It is known that a semisimple Banach algebra A whose multiplication is continuous (in both variables) with respect to the weak topology, is finite dimensional (see M. Akkar, E. Albrecht, L. Oubbi; A further characterization of finite dimensional Banach algebras; Preprint 1997). It is also known that, in a radical Banach algebra, the multiplication may be weakly continuous (any Banach space with the trivial multiplication). It may also happen that the multiplication in such an algebra is not weakly continuous (Take any infinite dimensional radical Banach algebra without any maximal ideals. see L. Oubbi, Weak topological algebras and P-algebra property; Mathematics Studies 4, Proceedings of ICTAA 2008, Estonian Mathematical Society, Tartu 2008, pp.73-79). Therefore the following question occurs : Which radical Banach algebras have a weakly continuous multiplication?