Abstract
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⋊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.
Highlights
We prove that the crossed product of a finite Hopf C ∗ -algebra and a
We focus on the crossed product that arises from the action of finite
Hopf C ∗ -algebra on a C ∗ -algebra
Summary
A Hopf C ∗ -algebra can be considered as a non-commutative locally compact quantum group. There is a possible approach to consider the crossed product algebras in the finite dimensional case. J. et al [7] They demonstrated that when H is a finite dimensional, semisimple Hopf algebra, and A is a semisimple Artinian algebra, the crossed product. Our work highlights the method of proof, which is to construct a faithful positive linear functional, to show the crossed product A o H is a C ∗ -algebra. Liu et al [10] established a faithful positive linear functional, and showed that the quantum double of the pairing of two finite Hopf C ∗ -algebras is a C ∗ -algebra.
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