Let G be a locally compact abelian group with dual group \(\hat G,\) and let \(\Sigma\) be a fixed sub-semigroup of \(\hat G.\) The author constructs a symbol map for the \(C^*\)-algebra generated by Toeplitz and compact operators on the space \(H^ 2(\Sigma)(\subset L^ 2(G))\) associated with \(\Sigma\).
As a consequence it follows that the essential range of the symbol is contained in the essential spectrum of the corresponding Toeplitz operator.