Exact Structures for Operator Modules

TL;DR

This paper introduces a framework for applying exact structures to the category of right operator modules over an operator algebra, enabling the study of global dimension using properties of the Haagerup tensor product.

Contribution

It establishes a method to define exact structures on operator modules, facilitating homological analysis in operator algebra theory.

Findings

Exact structures can be imposed on the category of operator modules.

The Haagerup tensor product is crucial for these structures.

This approach enables the study of global dimension in operator algebras.

Abstract

We demonstrate how exact structures can be placed on the additive category of right operator modules over an operator algebra in order to discuss global dimension for operator algebras. The properties of the Haagerup tensor product play a decisive role in this.