Measures that are orthogonal to the algebra of functions that are generalized analytic in the sense of Arens-Singer (Q1842700)

Let \(\Gamma\) be a discrete subgroup of the additive group \(\mathbb{R}\) of real numbers and let \(G\) be the group of characters of \(\Gamma\). Each element \(a \in \Gamma\) generates a continuous character of the group \(G\) defined by the relation \(\chi^\alpha (\alpha) = \alpha (a)\), \(\alpha \in G\). The present paper describes the structure of a measure, orthogonal to the algebra \(A\), which is a uniform algebra on \(G\), generated by the characters \(\chi^\alpha\), \(a \in \Gamma_+ = \{a \in \Gamma : a \geq a\}\).