Ordered structures on operator spaces and injectivity

Postgraduate Thesis uoadl:2899071 247 Read counter

Κατεύθυνση Θεωρητικά Μαθηματικά
Library of the School of Science
Deposit date:
Petropoulos Argyrios
Supervisors info:
Αριστείδης Κοντογεώργης, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Αριστείδης Κατάβολος, Ομότιμος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Μιχάλης Ανούσης, Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Αιγαίου
Original Title:
Δομές διάταξης σε χώρους τελεστών και εμφυτευτικότητα
Translated title:
Ordered structures on operator spaces and injectivity
The object of study of this thesis is the interaction between the ordering defined on (self-adjoint) operators on Hilbert spaces and operator space theory. In particular, we study several matrix orderings defined on operator systems as well as the concepts of injectivity and injective envelope in the categories of operator spaces and operator systems.
In Chapter 1, we present the basic theory of positive and completely positive maps. Among others, we state and prove Schwarz inequality for 2-positive maps and Krein separation theorem for cones.
In Chapter 2, we study * ordered linear spaces and introduce the concept of an Archimedean space. In addition, we define the order norms in such spaces and connect them with the states of the space. Then, using these tools, we state and prove Kadison's characterisation of function systems.
In Chapter 3, we define the operator system structures. We state and prove the theorem of Choi-Effros and study the minimal and the maximal such structure. Then, we present the application of these concepts to the study of entanglement breaking maps.
Finally, in Chapter 4 is given an introduction in the theory of injective operator systems. We also prove the existence of injective envelopes of operator spaces. Finally, we define the C* envelope of an operator algebra and prove Hamana's characterisation for such algebras.
Main subject category:
Operator spaces, C* algebra
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