Quantum convolution inequalities on Frobenius von Neumann algebras

In this paper, we introduce Frobenius von Neumann algebras and study quantum convolution inequalities. In this framework, we unify quantum Young's inequality on quantum symmetries such as subfactors, and fusion bi-algebras studied in quantum Fourier analysis. Moreover, we prove quantum entropic convolution inequalities and characterize the extremizers in the subfactor case. We also prove quantum smooth entropic convolution inequalities. We obtain the positivity of comultiplications of subfactor planar algebras, which is stronger than the quantum Schur product theorem. All these inequalities provide analytic obstructions of unitary categoriflcation of fusion rings stronger than the Schur product criterion.