Vol. 302, No. 1, 2019

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Obstructions to lifting abelian subalgebras of corona algebras

Andrea Vaccaro

Vol. 302 (2019), No. 1, 293–307

Let A be a noncommutative, nonunital C-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 a set to the multiplier algebra M(A). 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, σ-unital C-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.

corona algebra, commuting self-adjoint elements, lifting
Mathematical Subject Classification 2010
Primary: 47C15
Secondary: 03E75
Received: 5 February 2017
Revised: 20 March 2019
Accepted: 21 March 2019
Published: 5 November 2019
Andrea Vaccaro
Department of Mathematics
Ben Gurion University of the Negev
Be’er Sheva