Elements in a commutative Banach algebra determining the norm topology (Q2701582)

Let \((A, \|\cdot\|)\) be a commutative complex Banach algebra. An element \(a\in A\) is said to \textit{determine the norm topology of} \(A\) if every norm \(|\cdot|\) on \(A\) for which it becomes a Banach space and which makes the multiplication operator by \(a\) from \((A, |\cdot|)\) to itself continuous is equivalent to \(\|\cdot\|\). The element \(a\) is said to almost determine the norm topology of \(A\) if the following property holds: for every complete norm on \(A\) making the multiplication by \(a\) from \((A, |\cdot|)\) to itself continuous there exists an idempotent \(e\in A\) such that \(\dim eA<\infty\) and the quotient norms of \(|\cdot|\) and \(\|\cdot\|\) on \(A/(eA)\) are equivalent. The author gives some conditions when an element \(a\in A\) determines the norm topology, almost determines the norm topology or does not determine the norm topology of \(A\).