The noncommutative weak Extension Principle

TL;DR

This paper introduces the noncommutative weak Extension Principle, a new lifting principle for $^*$-homomorphisms between coronas of nonunital separable C*-algebras, and explores its validity under various set-theoretic axioms.

Contribution

It defines the noncommutative weak Extension Principle and investigates its consistency under different set-theoretic assumptions, also introducing nonmeagre ideals in noncommutative C*-algebras.

Findings

The principle holds under Open Colouring Axiom and Martin's Axiom.

The principle fails under the Continuum Hypothesis.

Introduces nonmeagre ideals in multipliers and coronas of noncommutative C*-algebras.

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 $\mathcal P(\mathbb N)$.