The noncommutative weak Extension Principle
Alessandro Vignati, Deniz Yilmaz
TL;DR
The paper addresses the classification of $^*$-homomorphisms between coronas of separable nonunital $\mathrm{C^*}$-algebras by introducing the noncommutative weak Extension Principle ($\mathsf{ncwEP}$). It proves that $\mathsf{ncwEP}$ holds under forcing axioms $\mathsf{OCA}$ and $\mathsf{MA}_{\aleph_1}$, by decomposing any map into a tractable, Borel part and a large-kernel part controlled via liftings and nonmeagre ideals, using advanced forcing-lifting techniques. The work also develops the theory of nonmeagre ideals in multipliers and coronas, showing such ideals are constrained (e.g., none exist in coronas of stable algebras) and deriving consequences for endomorphisms and embeddings under $\mathsf{OCA}$+$\mathsf{MA}_{\aleph_1}$. Finally, it discusses noncommutative dimension phenomena, connecting to commutative dimension results and outlining open problems about piecewise elementary maps and tensorial factorizations of coronas. Overall, the paper establishes a robust rigidity framework for noncommutative coronas under forcing axioms and clarifies the role of ideals and dimension in this setting.
Abstract
We introduce and study the noncommutative weak Extension Principle, a lifting principle aiming to characterise $^*$-homomorphisms between coronas of nonunital separable $\mathrm{C}^*$-algebras. While this principle fails if the Continuum Hypothesis is assumed, we show that this principle holds under mild forcing axioms such as the Open Colouring Axiom and Martin's Axiom. Further, we introduce and study the notion of nonmeagre ideals in multipliers and coronas of noncommutative $\mathrm{C}^*$-algebras, generalising the usual notion of nonmeagre ideals in $\mathcal P(\mathbb N)$.