Free Probability Seminar

Date: March 2, 2015

Time: 11:30AM - 12:20PM

Location: BLOC 628

Speaker: Igor Klep, University of Auckland

Title: Dilations, inclusion of free spectrahedra and beta distributions

Abstract: Given a tuple A=(A₁,...,A_g) of symmetric matrices of the same size, consider the affine linear matrix polynomial L(x):=I - ∑ A_j x_j. The solution set S_L of the corresponding linear matrix inequality, consisting of those x in R^g for which L(x) is positive semidefinite (PsD), is called a spectrahedron. The set D_L of tuples X=(X₁,...,X_g) of symmetric matrices (of the same size) for which L(X):=I - ∑ A_j ⊗ X_j is PsD, is a free spectrahedron. We explain that any tuple X of symmetric matrices in a bounded free spectrahedron D_L dilates, up to a scale factor, to a tuple T of commuting self-adjoint operators with joint spectrum in the corresponding spectrahedron S_L. From another viewpoint, this scale factor measures the extent that a positive map can fail to be completely positive. In the case when S_L is the hypercube [-1,1]^g, we derive an analytical formula for the scale factor, which as a by-product gives new probabilistic results for the binomial and beta distributions.