Standard generalized vectors for algebras of unbounded operators
The notion of (full) standard generalized vectors which makes it possible to develop the Tomita-Takesaki theory in \(\text{O}^*\)-algebras is defined and studied. Let \(\lambda\) be a standard generalized vector for a generalized von Neumann algebra \(\mathcal M\) on \(\mathcal D\) which is a generalization of von Neumann algebras to \(\text{O}^*\)-algebras. A one-parameter group \(\{\sigma^\lambda_t\}_{t\in R}\) of \(*\)-automorphisms of \(\mathcal M\) is defined and the quasi-weight \(\varphi_\lambda\) defined by \(\lambda\) satisfies the KMS-condition with respect to \(\{\sigma^\lambda_t\}_{t\in R}\). The Connes cocycle theorem and the Pedersen-Takesaki Radon-Nikodým theorem for von Neumann algebras are generalized to for generalized von Neumann algebras. Let \(\lambda\) and \(\mu\) be full standard generalized vectors for \(\mathcal M\). There uniquely exists a strongly continuous map \(t\in R\to U_t\in{\mathcal M}_u\equiv \{U\in{\mathcal M};\;\overline U\) is unitary\} such that \(U_{s+t}= U_t\sigma^\lambda_s(U_s)\) and \(\sigma^\mu_t(X)= U_t\sigma^\lambda_t(X) U^*_t\) for all \(s,t\in R\) and \(X\in{\mathcal M}\). It is shown that \(\{U_t\}_{t\in R}\) is a one-parameter group if and only if the domain \(D(\mu)\) of \(\mu\) is \(\{\sigma^\lambda_t\}\)-invariant and \(|\mu(\sigma^\lambda_t(X))|= |\mu(X)|\) for all \(X\in D(\mu)\) and \(t\in R\) if and only if \(\{U_t\}_{t\in R}\subset{\mathcal M}^{\sigma^\lambda}_b\equiv \{A\in{\mathcal M};\;\overline A\) is bounded and \(\sigma^\lambda_t(A)= A\) for all \(t\in R\}\). Furthermore, \(\mu\) satisfies the KMS-condition with respect to \(\{\sigma^\lambda_t\}\) if and only if \(\sigma^\mu_t= \sigma^\lambda_t\) for all \(t\in R\).
Tomita-Takesaki theory in \(\text{O}^*\)-algebras
scientific article; zbMATH DE number 844456
scientific article