arXiv subjects
Publications and source records attributed to Marzieh Forough.
Let $G$ be a locally compact, Hausdorff, second countable groupoid and $A$ be a separable, $C_0(G^{(0)})$-nuclear, $G$-$C^*$-algebra. We prove the existence of quasi-invariant, completely positive and contractive lifts for equivariant, completely positive and contractive maps from $A$ into a separable, quotient $C^*$-algebra. Along the way, we construct the Busby invariant for $G$-actions.
In this paper we show that the Cuntz--Pimsner algebras associated to minimal homeomorphisms twisted by line bundles, along with their orbit-breaking subalgebras, are $\mathcal{Z}$-stable whenever the underlying dynamical system has the small boundary property. This entails that this class is classified by the Elliott invariant. Furthermore, we show that the tensor product of two such $\mathrm{C}^*$-algebras is always $\mathcal{Z}$-stable, without assuming the small boundary property. In particular this applies to $\mathrm{C}^*$-algebras arising from systems with positive mean dimension.
Let $A$ and $B$ be $C^*$-algebras with $A$ separable, let $I$ be an ideal in $B$, and let $ψ\colon A\to B/I$ be a completely positive contractive linear map. We show that there is a continuous family $Θ_t\colon A\to B$, for $t\in [1,\infty)$, of lifts of $ψ$ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If $ψ$ is of order zero, then $Θ_t$ 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 $Θ_t$ can be chosen to be asymptotically equivariant. If a linear completely positive lift for $ψ$ exists, we can arrange that $Θ_t$ is linear and completely positive for all $t\in [1,\infty)$. 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.
In this paper, we construct a recursive subhomogeneous decomposition for the Cuntz--Pimsner algebras obtained from breaking the orbit of a minimal Hilbert $C(X)$-bimodule at a subset $Y \subset X$ with non-empty interior. This generalizes the known recursive subhomogeneous decomposition for orbit-breaking subalgebras of crossed products by minimal homeomorphisms.
In this paper we study Cuntz--Pimsner algebras associated to $\mathrm{C}^*$-correspondences over commutative $\mathrm{C}^*$-algebras from the point of view of the $\mathrm{C}^*$-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 $\mathrm{C}^*$-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 $\mathcal{Z}$-stable and hence classified by the Elliott invariant.
For a locally compact group $G$ and a strongly self-absorbing $G$-algebra $(\mathcal{D},δ)$, we obtain a new characterization of absorption of a strongly self-absorbing action using almost equivariant completely positive maps into the underlying algebra. The main technical tool to obtain this characterization is the existence of almost equivariant lifts for equivariant completely positive maps, proved in recent work of the authors. This characterization is then used to show that an equivariant $C_0(X)$-algebra with $\mathrm{dim}_{\mathrm{cov}}(X)<\infty$ is $(\mathcal{D},δ)$-stable if and only if all of its fibers are, extending a result of Hirshberg, Rørdam and Winter to the equivariant setting. The condition on the dimension of $X$ is known to be necessary, and we show that it can be removed if, for example, the bundle is locally trivial.
We develop the concept of weak tracial Rokhlin property for finite group actions on simple (not necessarily unital) C*-algebras and study its properties systematically. In particular, we show that this property is stable under restriction to invariant hereditary C*-algebras, minimal tensor products, and direct limits of actions. Some of these results are new even in the unital case and answer open questions asked by N. C. Phillips in full generality. We present several examples of finite group actions with the weak tracial Rokhlin property on simple stably projectionless C*-algebras. We prove that if $α\colon G \rightarrow \mathrm{Aut}(A)$ is an action of a finite group $G$ on a simple C*-algebra $A$ with tracial rank zero and $α$ has the weak tracial Rokhlin property, then the crossed product $A \rtimes _α G$ and the fixed point algebra $A^α$ are simple with tracial rank zero. This extends a result of N. C. Phillips to the nonunital case. We use the machinery of Cuntz subequivalence to work in this nonunital setting.
We prove that unital extensions of Kirchberg algebras by separable stable AF algebras have nuclear dimension one. The title follows.
Let $A$ be a simple, exact, separable, unital $C^*$-algebra and let $α\colon G \rightarrow Aut(A)$ be an action of a finite group $G$ with the weak tracial Rokhlin property. We show that every trace on $A \rtimes_α G$ is quasidiagonal provided that all traces on $A$ are quasidiagonal. As an application, we study the behavior of finite decomposition rank under taking crossed products by finite group actions with the weak tracial Rokhlin property. Moreover, we discuss the stability of the property that all traces are quasidiagonal under taking crossed products of finite group actions with finite Rokhlin dimension with commuting towers.
Let $A$ and $B$ be arbitrary $C^*$-algebras, we prove that the existence of a Hilbert $A$-$B$-bimodule of finite index ensures that the WEP, QWEP, and LLP along with other finite-dimensional approximation properties such as CBAP and (S)OAP are shared by $A$ and $B$. For this, we first study the stability of the WEP, QWEP and LLP under Morita equivalence of $C^*$-algebras. We present examples of Hilbert $A$-$B$-bimodules which are not of finite index, while such properties are shared between $A$ and $B$. To this end, we study twisted crossed products by amenable discrete groups.
We study the stability of Fredholm property for regular operators on Hilbert $C^*$-modules under some certain perturbations. We treat this problem when perturbing operators are (relatively) bounded or relatively compact. We also consider the perturbations of regular Fredholm operators in terms of the gap metric. In particular, we prove that the space of all regular Fredholm operators on a Hilbert $C^*$-module $E$ is open in the space of all regular operators on $E$ with respect to the gap metric. As an application, we construct some continuous paths of selfadjoint regular Fredholm operators with respect to the gap metric.