Symmetry and inverse-closedness of some $p$-Beurling algebras

Abstract

Let $(G,d)$ be a metric space with the counting measure $\mu$ satisfying some growth conditions. Let $\omega (x,y)=(1+d(x,y){)}^{\delta }$ for some $0<\delta \le 1$. Let $0<p\le 1$. Let ${\mathcal{A}}_{p\omega }$ be the collection of kernels $K$ on $G\times G$ satisfying $\max \{{\sup }_x{\sum }_y|K(x,y){|}^p\omega (x,y{)}^p,{\sup }_y{\sum }_x|K(x,y){|}^p\omega (x,y{)}^p\}<\infty$. Each $K\in {\mathcal{A}}_{p\omega }$ defines a bounded linear operator on ${\ell }^2(G)$. If in addition, $\omega$ satisfies the weak growth condition, then we show that ${\mathcal{A}}_{p\omega }$ is inverse closed in $B({\ell }^2(G))$. We shall also discuss inverse-closedness of $p$-Banach algebra of infinite matrices over ${\mathbb{Z}}^d$ and the $p$-Banach algebra of weighted $p$-summable sequences over ${\mathbb{Z}}^{2d}$ with the twisted convolution. In order to show these results, we prove Hulanicki's lemma and Barnes' lemma for $p$-Banach algebras.