Core Concepts

This research thesis investigates the relationship between the geometry of analytic discs and the algebraic structure of their associated multiplier algebras, aiming to determine when two such algebras are isomorphic and exploring the concept of embedding dimension for complete Pick spaces.

Abstract

**Bibliographic Information:**Mironov, M. (2024).*Hilbert function spaces and multiplier algebras of analytic discs*[Master's thesis, Technion - Israel Institute of Technology].**Research Objective:**This thesis aims to address two related problems: (1) the isomorphism problem for analytic discs, which investigates when two analytic discs have isomorphic multiplier algebras, and (2) the embedding dimension problem for complete Pick spaces, which seeks to determine the smallest dimension in which a given complete Pick space can be realized.**Methodology:**The thesis employs tools from functional analysis, complex analysis, and operator theory. It leverages properties of reproducing kernel Hilbert spaces, particularly the Drury-Arveson space, and explores the behavior of functions and multipliers on analytic discs.**Key Findings:**- The thesis establishes that for analytic discs attached to the unit sphere, an algebraic isomorphism between their multiplier algebras implies that the discs possess the same self-crossing structure up to a unit disc automorphism.
- For analytic discs with a single self-crossing, the isomorphism condition is rigid, with only two possible candidates for the isomorphism-inducing map.
- The thesis provides a characterization of the embedding dimension for rotation-invariant complete Pick spaces on the unit disc, linking it to the polynomial nature of a specific function derived from the reproducing kernel.
- It demonstrates that certain weighted Hardy-type spaces, specifically those with negative exponent, have infinite embedding dimension.

**Main Conclusions:**- The self-crossing structure of analytic discs plays a crucial role in determining the isomorphism class of their multiplier algebras.
- The embedding dimension, a measure of complexity for complete Pick spaces, can be effectively analyzed using the provided characterization, revealing distinctions between spaces with finite and infinite embedding dimensions.

**Significance:**This research contributes to the understanding of the interplay between geometric and algebraic properties of analytic objects. It sheds light on the structure of multiplier algebras, which are fundamental in operator theory and function theory, and provides insights into the realization of complete Pick spaces.**Limitations and Future Research:**The thesis primarily focuses on analytic discs attached to the unit sphere. Further research could explore more general varieties and their multiplier algebras. Additionally, investigating the sufficiency of the self-crossing condition for isomorphism and exploring the embedding dimension for broader classes of complete Pick spaces are promising avenues for future work.

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by Mikhail Miro... at **arxiv.org** 10-15-2024

Deeper Inquiries

Extending the theory developed in this thesis to more general Riemann surfaces beyond the unit disc presents exciting challenges and potential rewards. Here's a breakdown of the key considerations and potential avenues for generalization:
Challenges:
Complex Structure: The unit disc enjoys a simple complex structure, being simply connected. General Riemann surfaces can have much more intricate topologies, requiring more sophisticated tools from complex analysis and algebraic geometry.
Canonical Embeddings: The unit disc has a natural embedding into the complex plane. For general Riemann surfaces, canonical embeddings might not exist or might be much harder to work with. This impacts the notion of "analytic discs attached to the unit sphere," which needs to be redefined.
Multiplier Algebras: Characterizing multiplier algebras for function spaces on general Riemann surfaces can be significantly more complex. The interplay between the geometry of the surface and the properties of the function space (e.g., Hardy spaces, Bergman spaces) becomes more intricate.
Potential Avenues for Generalization:
Uniformization Theorem: Leverage the Uniformization Theorem to represent a Riemann surface as a quotient of the unit disc, upper half-plane, or Riemann sphere. This could provide a framework to transfer some results from the unit disc setting.
Sheaf Theory: Employ sheaf-theoretic techniques to study function spaces and multiplier algebras locally on the Riemann surface. This could help to understand how local properties of the surface influence the global structure of these algebras.
Geometric Function Theory: Utilize tools from geometric function theory, such as extremal length and harmonic measure, to study the boundary behavior of functions in these spaces and characterize their multiplier algebras.
Specific Considerations for the Isomorphism Problem:
Teichmüller Spaces: For compact Riemann surfaces, the moduli space of complex structures (Teichmüller space) plays a crucial role. The isomorphism problem for multiplier algebras could be related to the geometry of this moduli space.
Riemann Surface Automorphisms: The role of disc automorphisms in the unit disc case might be replaced by the group of automorphisms of the Riemann surface. Understanding the action of this group on the multiplier algebra could be key.

Yes, alternative characterizations of the embedding dimension could offer valuable geometric or analytic insights. Here are some possibilities:
Operator-Theoretic Characterizations:
Minimal Number of Generators: Relate the embedding dimension to the minimal number of generators for the multiplier algebra as a W*-algebra. This could connect to the representation theory of the algebra.
Completely Positive Maps: Explore characterizations based on the structure of completely positive maps on the multiplier algebra. This could tie into dilation theory and operator systems.
Geometric Characterizations:
Curvature: Investigate if the embedding dimension can be bounded or determined by the curvature properties of the image of the embedding in the unit ball.
Degree of Algebraic Varieties: For varieties defined by polynomial equations, relate the embedding dimension to the degree or other invariants of these varieties.
Analytic Characterizations:
Growth of Functions: Explore connections between the embedding dimension and the growth rates of functions in the reproducing kernel Hilbert space.
Interpolation Problems: Characterize the embedding dimension through the complexity of interpolation problems that can be solved in the multiplier algebra.

The findings in this thesis have interesting implications for the study of operator algebras and their representations, particularly in the context of reproducing kernel Hilbert spaces and multiplier algebras:
Structure of Multiplier Algebras: The results shed light on the intricate relationship between the geometry of a variety and the algebraic structure of its associated multiplier algebra. This is a recurring theme in the study of non-self-adjoint operator algebras, where geometric considerations often play a crucial role.
Representations on RKHS: The study of analytic discs and their multiplier algebras provides concrete examples of representations of operator algebras on reproducing kernel Hilbert spaces. These examples can serve as test cases for developing general theories about such representations.
Non-Commutative Function Theory: The theory of complete Pick spaces and their multiplier algebras can be viewed as a form of "non-commutative function theory." The results in the thesis contribute to this perspective by highlighting the geometric aspects of this theory.
Connections to Complex Geometry: The embedding dimension problem establishes a link between operator-theoretic notions and geometric properties of varieties in the unit ball. This suggests deeper connections between operator theory and complex geometry that warrant further exploration.
Operator Model Theory: The results could potentially be applied to operator model theory, which seeks to represent abstract operator algebras concretely as algebras of operators on Hilbert spaces. Understanding the structure of multiplier algebras arising from analytic discs could lead to new operator models.

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