Research Projects

The theory of extreme contractions, which are the extreme points of the unit ball of the space of bounded linear operators between Banach spaces, has been a central topic in functional analysis since the seminal work of Kadison (1951). Although considerable progress has been made over the decades, a complete characterization remains elusive even for finite-dimensional Banach spaces. This proposal aims to advance the study of extreme contractions from both analytical and combinatorial perspectives. A particular focus will be given to polyhedral Banach spaces, where the unit ball has finitely many extreme points. These spaces offer a rich framework for exact enumeration and structural analysis of extreme contractions. Building on recent developments, including the use of Birkhoff-James orthogonality and the concept of k-smoothness of operators, the project seeks to obtain new characterizations of extreme contractions. In an n-dimensional polyhedral Banach space, a point is extreme if and only if it is n-smooth. Applying the results of k-smoothness of operators in the study of extreme contraction is a key component of this work. The role of operator rank, particularly in relation to convex combinations and the newly introduced rank-invariant Krein-Milman property, will also be studied in depth. This study will enrich the understanding the geometry of the operator spaces. In addition, the project explores the so-called L-P and weak L-P properties, which describe the behavior of extreme contractions with respect to the extreme points of the domain and codomain unit balls. Under these properties, extreme contractions exhibit particularly nice structures, which help to obtain clearer classifications and facilitate the explicit determination of extreme contractions in many cases. A major aspect of the project is the investigation of the space $L(\ell_\infty^n, \ell_1^n)$, which holds special interest due to its connection with the Grothendieck constant. while a characterization of rank 1 extreme contraction in this space has been already obtained, higher-rank extreme contractions exist remains unknown. Understanding the structure and enumeration of extreme contractions in this space could contribute directly to the determination of the exact value of the Grothendieck constant, a long-standing open problem in functional and tensor analysis. Overall, this project aims to develop a systematic understanding of operator space geometry through the study of extreme contractions.