On Banach subalgebras of the Dirichlet Hardy algebra H^∞ consisting of lacunary Dirichlet series

Let H^∞ H ∞ be the set of all Dirichlet series f\!=\!∑ ₙ₌₁^∞ aₙn⁻ˢ f = ∑ n = 1 ∞ a n n - s (where aₙ\!∈ C a n ∈ C for all n\!∈ \! N\!=\!\1,2,3,⋯ \ n ∈ N = { 1 , 2 , 3 , ⋯ } ) that converge at each s in the half-plane C₀\!:=\!\!∈ \! C\!:\! Re(s)\!>\!0\ C 0 : = { s ∈ C : Re ( s ) > 0 } , such that f ∞\!=\! s∈ C₀\!|f(s)|\!<\!∞ ‖ f ‖ ∞ = sup s ∈ C 0 | f ( s ) | < ∞ . Then H^∞ H ∞ is a Banach algebra with pointwise operations and the supremum norm · _∞ ‖ · ‖ ∞