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

TL;DR

This paper extends the theory of weak*-rigged modules over dual operator algebras, exploring their tensor products and the structure of their Picard groups, including explicit calculations for certain function algebras.

Contribution

It introduces new results on weak*-rigged modules, their tensor products, and the structure of their Picard groups, including explicit computations for specific algebras.

Findings

Computed the weak Picard group of H^{ Infinity}(D)

Showed the weak Picard group of a weak* closed algebra is a semidirect product

Extended the theory of weak*-rigged modules with new tensor product results

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.