Linear isometries of the domain \(\mathfrak D(\delta^2)\) of the square of a closed-derivation \(\delta\) (Q819666)

The authors continue their earlier work from a decade ago [J.\ Math.\ Soc.\ Jpn.\ 48, No.~2, 229--254 (1996; Zbl 0860.46013)] on the description, along the lines of the classical Banach--Stone theorem, of surjective isometries of the domain of powers of a closed *-derivation \(\delta\) on a \(C(K)\) space. Let \(D(\delta^2) = \{f \in D(\delta) : \delta(f) \in D(\delta)\}\). This is a unital *-subalgebra and is a Banach space with the norm \(\| f\| = \max~\{\| f\| _{\infty}, \| \delta^2(f)\| _{\infty}\}\). The authors first obtain a description of the extreme points of the dual unit ball under the assumptions that \(D(\delta^2)\) is norm dense in \(C(K)\) and \(\delta(D(\delta^2)) = D(\delta)\). Using this, the main result of the paper (Theorem 3.1) gives a complete description of surjective isometries between \(D(\delta_1 ^2)\) and \(D(\delta_2 ^2)\).