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

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