Heat semigroups on quantum automorphism groups of finite dimensional C*-algebras

In this paper, we investigate heat semigroups on a quantum automorphism group Aut + ( B ) of a finite dimensional C*-algebra B and its Plancherel trace. We show ultracontractivity, hypercontractivity, and the spectral gap inequality of the heat semigroups on Aut + ( B ) . Furthermore, we obtain the sharpness of the Sobolev embedding property and the Hausdorff–Young inequality of Aut + ( B ) .