arXiv subjects

Ali Imad Raad

Publications and source records attributed to Ali Imad Raad.

5 recordsLinked to original sources

arXiv2026

AF-action groupoid models for diagonal AH-algebras

We show that an inductive AH-system with diagonal connecting maps describes an action of the canonical AF-groupoid on the spectrum of the canonical C$^*$-diagonal, and that the canonical groupoid model is given by the transformation groupoid associated to this action. This divides the groupoid structure into two distinct aspects: the acting AF-groupoid (which is well-studied) and the unit space on which it acts. We describe the unit space by enriching the Bratteli diagram with extra topological information, and apply these descriptions to examples of interest, including Villadsen algebras of the first kind.

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arXiv2025

C$^*$-diagonals in AH-algebras arising from generalized diagonal connecting maps: spectrum and uniqueness

We associate a Bratteli-type diagram to AH-algebras arising from generalized diagonal connecting maps. We use this diagram to give an explicit description of the connected components of the spectrum of an associated canonical C$^*$-diagonal. We introduce a topological notion on these connected components, that of being spectrally incomplete, and use it as a tool to show how various classes of AI-algebras, including certain Goodearl algebras and AH-algebra models for dynamical systems $([0,1],σ)$, do not admit unique inductive limit Cartan subalgebras. We focus on a class of spectrally complete C$^*$-algebras, namely the AF-algebras, and discuss the uniqueness of their inductive limit Cartan subalgebras.

math.OA↗

arXiv2023

Constructing C*-diagonals in AH-algebras

We construct Cartan subalgebras and hence groupoid models for classes of AH-algebras. Our results cover all AH-algebras whose building blocks have base spaces of dimension at most one as well as Villadsen algebras, and thus go beyond classifiable simple C*-algebras.

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arXiv2023

Inductive Limits of Noncommutative Cartan Inclusions

We prove that an inductive limit of aperiodic noncommutative Cartan inclusions is a noncommutative Cartan inclusion whenever the connecting maps are injective, preserve normalisers and entwine conditional expectations. We show that under the additional assumption that the inductive limit Cartan subalgebra is either essentially separable, essentially simple or essentially of Type I we get an aperiodic inclusion in the limit. Consequently, we subsume the case where the building block Cartan subalgebras are commutative and provide a proof of a theorem of Xin Li without passing to twisted étale groupoids.

math.OA↗

arXiv2022

A Generalization of Renault's Theorem for Cartan Subalgebras

We prove a generalized version of Renault's theorem for Cartan subalgebras. We show that the original assumptions of second countability and separability are not needed. This weakens the assumption of topological principality of the underlying groupoid to effectiveness.

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