arXiv subjects

Md Amir Hossain

Publications and source records attributed to Md Amir Hossain.

13 recordsLinked to original sources

arXiv2026

Ground states on the $\mathrm{C}^*$-algebras of Fell bundles over étale groupoids

Let $p\colon \mathcal{A}\to G$ be a Fell bundle over a locally compact Hausdorff second countable étale groupoid $G$ and let $\mathrm{C}^*(G; \mathcal{A})$ be the associated Fell bundle $\mathrm{C}^*$-algebra. Suppose $σ^c$ is the real dynamics on $\mathrm{C}^*(G;\mathcal{A})$ induced by a real-valued $1$-cocycle $c$. We establish an affine homeomorphism between the $σ^c$-ground state space of $\mathrm{C}^*(G;\mathcal{A})$ and the state space of $\mathrm{C}^*(G(Z); \mathcal{A}|_{G(Z)})$, where $Z$ is the boundary set of $c$ and $G(Z)$ is the boundary groupoid. In particular, $\mathrm{C}^*(G; \mathcal{A})$ admits $σ^c$-ground states if and only if $Z\neq \emptyset$. We further investigate the relation between ground states and KMS$_{\infty}$ states and give a crystallization interpretation of the boundary $\mathrm{C}^*$-algebra $\mathrm{C}^*(G(Z); \mathcal{A}|_{G(Z)})$ when the cocycle is locally constant. Finally, we apply our results to several classes of examples, including twisted $\mathrm{C}^*$-algebras of Deaconu--Renault groupoids and twisted higher-rank graph $\mathrm{C}^*$-algebras, obtaining explicit descriptions of their ground states.

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arXiv2026

Some remarks on Reduced $C^*$-algebras of semigroup dynamical systems and product systems

We study the exactness of the reduced crossed product of a semigroup dynamical system and the reduced $C^{*}$-algebra of a product system. We show that for a semigroup dynamical system $(A, P,α)$, under reasonable hypotheses (e.g., $P$ is abelian and finitely generated), the reduced crossed product $A \rtimes_{red} P$ is exact if and only if $A$ is exact. This strengthens our earlier result (\cite{Amir_Sundar-product-system}), where it was assumed that the action of $P$ on $A$ is by injective endomorphisms. We also compare the groupoid crossed product described in \cite{Amir_Sundar-product-system} and the Fell bundle constructed in \cite{Rennie_Sims} for a product system, and show that they are equivalent as Fell bundles.

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arXiv2026

Inclusions of Fell bundles $\mathrm{C}^*$-algebras and coaction crossed products

Let $p \colon \mathcal{A} \to G$ be a Fell bundle over a locally compact Hausdorff second countable groupoid $G$ equipped with a Haar system, and let $Γ$ be a discrete group. Given a continuous $1$-cocycle $c \colon G \to Γ$, we show that the $\mathrm{C}^*$-algebra of the restricted Fell bundle $\mathcal{A}|_{G_e}$ embeds isometrically into $\mathrm{C}^*(G;\mathcal{A})$, where $G_e = c^{-1}(e)$ is the clopen subgroupoid corresponding to the identity element. We exploit this embedding to show that $\mathrm{C}^*(G;\mathcal{A})$ admits a natural structure of a topologically graded $\mathrm{C}^*$-algebra in the sense of Exel. As a consequence, we obtain a canonical coaction $δ$ of $Γ$ on $\mathrm{C}^*(G; \mathcal{A})$. We further show that the associated coaction crossed product $\mathrm{C}^*(G; \mathcal{A})\rtimes_δΓ$ is naturally isomorphic to the $\mathrm{C}^*$-algebra of a Fell bundle constructed from the cocycle data.

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arXiv2026

On the Haagerup property for partial crossed products

Let $(A,G,α)$ be a partial dynamical system and let $A\rtimes_{α,r} G$ denote the associated reduced partial crossed product. In this article, we introduce the Haagerup property for partial actions of discrete groups on $C^*$-algebras. We prove that the partial crossed product $A\rtimes_{α,r} G$ has the Haagerup property if and only if both $A$ and the partial action $α$ have the Haagerup property. As a consequence, we obtain an equivalence between the Haagerup property of the partial crossed product and that of the underlying $C^*$-algebra and the acting group. We also show that the Haagerup property is preserved under inductive limits and apply this result to study the Haagerup property of inductive limits of partial crossed products.

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arXiv2025

Inductive limits of partial crossed products

Let $\big((A^{(i)}, G, α^{(i)}), ϕ_i\big)_{i \in \mathbb{N}}$ be an inductive sequence of partial dynamical systems. We prove the existence of an induced partial action $α$ of $G$ on the inductive limit $A=\varinjlim A^{(i)}$. We call $α$ the inductive limit partial action. Furthermore, we show the corresponding partial crossed product $A\rtimes_αG$ is canonically isomorphic to $\varinjlim A^{(i)}\rtimes_{α^{(i)}}G$. We also study the globalization of the inductive limit partial action $α$, its finite Rokhlin dimension and tracial states on $A\rtimes_αG$.

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arXiv2025

Extreme points of unital completely positive maps invariant under partial action

The classical Choquet theorem establishes a barycentric decomposition for elements in a compact convex subset of a locally convex topological vector space. This decomposition is achieved through a probability measure that is supported on the set of extreme points of the subset. In this work, we consider a partial action $τ$ of a group $G$ on a $C^\ast$-algebra $\mathcal{A}$. For a fixed Hilbert space $\mathcal{H}$, we consider the set of all unital completely positive maps from $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$ that are invariant under the partial action $τ$. This set forms a compact convex subset of a locally convex topological vector space. To complete the picture of the barycentric decomposition provided by the classical Choquet theorem, we characterize the set of extreme points of this set.

