[Paper Review] Functional Mellin Transforms

J. LaChapelle|arXiv (Cornell University)|Aug 5, 2013

Advanced Operator Algebra Research19 references3 citations

TL;DR

This paper introduces a generalized framework for functional integrals using locally compact topological groups and Banach-valued Haar measures, enabling the construction of functional Mellin transforms. These transforms facilitate the definition of functional traces, logarithms, and determinants, offering new tools for analyzing function spaces and C*-algebras.

ABSTRACT

Functional integrals are defined in terms of locally compact topological groups and their associated Banach-valued Haar integrals. This approach generalizes the functional integral scheme of Cartier and DeWitt-Morette. The definition allows a construction of functional Mellin transforms. In turn, the functional Mellin transforms can be used to define functional traces, logarithms, and determinants. The associated functional integrals are useful tools for probing function spaces in general and $C^\ast$-algebras in particular. Several interesting aspects are explored.

Motivation & Objective

• To generalize the functional integral approach of Cartier and DeWitt-Morette using topological groups and Banach-valued measures.
• To establish a rigorous foundation for functional Mellin transforms within this generalized framework.
• To enable the definition of functional traces, logarithms, and determinants through the Mellin transform construction.
• To provide new analytical tools for studying function spaces and C*-algebras via these generalized integrals.
• To explore structural and operational properties of functional integrals in non-abelian and non-compact group settings.

Proposed method

• Define functional integrals using Haar measures on locally compact topological groups with values in Banach spaces.
• Construct functional Mellin transforms as integrals over group representations, extending classical Mellin transform concepts.
• Utilize the Mellin transform to define functional traces via contour integration and analytic continuation.
• Derive functional logarithms and determinants as derived operations from the Mellin transform framework.
• Apply the formalism to C*-algebras by interpreting group actions and representations as operators in the algebra.
• Ensure consistency and convergence through topological and measure-theoretic properties of the underlying groups.

Experimental results

Research questions

• RQ1 How can functional integrals be systematically defined using topological groups and Banach-valued measures?
• RQ2 What is the role of the Mellin transform in generalizing trace, logarithm, and determinant operations for functionals?
• RQ3 In what ways do these generalized integrals enhance the analysis of C*-algebras and function spaces?
• RQ4 How does the framework extend prior work by Cartier and DeWitt-Morette in the context of non-abelian or non-compact groups?
• RQ5 What are the structural and analytical properties of the resulting functional Mellin transforms?

Key findings

• The functional Mellin transform is successfully constructed as a generalization of classical Mellin transforms within a group-theoretic and Banach-space-valued framework.
• Functional traces, logarithms, and determinants are defined through the Mellin transform, enabling new analytical tools in operator theory.
• The framework generalizes the functional integral scheme of Cartier and DeWitt-Morette by incorporating topological group structure and Banach-valued measures.
• The method provides a consistent and rigorous approach to defining spectral operations on function spaces and C*-algebras.
• The use of Haar measures ensures invariance and compatibility with group symmetries, enhancing the formalism's applicability.
• The construction supports analytical continuation and contour integration techniques, crucial for defining logarithmic and trace-like functionals.