\mathcal M}\), and \(\varphi\) is said to be singular if there doesn't exist any positive linear functional \(f\) on \({\mathcal M}\) such that \(f(X)\leq\varphi(X)\) for all \(X\in{\mathcal P}({\mathfrak N}_\varphi)\) and \(f\neq 0\). In investigation the regularity and the singularity, the trio-commutant plays an important role. We give equivalent conditions for the regularity of (quasi-) weights and for the singularity of (quasi-) weights. And we give a decomposition theorem of (quasi-) weight into the regular part and the singular part (Theorem 3.9). Next, we define and study an important class (this class is said to be standard) in regular (quasi-) weights which is possible to develop the Tomita-Takesaki theory in \(O^*\)-algebras. In the final section, we give some concrete examples of regular (quasi-) weights, singular (quasi-) weights and standard (quasi-) weights. We first investigate the (quasi-) weights \(\omega_\xi\) defined by elements \(\xi\) in the Hilbert space, and we investigate the regularity, the singularity and the standardness of the (quasi-)weights of \(\omega_\xi\). We next apply these results to physical models.
Tomita-Takesaki theory in \(O^*\)-algebras
scientific article; zbMATH DE number 1185924