Dense Banach subalgebras of the null sequence algebra which do not   satisfy a differential seminorm condition
  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 $\|ab \|_A \leq K(\|a\|_{\infty} \|b\|_A + \|a \|_A \|b \|_{\infty})$, for all $a, b \in A$. The sequences in $A$ vanish in a skewed manner with respect to an unbounded function $\sigma\colon {\mathbb N} \rightarrow [1, \infty)$. 
Larry B. Schweitzer
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