Groups and Operator Algebras Seminar

Speaker: Douglas Farenick (University of Regina)

Title: Geometry of essential matrix ranges and the Smith-Ward problem for operator systems

Abstract:

A $d$-tuple of bounded linear selfadjoint operators acting on an infinite-dimensional separable Hilbert space is said to have the Smith-Ward property if the essential matrix range of the $d$-tuple agrees with the matrix range of a compact perturbation of the $d$-tuple. The Smith-Ward problem is to determine which $d$-tuples of selfadjoint operators exhibit the Smith-Ward property. Stated differently, the Smith-Ward problem is to determine whether the identity map on a given finite-dimensional operator subsystem of the Calkin algebra has a completely positive lift. Therefore, we say a finite-dimensional operator system of bounded linear operators acting on an infinite-dimensional separable Hilbert space has the Smith-Ward property if the identity map of the image of the operator system in the Calkin algebra has a completely positive lift. 

This lecture focuses upon noncommutative geometric properties of a finite-dimensional operator system, as captured by its matrix state space, with the goal of understanding how geometric information encoded by the essential matrix range of a spanning set of linear basis for the operator system implies the Smith-Ward property. 

This talk is based on joint work with Chi-Kwong Li and Sushil Singla.