Postgraduate Thesis uoadl:2864927 849 Read counter

Κατεύθυνση Θεωρητικά Μαθηματικά

Library of the School of Science

Library of the School of Science

2019-03-05

2019

Papapetros Evaggelos

Δημήτρης Γατζούρας, Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Αθηνών

Μιχάλης Ανούσης, Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Αιγαίου

Απόστολος Γιαννόπουλος, Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Αθηνών

Μιχάλης Ανούσης, Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Αιγαίου

Απόστολος Γιαννόπουλος, Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Αθηνών

Εισαγωγή στη Θεωρία Χώρων Τελεστών και Θεωρήματα Επέκτασης

Greek

An introduction to the theory of Operator Spaces and Extension Theorems

The object of study of this thesis is the operators and especially the linear and bounded operators from a Hilbert space to itself. We also study operator spaces which are defined as subspaces of a $C^{\star}$ algebra $\mathcal{A}.$ In particular, we study operator systems, that is, selfadjoint subspaces containing the unit of the algebra.

Then we study maps of operator systems to a given $C^{\star}$ algebra and we deal with the positive, completely positive and completely bounded maps. We prove basic theorems such as those of Stinespring, Arveson and Wittstock.

In Chapter 1, we present basic concepts of the theory of tensor products. We define the Algebraic Tensor Product of Linear Spaces, the Projective Tensor Product of Banach Spaces, the tensor product of Hilbert Spaces and tensor products of $C^{\star}$ algebras.

In Chapter 2, we define the $C^{\star}$ algebras $\mathbb{M}_{n}(\mathcal{A})$, where $\mathcal{A}$ is a $C^{\star}$ algebra, we present dilation theorems of bounded linear operators on Hilbert spaces as well as basic concepts of positive maps.

In Chapter 3, we define completely positive maps of an operator system to a $C^{\star}$ algebra and we prove the theorem of Choi.

In Chapter 4, we state and prove Stinespring's Theorem, which characterizes the completely positive maps of a $C^{\star}$ algebra $\mathcal{A}$ to the $C^{\star}$ algebra $\mathbb{B}(H)$ of a Hilbert space $H.$

In Chapter 5, we study completely positive maps taking values in the $C^{\star}$ algebra $\mathbb{M}_{n}(\mathbb{C})$ and

we prove a theorem of extension. The basic Arveson Extension Theorem is presented in Chapter 6.

Finally, Chapter 7 presents two major theorems of Wittstock, an extension theorem and a decomposition theorem,

from which we conclude that the linear span of the set of completely positive maps is the space of completely bounded maps.

Then we study maps of operator systems to a given $C^{\star}$ algebra and we deal with the positive, completely positive and completely bounded maps. We prove basic theorems such as those of Stinespring, Arveson and Wittstock.

In Chapter 1, we present basic concepts of the theory of tensor products. We define the Algebraic Tensor Product of Linear Spaces, the Projective Tensor Product of Banach Spaces, the tensor product of Hilbert Spaces and tensor products of $C^{\star}$ algebras.

In Chapter 2, we define the $C^{\star}$ algebras $\mathbb{M}_{n}(\mathcal{A})$, where $\mathcal{A}$ is a $C^{\star}$ algebra, we present dilation theorems of bounded linear operators on Hilbert spaces as well as basic concepts of positive maps.

In Chapter 3, we define completely positive maps of an operator system to a $C^{\star}$ algebra and we prove the theorem of Choi.

In Chapter 4, we state and prove Stinespring's Theorem, which characterizes the completely positive maps of a $C^{\star}$ algebra $\mathcal{A}$ to the $C^{\star}$ algebra $\mathbb{B}(H)$ of a Hilbert space $H.$

In Chapter 5, we study completely positive maps taking values in the $C^{\star}$ algebra $\mathbb{M}_{n}(\mathbb{C})$ and

we prove a theorem of extension. The basic Arveson Extension Theorem is presented in Chapter 6.

Finally, Chapter 7 presents two major theorems of Wittstock, an extension theorem and a decomposition theorem,

from which we conclude that the linear span of the set of completely positive maps is the space of completely bounded maps.

Science

Operator Spaces, Operator Systems, Completely positive maps, Dilation Theorems, Extension Theorems

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