Dilations, Completely Positive Maps and Geometry

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Dilation for One Operator

   B. V. Rajarama Bhat, Tirthankar Bhattacharyya

   Abstract

   This chapter deals with the Hardy space on the unit disc and dilation of a single contraction. These two topics are intimately related and have been greatly studied over several decades. Hence, the material presented is classical. The presentation is from a modern viewpoint. We give several proofs of dilation, von Neumann inequality, a thorough introduction to the characteristic function, a functional model and the Berger-Coburn-Lebow Theorem.

3. Chapter 2. -Algebras and Completely Positive Maps

   B. V. Rajarama Bhat, Tirthankar Bhattacharyya

   Abstract

   In the first two sections of this chapter, we briefly describe some basics of the Banach and the \(C^*\)-algebra theory. An elementary proof of the spectral radius formula, without using complex analysis, has been presented. The second section has the all important Gelfand-Naimark- Segal (GNS) construction for states. Here we provide several simple examples to illustrate the construction. In the last section, we study completely positive maps, a very useful class of maps on \(C^*\)-algebras. These maps are characterized using Stinespring’s theorem.

4. Chapter 3. Dilation Theory in Two Variables—The Bidisc

   B. V. Rajarama Bhat, Tirthankar Bhattacharyya

   Abstract

   Study of dilation of a tuple of commuting bounded operators is begun in this chapter. Here, we deal with a pair of contractions. Apart from Ando’s Theorem, we delve into functional models. This connects with algebraic varieties. More specifically, the distinguished varieties play a role. The realization formula for a bounded holomorphic function on the unit disc is proved.

5. Chapter 4. Dilation Theory in Several Variables—The Euclidean Ball

   B. V. Rajarama Bhat, Tirthankar Bhattacharyya

   Abstract

   We continue the study of dilation of a tuple. Now, we have a contractive tuple. The dilation theory involves the full Fock space in the non-commuting case and the symmetric Fock space in the commuting case. Hence these spaces along with their creation operators are studied. Then, we apply Stinespring’s dilation theorem developed in Chapter 2 to construct dilations. The structure of the dilation tuples are thoroughly decoded.

6. Chapter 5. The Euclidean Ball—The Drury Arveson Space

   B. V. Rajarama Bhat, Tirthankar Bhattacharyya

   Abstract

   This chapter treats a commuting contractive tuple from the function theoretic point of view. This means introducing the Drury-Arveson space and studying its multipliers. One of the inner multipliers is the characteristic function of a pure commuting contractive tuple. We study that and develop a functional model for a pure commuting contractive tuple.

7. Chapter 6. Dilation Theory in Several Variables—The Symmetrized Bidisc

   B. V. Rajarama Bhat, Tirthankar Bhattacharyya

   Abstract

   This chapter deals with the highly successful theory of dilation on the symmetrized bidisc. This involves the joint spectrum which we begin with. We then describe the concept of a spectral set. This leads us to the symmetrized bidisc. Its properties are studied. In preparation for dilation, we study an operator valued version of the Fejer-Riesz theorem. Then a pair of operators for which the symmetrized bidisc is a spectral set is considered and its dilation is explicitly constructed. A large class of examples is given.

8. Chapter 7. An Abstract Dilation Theory

   B. V. Rajarama Bhat, Tirthankar Bhattacharyya

   Abstract

   Here we obtain several dilation theory results in a much simpler setting with very little structure. We replace Hilbert spaces by sets and bounded operators by arbitrary functions. Injective maps would play the role of isometries and bijective maps replace unitaries. Direct sums of Hilbert spaces would be replaced by disjoint unions of sets. Finally what we have is not much of operator theory but just set theory. But we believe that this showcases some essential features of dilation theory.

9. Backmatter