Department Colloquium

Location: RI 209

Speaker: Sushil Singla, University of Regina

Title:  Matrix convexity and operator systems

Abstract:

A matrix convex set in \(\mathbb C^d\) is a graded subset of \(M_n(\mathbb C)^d\) for all \(n\), that consists of all its matrix-convex combinations. For any classical compact convex set \(K\), a matrix compact convex set \(K_{\rm nc}\) is said to be a noncommutative realisations of \(K\) if \(K_{\rm nc}\) agrees with \(K\) at the ground level of the grading. Associated with \(K\), there are two extreme matrix compact convex sets known as {maximal} and {minimal} noncommutative realisations of \(K\). The minimal noncommutative realisations are related to the dilation theory, and the maximal noncommutative realisations are related to the joint numerical ranges of matrices. A theorem of Webster-Winkler proves a duality between the category of matrix compact convex sets and the category of operator systems, which are \(\ast\)-closed subspaces of the space of bounded linear operators on a Hilbert space. The main aim of this talk is to explore these connections via the noncommutative realisations of cubes, polydiscs, ellipses, complex Euclidean balls, and the prisms. This is joint work with D. Farenick, R. Maleki, and S. Medina Varela.