arXiv subjects

K. N. Sridharan

Publications and source records attributed to K. N. Sridharan.

4 recordsLinked to original sources

arXiv2026

Weak containment of representation on topological groupoids

Let $G$ be a second-countable, locally compact Hausdorff groupoid equipped with a Haar system. This paper investigates the weak containment of continuous unitary representations of groupoids. We show that both induction and inner tensor product of representations preserve weak containment. Additionally, we introduce the notion of a topological invariant mean on $G/H$ and explore its connection to amenability. With that, we establish a groupoid analogue of Greenleaf's theorem. Finally, we provide independent results concerning the restriction of induced representations for continuous unitary representations of relatively clopen wide subgroupoids $H\subseteq G$ with discrete unit space and closed transitive wide subgroupoids of compact transitive groupoids.

math.FA↗

arXiv2026

Amenable unitary representations of locally compact groupoids

Let $G$ be a second countable locally compact groupoid equipped with a Haar system $λ$.In this work, we introduce and develop the notion of amenability for continuous unitary representations of $G$, formulated in terms of Hilbert bundles over the unit space $G^{0}$. We prove that $G$ is amenable if and only if its left regular representation is amenable, thereby extending Bekka's characterisation of amenable unitary representations from groups to groupoids. We further investigate the amenability of induced representations of $G$ and also study the representation of properly amenable groupoids. We define a topological invariant mean associated with a representation, constructed by utilising the theory of operator-valued vector measures on the unit space $G^{0}$, to characterise amenability. Finally, we investigate the relationship between the amenability of a representation and the amenability of its restriction to isotropy subgroups with an emphasis on HLS and certain transitive groupoids.

math.OA↗

arXiv2026

Continuous field of Orlicz space on locally compact groupoids and related results

Let $G$ be a locally compact second countable groupoid with a fixed Haar system $λ=\{λ^{u}\}_{u\in G^{0}}$ and $(Φ,Ψ)$ be a complementary pair of $N$-functions satisfying $Δ_{2}$-condition. In this article, we introduce the continuous field of Orlicz space $(L^Φ_{0},Δ_{1})$ and provide a sufficient condition for the space of continuous sections vanishing at infinity, denoted $E^Φ_{0}$, to be an Banach algebra under a suitable convolution. Further, the condition for a closed $C_{b}(G^{0})$-submodule $I$ of $E^Φ_{0}$ to be a left ideal is established. Moreover, we provide a groupoid analogue of the characterization of the space of convolutors of Morse-Transue space for locally compact groups.

math.FA↗

arXiv2026

Induced Representation of Topological groupoids

Let $G$ be a locally compact second countable groupoid with a Haar system. In this article, we introduce the induced representation of $G$ from a continuous unitary representation of a closed wide subgroupoid $H$ with a Haarsystem provided there exists a full equivariant system of measures $μ=\{μ^{u}\}_{u\in G^{0}}$ on $G/H$. We prove some basic properties of induced representation and a theorem on induction in stages. A groupoid version of Mackey's tensor product theorem is also provided. We also prove a groupoid version of Frobenius Reciprocity theorem on compact transitive groupoids.

math.OA↗