Rigged modules I: modules over dual operator algebras and the Picard group

Abstract

In a previous paper we generalized the theory of W*-modules to the setting of modules over nonselfadjoint dual operator algebras, obtaining the class of weak*-rigged modules. At that time we promised a forthcoming paper devoted to other aspects of the theory. We fulfill this promise in the present work and its sequel "Rigged modules II", giving many new results about weak*-rigged modules and their tensor products. We also discuss the Picard group of weak* closed subalgebras of a commutative algebra. For example, we compute the weak Picard group of $H^{\infty}(D)$, and prove that for a weak* closed function algebra A, the weak Picard group of A is a semidirect product of the automorphism group of A, and the subgroup consisting of symmetric equivalence bimodules.