Let \(\mathscr {H}^\infty \) be the set of all Dirichlet series \(\textstyle f\!=\!{{\sum \limits _{n=1}^\infty }} a_nn^{-s}\) (where \(a_n\!\in \mathbb {C}\) for all \(n\!\in \! \mathbb {N}\!=\!\{1,2,3,\cdots \}\) ) that converge at each s in the half-plane \(\mathbb {C}_0\!:=\!\{s\!\in \! \mathbb {C}\!:\! \text {Re}(s)\!>\!0\}\) , such that \(\Vert f\Vert _{\infty }\!=\!\sup _{s\in \mathbb {C}_0}\!|f(s)|\!<\!\infty \) . Then \(\mathscr {H}^\infty \) is a Banach algebra with pointwise operations and the supremum norm \(\Vert \cdot \Vert _\infty \) , and has been studied in earlier works. The article introduces a new family of Banach subalgebras \(\mathscr {H}^\infty _{S}\) of \(\mathscr {H}^\infty \) . For \(S\!\subset \! \mathbb {N}\) , let \(\mathscr {H}^\infty _{S}\) be the set of all elements \(\textstyle {{\sum \limits _{n=1}^\infty }} a_nn^{-s}\in \mathscr {H}^\infty \) such that for all \(n\in \mathbb {N}\setminus S\) , \(a_n\!=\!0\) . Then \(\mathscr {H}^\infty _{S}\) is a unital Banach subalgebra of \(\mathscr {H}^\infty \) with the \(\Vert \cdot \Vert _\infty \) norm if and only if S is a multiplicative subsemigroup of \(\mathbb {N}\) containing 1. It is shown that for such S, \(\mathscr {H}^\infty _{S}\) is the multiplier algebra of \(\mathscr {H}^2_S\) , where \(\mathscr {H}^2_S\) is the Hilbert space of all \( \textstyle f\!=\!{{\sum \limits _{n\in S}}} a_nn^{-s}\) such that \(\Vert f\Vert _2\!:=\!({{\sum \limits _{n\in S}}} |a_n|^2)^{\frac{1}{2}}\!<\!\infty \) . A characterisation of the group of units in \(\mathscr {H}^\infty _{S}\) is given, by showing an analogue of the Wiener 1/f theorem for \(\mathscr {H}^\infty _{S}\) . If S has an infinite set of generators allowing a unique representation of each element of S, then it is shown that the Bass stable rank of \(\mathscr {H}^\infty _S\) is infinite.
