Alladi Ramakrishnan Hall

Continuous Minmax Theorems (joint work with V. S. Sunder)

Madhusree Basu

IMSc

In this talk I will extend certain extremal characterizations of eigenvalues of Hermitian matrices to a `continuous' context, involving distributions of self adjoint elements of von Neumann algebras equipped with faithful normal tracial states. I will first prove an extension of the classical minmax theorem of Ky Fan's (characterizing sum of smallest $k$ eigenvalues of an $n \times n$ scalar Hermitian matrix for all $1 \le k \le n$), in a von-Neumann algebraic setting. Then, as an application of the above, I will prove an exact analogue of the Courant-Fischer-Weyl minmax theorem for self adjoint elements having no eigenvalues, inside $II_1$ factors and discuss some of its applications.