‎representation of a class of semisimple f-algebras as a topological product of primitive algebras

‎it is a classical theorem of bonsall-duncan that every semisimple banach algebra can be represented as a product of primitive banach algebras‎. ‎in this paper‎, ‎we investigate the extent to which this result remains valid beyond the banach algebra setting‎. ‎first, we show that the usual product topology fails to preserve several important topological properties‎, ‎such as local boundedness‎, ‎metrizability‎, ‎and strong sequentiality‎. ‎nevertheless‎, ‎we establish a weaker version of the theorem for a class of semisimple f-algebras‎, ‎namely the class of ffss-algebras‎, ‎which properly contains the class of locally bounded complete algebras‎. ‎to obtain a stronger analogue of the classical theorem‎, ‎we focus on semisimple locally bounded complete algebras‎. ‎by introducing a suitable product structure endowed with a different topology (the supremum norm topology)‎, ‎we establish a complete counterpart of the bonsall-duncan theorem for this class‎.