Vector-valued holomorphic and harmonic functions

Holomorphic and harmonic functions with values in a Banach space are investigated. Following an approach given in a joint article with Nikolski [4] it is shown that for bounded functions with values in a Banach space it suffices that the composition with functionals in a separating subspace of the dual space be holomorphic to deduce holomorphy. Another result is Vitali’s convergence theorem for holomorphic functions. The main novelty in the article is to prove analogous results for harmonic functions with values in a Banach space.

Continuity of Lie derivations on Banach algebras

The separating subspace of any Lie derivation on a semisimple Banach algebra A is contained in the centre of A.