Obstructions to lifting abelian subalgebras of corona algebras

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.