The study of boundary representations and hyperrigidity of operator spaces and operator Systems.

Abstract

The title of the thesis is “A STUDY OF BOUNDARY REPRESENTATIONS AND HYPERRIGIDITY OF OPERATOR SPACES AND OPERATOR SYSTEMS”. The thesis begins with a chapter surveying the literature and describing the structure of the thesis. This is followed by a chapter setting up the necessary preliminaries about C*-algebras and CP-maps. It also summarises well- understood work on boundary representations and hyper-rigidity. The main results of the thesis are contained in the next three chapters. In Chapter 3, the main objects of study are the notions of weak boundary representations and quasi-hyper-rigidity originally introduced by Namboodiri, Pramod, Shankar and Vijayarajan. It is shown that weak boundary representations for operator systems in unital C*-algebras are characterised by their amplifications also being so. This implies that the corresponding result for quasi-hyper- rigidity of operator systems holds. Chapter 4 first characterises boundary representations of operator spaces through that of their associated Paulsen systems and thereby shows that the term “boundary” is a natural one. We proved that there is a one to one correspondence between boundary representations of the operator space with that of the Paulsen system. Subsequently weak boundaries for operator spaces are also introduced and studied, again generalising from weak boundaries of operator systems. Finally rectangular hyper-rigidity for operator spaces in ternary rings of operators is introduced and a finite-dimensional version of Saskin’s theorem is proved. In Chapter 5, the notion of a non-commutative Choquet boundary is introduced for some spaces of unbounded operators and even more generally for locally C*-algebras and an analogue of Arveson’s extension theorem is established. Then the unique extension property and boundary representations are introduced in this generality and studied with appropriate examples. The last chapter concludes with a well-written summary of the thesis and poses several interesting questions for future work.