TECHNOLOGICAL BASIS OF “INDUSTRY 4.0”
Trace Reciprocity under Measurable Decomposition for Trace-Class Endofunctors. Introduction to Trace Reciprocity Framework
- 1 Technical University of Sofia, Bulgaria
- 2 Digital Fabrica Theory
Abstract
Trace theory is a fundamental tool in linear algebra, spectral theory, homological algebra, and related areas of modern mathematics. While the additivity of trace is classical in finite-dimensional settings, a unified structural principle for its behavior under measurable decomposition is less explicitly formulated across categorical and analytic frameworks. [8,9].
The paper introduces a compact trace reciprocity framework for trace-class exact endofunctors admitting measurable decomposition over a σ-finite measure space. Under suitable measurability, integrability, and boundedness assumptions, the global trace can be recovered from the local traces of the measurable components. The report summarizes the formal setting, states the main theorem, and outlines the proof strategy based on approximation by simple families, trace additivity, and convergence arguments. [6,4]
Representative examples are included to show how the framework recovers familiar identities in finite-dimensional and operator-theoretic settings. The goal is to provide a concise conference-style presentation of the main reciprocity principle, its mathematical scope, and its structural significance. [7,3]
Keywords – Trace theory; trace reciprocity; measurable decomposition; trace-class endofunctors; stable categories; exact endofunctors; categorical invariants; operator theory.
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