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arXiv2025

Reduced $C^{*}$-algebras of Product Systems -- an $E_0$-semigroup and a Groupoid perspective

For Ore semigroups $P$ with an order unit, we prove that there is a bijection between $E_0$-semigroups over $P$ and product systems of $C^{*}$-correspondences over $P^{op}$. We exploit this bijection and show that the reduced $C^{*}$-algebra of a proper product system is Morita equivalent to the reduced crossed product of the associated semigroup dynamical system given by the corresponding $E_0$-semigroup. We appeal to the groupoid picture of the reduced crossed product of a semigroup dynamical system derived in [47] to prove that, under good conditions, the reduced $C^{*}$-algebra of a proper product system is nuclear/exact if and only if the coefficient algebra is nuclear/exact. We also discuss the invariance of $K$-theory under homotopy of product systems.

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arXiv2025

Metrics on $C^{\ast}$-algebras of Étale groupoids from length functions

We show that for an étale groupoid with compact unit space, the natural Dirac type operator from a continuous length function produces a natural pseudo-metric on the state space of the corresponding reduced $C^{\ast}$-algebra. For a transformation groupoid with a continuous, proper length function with rapid decay, the state space decomposes into genuine metric spaces with a uniform finite diameter fibred over the state space of the compact unit space. Moreover, when the unit space of the transformation groupoid has finitely many points, the metric on each fibre metrizes the weak*-topology.

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arXiv2023

C$^*$-algebras of Fell bundles over étale groupoids

We describe a construction for the full C$^*$-algebra of a possibly unsaturated Fell bundle over a possibly non-Hausdorff locally compact étale groupoid without appealing to Renault's disintegration theorem. This construction generalises the standard construction given by Muhly and Williams.

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arXiv2023

KMS states on the $\mathrm{C}^*$-algebras of Fell bundles over {é}tale groupoids

Let $p\colon \mathcal{A} \to G$ be a saturated Fell bundle over a locally compact, Hausdorff, second countable, {é}tale groupoid~$G$, and let $\mathrm{C}^*(G;\mathcal{A})$ denote its full $\mathrm{C}^*$-algebra. We prove an integration-disintegration theorem for KMS states on $\mathrm{C}^*(G;\mathcal{A})$ by establishing a one-to-one correspondence between such states and fields of measurable states on the $\mathrm{C}^*$-algebras of the Fell bundles over the isotropy groups. This correspondence is established for certain states on $\mathrm{C}^*(G;\mathcal{A})$ also. While proving this main result, we construct an induction $\mathrm{C}^*$-correspondence between~$\mathrm{C}^*(G;\mathcal{A})$ and the $\mathrm{C}^*$-algebra of an isotropy Fell bundle. We demonstrate our results through many examples such as groupoid crossed products, twisted groupoid crossed products, $G$-spaces and matrix algebras~$\mathrm{M}_n(\mathrm{C}(X))\otimes A$. While studying the matrix algebra~$\mathrm{M}_n(\mathrm{C}(X))$, we propose a groupoid model for it. While demonstrating our main result for this groupoid model, we provide a solution to the Radon--Nikodym problem for the groupoid used in this model.

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arXiv2023

Topological fundamental groupoid. I

We show that the fundamental groupoid~$Π_1(X)$ of a locally path connected semilocally simply connected space~$X$ can be equipped with a \emph{natural} topology so that it becomes a topological groupoid; we also justify the necessity and minimality of these two hypotheses on~$X$ in order to topologise the fundamental groupoid. We find that contrary to a belief -- especially among the Operator Algebraists -- the fundamental groupoid is not {\etale}. Further, we prove that the fundamental groupoid of a topological group, in particular a Lie group, is a \emph{transformation groupoid}; again, this result disproves a standard belief that the fundamental groupoids are \emph{far} away from being transformation groupoids. We also discuss the point-set topology on the fundamental groupoid with the intention of making it a locally compact groupoid.

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arXiv2023

Topological fundamental groupoid. III. Haar systems on the fundamental groupoid

Let $X$ be a path connected, locally path connected and semilocally simply connected space; let $\tilde{X}$ be its universal cover. We discuss the existence and description of a Haar system on the fundamental groupoid $Π_1(X)$ of $X$. The existence of a Haar system on $Π_1(X)$ is justified when $X$ is a second countable, locally compact and Hausdorff. We provide equivalent criteria for the existence of the Haar system on a locally compact (locally Hausdorff) fundamental groupoid in terms of certain measures on $X$ and $\tilde{X}$. $\mathrm{C}^*(Π_1(X))$ is described using a result of Muhly, Renault and Williams. Finally, two formulae for the Haar system on $Π_1(X)$ in terms of measures on $X$ or $\tilde{X}$ are given.

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arXiv2023

Topological Fundamental Groupoid. II. An action category of the fundamental groupoid

For a path connected, locally path connected and semilocally simply connected space $X$, let $Π_1(X)$ denote its topologised fundamental groupoid as established in the first article of this series. Let $\mathcal{E}$ be the category of $Π_1(X)$-spaces in which the momentum maps are local homeomorphisms. We show that this category is isomorphic to that of covering spaces of $X$. Using this, we give different characterisations for free or proper actions of the fundamental groupoid in $\mathcal{E}$.

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