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

Abstract

Let $G$ be a locally compact second countable groupoid with a fixed Haar system $\lambda=\{\lambda^{u}\}_{u\in G^{0}}$ and $(\Phi,\Psi)$ be a complementary pair of $N$-functions satisfying $\Delta_{2}$-condition. In this article, we introduce the continuous field of Orlicz space $(L^{\Phi}_{0},\Delta_{1})$ and provide a sufficient condition for the space of continuous sections vanishing at infinity, denoted $E^{\Phi}_{0}$, to be an Banach algebra under a suitable convolution. Further, the condition for a closed $C_{b}(G^{0})$-submodule $I$ of $E^{\Phi}_{0}$ 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.