Topology and Geometry Seminar

Location: AH 347

Speaker: Sushil Singla, University of Regina

Title: The noncommutative geometry of cubes and prisms

Abstract:  

A map \(f: K\rightarrow L\) between two compact convex sets is affine if it respects convex combinations, meaning that \[f(tx+(1-t)y)=tf(x)+(1-t)f(y),\] for all \(x,y\in K\) and \(0\leq t \leq 1\). The collection of compact convex sets and continuous affine maps forms a category, and the classical representation theorem of Kadison (1951) states that it is dual to the category of Archimedean ordered \(\ast\)-vector spaces, where morphisms are unital positive maps.

A non-commutative (nc) compact convex set is a graded set \(K_{\rm nc}=\bigsqcup_{n\leq \aleph}K_n\), where \(K_n\) denotes a compact collection of bounded linear operators acting on an \(n\)-dimensional Hilbert space, for possibly infinite cardinals \(n \leq \aleph\). Furthermore, \(K_{\rm nc}\) is required to be closed under a non-commutative analogue of convex combinations. The collection of nc compact convex sets also forms a category, where the morphisms are the nc continuous affine maps. Very recently, Davidson and Kennedy (2025) proved that this category is dual to the category of operator systems, consisting of Archimedean matrix ordered \(\ast\)-vector spaces and unital completely positive maps.

Given a compact convex set \(K\subseteq \mathbb C^d\), a non-commutative realization is defined to be a nc compact convex set \(K_{\rm nc}\) whose first level is \(K_{1} = K\). In this talk, I will describe the non-commutative realizations of three classical geometric objects: cubes, polydiscs, and prisms. I will describe their various geometric and algebraic properties, such as their non-commutative extreme points, and I will apply two classical dilation theorems of Halmos and Mirman to give a complete description of the nc triangular prism in terms of joint unitary dilations. This is joint work with D. Farenick, R. Maleki, and S. Medina Varela.

This event is sponsored by PIMS.