The crossed product of finite hopf C*-algebra and C*-algebra

Let H be a finite Hopf C*-algebra and A a C*-algebra of finite dimension. In this paper, we focus on the crossed product A o H arising from the action of H on A, which is a *-algebra. In terms of the faithful positive Haar measure on a finite Hopf C*-algebra, one can construct a linear functional on the *-algebra A H, which is further a faithful positive linear functional. Here, the complete positivity of a positive linear functional plays a vital role in the argument. At last, we conclude that the crossed product A H is a C*-algebra of finite dimension according to a faithful *- representation.