C∗-extreme Maps and Nest Algebras/ Manish Kumar

Thesis (Ph.D.) - Indian Statistical Institute, 2022

Includes bibliographical references and index

Introduction --
1 Preliminaries -- 1.1 C∗-algebras -- 1.2 Completely positive maps -- 1.3 Positive operator valued measures -- 1.4 Correspondence between CP maps and POVMs -- 1.5 Nest algebras and factorization property --
2 C∗-convexity Structure of Generalized State Spaces -- 2.1 Definitions and general properties -- 2.2 Abstract characterizations of C∗-extreme maps -- 2.3 Direct sums of pure UCP maps -- 2.4 Krein-Milman type theorem for UCP maps on separable C∗-algebras
2.5 Examples and applications --
3 Normal C∗-extreme Maps --
3.1 Normal C∗-extreme maps on type I factors -- 3.2 Krein-Milman type theorem for UCP maps on type I factors -- 3.3 Examples of normal C∗-extreme maps --
4 C∗-extreme Positive Operator Valued Measures -- 4.1 General Properties of C∗-extreme POVMs -- 4.2 C∗-extreme POVMs with commutative ranges -- 4.3 Atomic C∗-extreme POVMs -- 4.4 Singular POVMs and their direct sums -- 4.5 Measure Isomorphic POVMs --
5 C∗-extreme Maps on Commutative C∗-algebras -- 5.1 Regular atomic and non-atomic POVMs -- 5.2 Regular C∗-extreme POVMs -- 5.3 Krein-Milman type theorem for PH(X) -- 5.4 Applications to UCP Maps on C(X) --
6 Logmodular Algebras -- 6.1 Definitions and examples -- 6.2 Lattices of logmodular algebras -- 6.3 Proof of the main result -- 6.4 Reflexivity of algebras with factorization --
Open Problems

Guided by Prof. B. V. Rajarama Bhat