Obstructions to lifting abelian subalgebras of corona algebras

TL;DR

This paper investigates the limitations in lifting large collections of commuting positive elements from the corona algebra to the multiplier algebra in certain non-commutative C*-algebras, revealing size-dependent obstructions.

Contribution

It demonstrates that for uncountably large collections, lifting to commuting elements is impossible, highlighting size as an obstruction in the lifting problem.

Findings

No obstacles for countable families of projections or orthogonal positive elements

Existence of uncountable orthogonal positive elements with no commuting lift subsets

Obstructions depend on the size of the collection and the algebra's properties

Abstract

Let $A$ be a non-commutative, non-unital $\mathrm{C}^\ast$-algebra. Given a set of commuting positive elements in the corona algebra $Q(A)$, we study some obstructions to the existence of a commutative lifting of such set to the multiplier algebra $M(A)$. Our focus are the obstructions caused by the size of the collection we want to lift. It is known that no obstacles show up when lifting a countable family of commuting projections, or of pairwise orthogonal positive elements. However, this is not the case for larger collections. We prove in fact that for every primitive, non-unital, $\sigma$-unital $\mathrm{C}^\ast$-algebra $A$, there exists an uncountable set of pairwise orthogonal positive elements in $Q(A)$ such that no uncountable subset of it can be lifted to a set of commuting elements of $M(A)$. Moreover, the positive elements in $Q(A)$ can be chosen to be projections if $A$ has real rank zero.