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Dense Banach subalgebras of the null sequence algebra which do not satisfy a differential seminorm condition. (English) Zbl 1369.46042

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.

MSC:

46J10 Banach algebras of continuous functions, function algebras
46L05 General theory of \(C^*\)-algebras

Citations:

Zbl 0808.46066