Elements of Tomita-Takesaki theory for embeddable \(AW^*\)-algebras
The subject of the paper is to describe elements of Tomita-Takesaki theory for embeddable AWß*-algebras M, possessing in one of their ``normal'' representations \(\pi\) (i.e., \(\pi (M)=\pi (M)'')\) on a faithful self-dual Hilbert Cß*-module \({\mathcal H}\) over a commutative AWß*-algebra A a cyclic-separating element \=x\(\in {\mathcal H}\). The proving technics are derived mainly from apaper of \textit{M. A. Rieffel} and \textit{A. van Daele} [Pac. J. Math. 69,187-221 (1976; Zbl 0347.46073)] with changes in the proofs and generalizations of the basic definitions. The classical Tomita-Takesaki theory on Hilbert space is included as a special case. The crucial rule in the proofs play special real subspaces \({\mathcal K}\) of \({\mathcal H}\) generated by the self-adjoint part of \(\pi\) (M) and by \=x\(\in {\mathcal H}\), as well as related to \({\mathcal K}\) operators. From them an order continuous A-linear unitary one-parameter group \(\{\Delta ^{it}:t\in {\mathbb{R}}\}\) and a conjugate-A-linear involution J arise being definedon \({\mathcal H}\) and satisfying the Tomita-Takesaki conditions \(J\pi (M)J=\pi (M)'\), \(\Delta ^{it}\pi (M)\Delta ^{- it}=\pi (M)\) (t\(\in {\mathbb{R}})\). Moreover, this one-parameter group fulfils a generalized K.M.S. condition with respect to \({\mathcal K}\). Additionally, a geometrical interpretation of these results in terms of locally trivial Hilbert bundles on stoneian basic spaces is given.
Tomita-Takesaki theory for embeddable \(AW^*\)-algebras
faithful self-dual Hilbert \(C^*\)-module
locally trivial Hilbert bundles on Stoneian basic spaces
scientific article; zbMATH DE number 4056329
scientific article