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 $c_0$ which are spectral-invariant but do not satisfy the $D_1$-condition $\Vert ab{\Vert }_A\le K(\Vert a{\Vert }_{\infty }\Vert b{\Vert }_A+\Vert a{\Vert }_A\Vert b{\Vert }_{\infty })$ for all $a,b\in A$. The sequences in $A$ vanish in a skewed manner with respect to an unbounded function $\sigma :\mathbb{N}\to [1,\infty )$.