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

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Let G be a locally compact abelian Hausdorff topological group which is non-compact and whose Pontryagin dual Γ is partially ordered. Let Γ⁺⊂Γ be the semigroup of positive elements in Γ. The Hardy space H²(G) is the closed subspace of L²(G) consisting of functions whose Fourier transforms are supported on Γ⁺. In this paper we consider the C*-algebra C*(T(G)∪ F(C(Γ⁺))) 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 Γ⁺ on the Hilbert-Hardy space H²(G). We characterize the character space of this C*-algebra using a theorem of Power.