Preservation of strict comparison under standard C*-algebraic constructions

Determine whether strict comparison is preserved under inductive limits, minimal tensor products, and reduced crossed products of C*-algebras.

Background

Strict comparison is a fundamental regularity property of C*-algebras and is closely related to almost unperforation of the Cuntz semigroup. The paper notes that strict comparison is equivalent, under appropriate hypotheses, to important regularity properties such as Z-stability and finite nuclear dimension. Despite its significance, the preservation of strict comparison under several natural C*-algebraic operations remains unresolved: inductive limits, minimal tensor products, and reduced crossed products. The paper contrasts this unresolved status with selflessness, a stronger property in the setting under consideration that is known to be preserved under inductive limits and, for exact C*-probability spaces, under minimal tensor products.