Marzieh Forough, Ph.D.

Article

10.1016/j.jfa.2024.110655

Department of Applied Mathematics

Let A and B be C⁎-algebras with A separable, let I be an ideal in B, and let ψ:A→B/I be a completely positive contractive linear map. We show that there is a continuous family Θt:A→B, for t∈[1,∞), of lifts of ψ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If ψ is of order zero, then Θt can be chosen to have this property asymptotically. If A and B carry continuous actions of a second countable locally compact group G such that I is G-invariant and ψ is equivariant, we show that the family Θt can be chosen to be asymptotically equivariant. If a linear completely positive lift for ψ exists, we can arrange that Θt is linear and completely positive for all t∈[1,∞). In the equivariant setting, if A, B and ψ are unital, we show that asymptotically linear unital lifts are only guaranteed to exist if G is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.

Article

10.1090/tran/8900

Department of Applied Mathematics

In this paper we study Cuntz-Pimsner algebras associated to C*-correspondences over commutative C*-algebras from the point of view of the C*-algebra classification programme. We show that when the correspondence comes from an aperiodic homeomorphism of a finite dimensional infinite compact metric space X twisted by a vector bundle, the resulting Cuntz- Pimsner algebras have finite nuclear dimension. When the homeomorphism is minimal, this entails classification of these C*-algebras by the Elliott invariant. This establishes a dichotomy: when the vector bundle has rank one, the Cuntz-Pimsner algebra has stable rank one. Otherwise, it is purely infinite. For a Cuntz-Pimsner algebra of a minimal homeomorphism of an infinite compact metric space X twisted by a line bundle over X, we introduce orbit breaking subalgebras. With no assumptions on the dimension of X, we show that they are centrally large subalgebras and hence simple and stably finite. When the dimension of X is finite, they are furthermore Z-stable and hence classified by the Elliott invariant.