Quantum inequalities and their applications

In recent years, various quantum inequalities have been established on quantum symmetries in the framework of quantum Fourier analysis. We provide a detailed introduction to quantum inequalities including Hausdorff-Young inequality, Young's inequality, uncertainty principles, entropic convolution inequalities etc on subfactors, an important type of quantum symmetries. We cite several applications of the complete positivity of the comultiplication in category theory and subfactor theory, which indicate the fundamental differences between quantum inequalities and non-commutative inequalities. We also review the Perron-Frobenius theorem together with the algebraic structures of eigenvector spaces.