Some notes on point derivations on semidirect products of Banach algebras

Let A and B be two Banach algebras such that B is a Banach A -bimodule with compatible algebra multiplication and module actions. The semidirect Banach algebra A nB is the 1-direct sum A B equipped with the algebra multiplication (a; b)(a0; b0) = (aa0; a:b0 + b:a0 + bb0) (a; a0 2 A ; b; b0 2 B) and 1 norm. In this note we investigate the continuity of point derivations on A n B at a character and establish some results in this respect. Particularly, as a result, we show that if point derivations on B are continuous, then point derivations on A nB at some character are continuous, provided that jB 6= 0.

کلیدواژه ها:

Banach algebra ، semidirect product ، point derivation ، continuity