Research Projects

Inclusions of operator algebras are ubiquitous in the theory of C*-algebras and von Neumann algebras. The theory of von Neumann algebras is oftentimes called noncommutative measure theory, while the C*-algebra theory is called the noncommutative topology. Algebras with finite index have wide applications in operator algebra theory, knot theory, low dimensional topology, mathematical physics, biology, to name a few. Our project aims to apply techniques of subfactor theory, pioneered by Vaughan Jones, to an inclusion of C*-algebras and vice versa. Inclusion of C*-algebras examines the relative position of one subalgebra inside the ambient algebra. However, if there are multiple subalgebras the overall structure becomes very complicated. In this project we focus on a pair of subalgebras with finite indices and investigate the quantum entropy, angle operator and Pimsner-Popa probabilistic constant. The key technical tool we use is the Fourier theory of the relative commutants of the inclusion. The project will have applications in various quantum Fourier theoretic inequalities, noncommutative uncertainty principle, noncommutative ergodic theory and quantum information theory. The project will open new research avenues in the theory of inclusion of C*-algebras with finite Watatani index and Jones subfactor theory.