My Research Overview

Recent Project Descriptions

1. Boundary Representations and C*-Envelopes of finite-dimensional operator systems:  The theory of operator systems, introduced by Arveson, and later treated formally by Choi and Effros, has advanced substantially over the last two decades, with the proof of the existence of the noncommutative Choquet boundary representing one of the theory's pinnacle achievements. This existence of the noncommutative Choquet boundary, established first by Arveson for separable operator systems and then completed later by Davidson and Kennedy for arbitrary operator systems, provides a new avenue towards determining the minimal C*-cover, which is also known as the C*-envelope of any given operator system. Our purpose in this project is to better understand boundary representations and minimal C*-covers of finite-dimensional operator systems using results and techniques involving noncommutative states and noncommutative convexity. A finite-dimensional operator system is generated by d-tuples of basis of the operator systems. Thus, study of finite-dimensional operator systems is equivalent to the study of operator systems generated by d-tuple of operators on a Hilbert space. Conversely, this approach lends itself well to the study of certain phenomena in single- and several-variable operator theory, such as the Smith-Ward property, where the analysis of a d-tuple of linearly independent operators can be achieved by using properties of the corresponding operator system.

2. Isomorphism and property recognition problems: How important is a given relation? The question is intentionally vague, and its answer depends very much on the context. One can measure the importance of the relation by using graphs: The vertices, typically (though not always), are all the elements of a given object and two vertices are connected by a directed edge if they are related. This approach concentrates on the relation alone and lefts aside how the relation interacts with other structures/operations inside the object. Having built the graphs on two objects within the same category, one can then examine how close the two objects are if the relation on them behaves in exactly the same way. Technically, if the two graphs are isomorphic, does it imply that the two objects are isomorphic? We call this an isomorphism problem. Currently, I am working with Bojan Kuzma on the relation of Birkhoff-James orthogonality in the category of normed spaces. Recently, we have solved the isomorphism of the real as well as complex finite-dimensional C*-algebra in the category of finite-dimensional C*-algebras ([1,2,3]). Another measure of the importance of the relation is whether it can classify elements with a particular property inside an object. We call this property recognition problem. We have already completed one project about property recognition problem with Alexander Guterman and Svetlana Zhilina which is available on DOI:10.1007/s43036-024-00321-0.

3. Linear Preserver Problems: Linear preserver problems concern the characterization of linear operators on the space of bounded linear operators that leave certain functions, subsets, relations, etc., invariant. We have characterized the bijective linear maps sending parallel pairs of matrices to parallel pairs of matrices with respect to the Ky-Fan k-norms (DOI:10.1016/j.laa.2024.01.018) and with respect to k-numerical radius (Arxiv:2408.16066), where two matrices A and B are said to be parallel with respect to a norm ||•|| if ||A+\mu B|| = ||A|| + ||B|| for some scalar \mu with |\mu| = 1.  This is a joint project with Bojan Kuzma; Chi-Kwong Li; and Edward Poon.