A note on zero Lie product determined nest algebras as Banach algebras

A Banach algebra \A is said to be zero Lie product determined Banach algebra if for every continuous bilinear functional \phi:\A \times \A\rightarrow \mathbb{C} the following holds: if \phi(a,b)=۰ whenever ab=ba, then there exists some \tau \in \A^* such that \phi(a,b)=\tau(ab-ba) for all a,b\in \A. We show that any finite nest algebra over a complex Hilbert space is a zero Lie product determined Banach algebra.