Song-Jian Han. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 72-73). / Acknowledgement --- p.i / Abstract --- p.ii / Introduction --- p.iii / Chapter 1 --- Topological Vector Spaces and Elemantary Duality Theory --- p.1 / Chapter 1.1 --- Locally Convex Spaces --- p.2 / Chapter 1.2 --- Bornological Spaces and Bornological Vector Spaces --- p.4 / Chapter 1.3 --- Elementary Properties of Dual Spaces --- p.6 / Chapter 1.4 --- Topological Injections and Surjections Bornological Injections and Surjections --- p.10 / Chapter 2 --- Locally Convex Riesz Spaces --- p.15 / Chapter 2.1 --- Ordered Vector Spaces --- p.15 / Chapter 2.2 --- Riesz Space --- p.18 / Chapter 2.3 --- Locally Convex Riesz Spaces --- p.20 / Chapter 3 --- Half-Full Injections and Half-Decomposable Surjections Half- Full Bornological Injections and Half-Decomposable Bornologi- cal Surjections --- p.24 / Chapter 4 --- Operator Modules between Locally Convex Riesz Spaces --- p.35 / Chapter 4.1 --- Preliminaries --- p.35 / Chapter 4.2 --- Operator Modules and Ideal Cones --- p.37 / Chapter 4.3 --- The Half-Full Injective Hull and the Half-Decomposable Bornolog- ical Surjective Hull of Operator Modules Between Locally Convex Riesz Spaces --- p.41 / Chapter 4.4 --- Extensions of Operator Modules and Ideal Cones --- p.57 / References --- p.72

No description available.

This Master's thesis will develops a modern approach to complex interpolation problems studied by Carath\'odory, Nevanlinna, Pick, and Schur in the early $20^{th}$ century. The fundamental problem to solve is as follows: given complex numbers $z_1,z_2,...,z_N$ of modulus at most $1$ and $w_1,w_2,...,w_N$ additional complex numbers, what is a necessary and sufficiency condition for the existence of an analytic function $f: \mathbb{D} \rightarrow \mathbb$ satisfying $f(z_i) = w_i$ for $1 \leq i \leq N$ and $\vert f(z) \vert \leq 1$ for each $z \in \mathbb{D}$? The key idea is to realize bounded, analytic functions (the algebra $H^\infty$) as the \emph of the Hardy class of analytic functions, and apply dilation theory to this algebra. This operator theoretic approach may then be applied to a wider class of interpolation problems, as well as their matrix-valued equivalents. This also yields a fundamental distance formula for $H^\infty$, which provides motivation for the study of completely isometric representations of certain quotient algebras. Our attention is then turned to a related interpolation problem. Here we require the interpolating function $f$ to satisfy the additional property $f'(0) = 0$. When $z_i =0$ for some $i$, we arrive at a special case of a problem class studied previously. However, when $0$ is not in the interpolating set, a significant degree of complexity is inherited. The dilation theoretic approach employed previously is not effective in this case. A more function theoretic viewpoint is required, with the proof of the main interpolation theorem following from a factorization lemma for the Hardy class of analytic functions. We then apply the theory of completely isometric maps to show that matrix interpolation fails when one imposes this constraint.

