Dense Banach subalgebras of the null sequence algebra which do not satisfy a differential seminorm condition

Abstract

We construct dense Banach subalgebras A of the null sequence algebra c0 which are spectral-invariant but do not satisfy the D1-condition ‖ab‖A≤K(‖a‖∞‖b‖A+‖a‖A‖b‖∞) for all a,b∈A. The sequences in A vanish in a skewed manner with respect to an unbounded function σ:N→[1,∞).