Obstructions to lifting abelian subalgebras of corona algebras

Andrea Vaccaro

Vol. 302 (2019), No. 1, 293–307

DOI: 10.2140/pjm.2019.302.293

Abstract

Let $A$ be a noncommutative, nonunital $C^{\ast }$-algebra. Given a set of commuting positive elements in the corona algebra $Q\left(A\right)$, we study some obstructions to the existence of a commutative lifting of such a set to the multiplier algebra $M\left(A\right)$. Our focus is on 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, nonunital, $\sigma$-unital $C^{\ast }$-algebra $A$, there exists an uncountable set of pairwise orthogonal positive elements in $Q\left(A\right)$ such that no uncountable subset of it can be lifted to a set of commuting elements of $M\left(A\right)$. Moreover, the positive elements in $Q\left(A\right)$ can be chosen to be projections if $A$ has real rank zero.

Keywords

corona algebra, commuting self-adjoint elements, lifting

Mathematical Subject Classification 2010

Primary: 47C15

Secondary: 03E75

Milestones

Received: 5 February 2017

Revised: 20 March 2019

Accepted: 21 March 2019

Published: 5 November 2019