13 papers
Intrinsic Geometry of Categorified Spectral Objects
Shih-Yu Chang
This paper develops the intrinsic geometry of the categorified spectral object associated with an admissible operator-semantic system in the Categorified S…
Operator Inequalities in -Product Tensor Algebras: Invariance and Transform Sensitivity
Shih-Yu Chang, Michael K. Ng
We study classical operator inequalities in -product tensor algebras (a transformation -based generalization of the -product framework) for third-order tensors. Although t…
Categorified Spectral Duality: From Operator Systems to Spectral Stacks and Back
Shih-Yu Chang
Classical Gelfand duality provides an equivalence between commutative C-star algebras and topological spaces, but fails to furnish a geometric object for noncommutative operator sy…
A Universal Theory of Spectral Propagation for Compositional Operator Networks
Shih-Yu Chang
Classical spectral theory lacks a framework for understanding how spectra propagate through compositional systems like deep neural networks, feedback control loops, and quantum cir…
Interaction Residues and Localized Spectral Defects in Stratified Operadic Systems
Shih-Yu Chang
We develop a framework for studying how global spectral structure emerges from interacting local sectors in stratified operadic systems. The central object is the interaction resid…
Spectral Operadic Calculus: Norm-Analytic Functor Calculus
Shih-Yu Chang
Classical spectral theory provides powerful tools for analyzing linear operators, but does not extend naturally to nonlinear or compositional settings. In particular, there is no g…
The Operadic Spectrum and Obstructions to Spectral Base Change
Shih-Yu Chang
We introduce an operadic notion of spectrum for algebras over colored operads in a symmetric monoidal category. The construction is defined via a canonical Hochschild-type object t…
The Unitary Conjugation Groupoid as a Universal Mediator of the Baum--Connes Assembly Map
Shih-Yu Chang
We show that the Baum--Connes assembly map factors canonically through the unitary conjugation groupoid, which serves as a universal mediator among groupoid models that are Morita…
An Index Theorem for Fredholm Operators via the Unitary Conjugation Groupoid
Shih-Yu Chang
In a previous paper we introduced the unitary conjugation groupoid associated to any unital separable Type I C*-algebra. This groupoid encodes the representation-theoretic structur…
The Unitary Conjugation Groupoid of a Type I C*-Algebra: Topology, Fell Continuity, and the Canonical Diagonal Embedding
Shih-Yu Chang
This paper introduces a canonical Polish groupoid associated to any separable unital C*-algebra, termed the unitary conjugation groupoid. It is defined as the semidirect product of…
Multicategorical Adjoints, Monadicity, and Quantum Resources
Shih-Yu Chang
This paper is the third part of a program aimed at building a unified operadic and multicategorical foundation for operator theory and quantum processes. Building on the multicateg…
Composition and Coherence: The Syntax of Operator Networks
Shih-Yu Chang
Coherence is a central issue in category theory and multicategory theory, ensuring that formally distinct compositions of morphisms, such as tensor reorderings or diagrammatic rewi…
HilbMult: A Banach-Enriched Multicategory for Operator Algebras
Shih-Yu Chang
Category and multicategory theory provide abstract frameworks for describing structures and their compositions, with multicategories extending traditional categories to handle mult…