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 g{cyrillic} is partially ordered. Let g{cyrillic}₊ ⊂ g{cyrillic} be the semigroup of positive elements in g{cyrillic}. The Hardy space H²(G) is the closed subspace of L²(G) consisting of functions whose Fourier transforms are supported on g{cyrillic}⁺. In this paper we consider the C*-algebra C*(T (G) ∪ F(C(g{cyrillic}₊))) 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 g{cyrillic}⁺ on the Hilbert-Hardy space H²(G). We characterize the character space of this C*-algebra using a theorem of Power.