Schweitzer, Larry B. Dense Banach subalgebras of the null sequence algebra which do not satisfy a differential seminorm condition. (English) Zbl 1369.46042 Ann. Funct. Anal. 7, No. 4, 686-690 (2016). A subalgebra \(A\) of a \(C^*\)-algebra \(B\) is called a dense Banach subalgebra if \(A\) is dense in \((B, \|\cdot\|_B)\) and \(A\) is a Banach algebra with respect to some norm \(\|\cdot\|_A\). In the present paper, the author constructs dense Banach subalgebras \(A\) of \(c_0\), the null sequence algebra, which are spectral-invariant but do not satisfy the \(D_1\)-condition \(\|ab\|_A\leq K(\|a\|_0 \|b\|_A+ \|a\|_A \|b\|_0)\) for all \(a, b \in A\). Subalgebras of a \(C^*\)-algebra satisfying property \((D_p)\), \(p\geq 1\), were introduced and studied by E. Kissin and V. S. Shulman [Proc. Edinb. Math. Soc., II. Ser. 37, No. 3, 399–422 (1994; Zbl 0808.46066)]. Such a condition gives rise to various differential properties of the \(C^*\)-algebra. Reviewer: Ying-Fen Lin (Belfast) MSC: 46J10 Banach algebras of continuous functions, function algebras 46L05 General theory of \(C^*\)-algebras Keywords:\(D_1\)-subalgebra; spectral invariance; null sequence algebra; differential structure in \(C^*\)-algebras Citations:Zbl 0808.46066 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid