Coordinates for triangular operator algebras. II
Write \({\mathcal M}\) for the full algebra of complex \(n\times n\) matrices, \({\mathcal A}\) for its subalgebra of diagonal matrices, and \({\mathcal P}\) for the intermediate algebra of upper-triangular matrices. Fix \(\delta\) in \({\mathcal A}\) with strictly decreasing real diagonal entries and define a group of automorphisms of \({\mathcal M}\) by \(\alpha_ t(a)=e^{it\delta}ae^{-it\delta}\). Then \({\mathcal P}\) can be recovered as the ``analytic algebra'' consisting of those a for which the maps \(t\to \rho \circ \alpha_ t(a)\) belong to the classical Hardy space \(H^{\infty}(R)\) for each \(\rho\in {\mathcal M}^*.\) The present paper studies the possibility of generalizing this construction. \({\mathcal M}\) is taken to be a von Neumann algebra, \({\mathcal A}\) a Cartan subalgebra of \({\mathcal M}\), and \({\mathcal P}\) a maximal subdiagonal algebra lying between them; the question is when \({\mathcal P}\) can be recovered as the analytic algebra corresponding to some one-parameter automorphism group of \({\mathcal M}\). The answer is given in terms of the representation studied in Part I of this paper [Ann. Math. 127, 245-278 (1988; Zbl 0649.47036)]: members of \({\mathcal M}\) are represented as matrices indexed by an equivalence relation R on a measure space, and the members of \({\mathcal P}\) are supported on some partial order \(P\subseteq R\). Then \({\mathcal P}\) is an analytic algebra iff \(P=d^{-1}(R^+)\) for some cocycle d. This is always the case when \({\mathcal M}\) is Type I, but the authors construct and examine several explicit examples where it fails.
analytic algebra corresponding to some one-parameter automorphism group
scientific article; zbMATH DE number 4155265