  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 $C^{*}(\mathcal{T}(G)\cup F(C(\dot{\Gamma^{+}})))$
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.