This Master's thesis will develops a modern approach to complex interpolation problems studied by Carath\'odory, Nevanlinna, Pick, and Schur in the early $20^{th}$ century. The fundamental problem to solve is as follows: given complex numbers $z_1,z_2,...,z_N$ of modulus at most $1$ and $w_1,w_2,...,w_N$ additional complex numbers, what is a necessary and sufficiency condition for the existence of an analytic function $f: \mathbb{D} \rightarrow \mathbb$ satisfying $f(z_i) = w_i$ for $1 \leq i \leq N$ and $\vert f(z) \vert \leq 1$ for each $z \in \mathbb{D}$? The key idea is to realize bounded, analytic functions (the algebra $H^\infty$) as the \emph of the Hardy class of analytic functions, and apply dilation theory to this algebra. This operator theoretic approach may then be applied to a wider class of interpolation problems, as well as their matrix-valued equivalents. This also yields a fundamental distance formula for $H^\infty$, which provides motivation for the study of completely isometric representations of certain quotient algebras. Our attention is then turned to a related interpolation problem. Here we require the interpolating function $f$ to satisfy the additional property $f'(0) = 0$. When $z_i =0$ for some $i$, we arrive at a special case of a problem class studied previously. However, when $0$ is not in the interpolating set, a significant degree of complexity is inherited. The dilation theoretic approach employed previously is not effective in this case. A more function theoretic viewpoint is required, with the proof of the main interpolation theorem following from a factorization lemma for the Hardy class of analytic functions. We then apply the theory of completely isometric maps to show that matrix interpolation fails when one imposes this constraint.

The following thesis is divided into two main chapters. In Chapter 2 we study isometric representations of product systems of correspondences over the semigroup 𝐍ᵏ which are minimal dilations of finite dimensional, fully coisometric representations. We show the existence of a unique minimal cyclic coinvariant subspace for all such representations. The compression of the representation to this subspace is shown to be a complete unitary invariant. For a certain class of graph algebras the nonself-adjoint WOT-closed algebra generated by these representations is shown to contain the projection onto the minimal cyclic coinvariant subspace. This class includes free semigroup algebras. This result extends to a class of higher-rank graph algebras which includes higher-rank graphs with a single vertex. In chapter 3 we move onto semicrossed product algebras. Let 𝒮 be the semigroup 𝒮=Σ𝒮ᵢ, where 𝒮ᵢ is a countable subsemigroup of the additive semigroup 𝐑₊ containing 0. We consider representations of 𝒮 as contractions {Tᵣ }ᵣ on a Hilbert space with the Nica-covariance property: Tᵣ*Tᵤ=TᵤTᵣ* whenever t^s=0. We show that all such representations have a unique minimal isometric Nica-covariant dilation. This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant representations of the action of 𝒮 on an operator algebra 𝒜 by completely contractive endomorphisms. We conclude by calculating the C*-envelope of the isometric nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis).

No description available.

The following thesis is divided into two main chapters. In Chapter 2 we study isometric representations of product systems of correspondences over the semigroup 𝐍ᵏ which are minimal dilations of finite dimensional, fully coisometric representations. We show the existence of a unique minimal cyclic coinvariant subspace for all such representations. The compression of the representation to this subspace is shown to be a complete unitary invariant. For a certain class of graph algebras the nonself-adjoint WOT-closed algebra generated by these representations is shown to contain the projection onto the minimal cyclic coinvariant subspace. This class includes free semigroup algebras. This result extends to a class of higher-rank graph algebras which includes higher-rank graphs with a single vertex. In chapter 3 we move onto semicrossed product algebras. Let 𝒮 be the semigroup 𝒮=Σ𝒮ᵢ, where 𝒮ᵢ is a countable subsemigroup of the additive semigroup 𝐑₊ containing 0. We consider representations of 𝒮 as contractions {Tᵣ }ᵣ on a Hilbert space with the Nica-covariance property: Tᵣ*Tᵤ=TᵤTᵣ* whenever t^s=0. We show that all such representations have a unique minimal isometric Nica-covariant dilation. This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant representations of the action of 𝒮 on an operator algebra 𝒜 by completely contractive endomorphisms. We conclude by calculating the C*-envelope of the isometric nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis).

Thesis--Bonn. / Includes bibliographical references (p. 84).

Thesis (M.C.S.)--Carleton University, 2001. / CD-ROM contains copies of the thesis (pdf and postscript) and the code the author wrote to do the research. Includes bibliographical references (p. 117-120). Also available in electronic format on the Internet.

Thesis--Bonn. / Includes bibliographical references (p. 84).