Crossed products of operator algebras: Applications of Takai duality

Using Takai duality we also continue our study of the Radical for the crossed product of an operator algebra and we solve open problems stemming from the earlier work of the authors. Among others we show that the crossed product of a radical operator algebra by a compact abelian group is a radical operator algebra. We also show that the permanence of semisimplicity fails for crossed products by R. A final section of the paper is devoted to the study of radically tight dynamical systems, i.e., dynamical systems (A,G,α) for which the identity Rad(A⋊αG)=(RadA)⋊αG persists. A broad class of such dynamical systems is identified.