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 of 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*(T(G) boolean OR F(C(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.