Exact Structures for Operator Modules

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.

Schanuel’s Lemma for Exact Categories

“…In [1], exact categories of sheaves of operator modules over C*-ringed spaces are studied. Relative cohomology and cohomological dimension for (not necessarily self-adjoint) operator algebras is the topic of [6], see also [8]. In view of this, it seems beneficial to establish an injective version of Schanuel's lemma for exact categories and show how it yields the injective dimension theorem.…”