The Picard groups for unital inclusions of unital $C^*$-algebras

TL;DR

This paper introduces and studies the Picard group associated with unital inclusions of unital $C^*$-algebras, exploring its properties, relationships, and providing concrete examples.

Contribution

It defines the Picard group for algebra inclusions, analyzes its properties, and relates it to the classical Picard group, with illustrative examples.

Findings

Established the basic properties of the Picard group for inclusions.

Explored the relationship between the inclusion Picard group and the classical Picard group.

Provided explicit examples of the Picard groups for unital inclusions.

Abstract

We shall introduce the notion of the Picard group for an inclusion of $C^*$-algebras. We shall also study its basic properties and the relation between the Picard group for an inclusion of $C^*$-algebras and the ordinary Picard group. Furthermore, we shall give some examples of the Picard groups for unital inclusions of unital $C^*$-algebras.