On the C*-algebra Generated by Toeplitz Operators and Fourier Multipliers on the Hardy Space of a Locally Compact Group

Abstract

Let $G$ be a locally compact abelian Hausdorff topological group which is non-compact and whose Pontryagin dual $\Gamma$ is partially ordered. Let ${\Gamma }^{+}\subset \Gamma$ be the semigroup of positive elements in $\Gamma$. The Hardy space $H^2(G)$ is the closed subspace of $L^2(G)$ consisting of functions whose Fourier transforms are supported on ${\Gamma }^{+}$. In this paper we consider the C*-algebra $C^{\ast }(\mathcal{T}(G)\cup F(C(\dot{{\Gamma }^{+}})))$ generated by Toeplitz operators with continuous symbols on $G$ which vanish at infinity and Fourier multipliers with symbols which are continuous on one point compactification of ${\Gamma }^{+}$ on the Hilbert-Hardy space $H^2(G)$. We characterize the character space of this C*-algebra using a theorem of Power.

arXiv comment11 pages

References in corpus (1)

• Essential Spectra of Quasi-parabolic Composition Operators on Hardy Spaces of Analytic Functions