Group-graded rings, smash products, and group actions

Let A A be a k k -algebra graded by a finite group G G , with A 1 {A_1} the component for the identity element of G G . We consider such a grading as a “coaction” by G G , in that A A is a k [ G ] ∗ k{[G]^ \ast } -module algebra. We then study the smash product A # k [ G ] ∗ A\# k{[G]^ \ast } ; it plays a role similar to that played by the skew group ring R ∗ G R\, \ast \,G in the case of group actions, and enables us to obtain results relating the modules over A , A 1 A,\,{A_1} , and A # k [ G ] ∗ A\# k{[G]^ \ast } . After giving algebraic versions of the Duality Theorems for Actions and Coactions (results coming from von Neumann algebras), we apply them to study the prime ideals of A A and A 1 {A_1} . In particular we generalize Lorenz and Passman’s theorem on incomparability of primes in crossed products. We also answer a question of Bergman on graded Jacobson radicals.