🤖 AI Summary
This work addresses the longstanding gap in formal mathematics concerning the continuous functional calculus for C*-algebras. Using Lean 4 and the Mathlib library, we present the first fully verified formalization of this theory in any proof assistant: we rigorously define the continuous functional calculus on arbitrary C*-algebras, construct a framework for continuous maps from compact subsets of ℂ to bounded operators, and formally verify its fundamental properties—including the spectral mapping theorem, algebra homomorphism property, and continuity. Our design balances mathematical naturalness with seamless integration into Mathlib’s existing infrastructure; all results have been merged into the main Mathlib repository. This formalization establishes a foundational cornerstone for the mechanized development of C*-algebra theory and spectral theory, while providing highly reusable interfaces that significantly facilitate subsequent formalizations—such as spectral decomposition and the classification of normal operators.
📝 Abstract
The continuous functional calculus is perhaps the most fundamental construction in the theory of operator algebras, especially $C^{*}$-algebras. Here we document our formalization of the continuous functional calculus in Lean, which constitutes the first such formalization in any proof assistant. Our implementation is already merged into Lean's mathematical library, Mathlib. We provide a brief introduction to the mathematical theory for those unfamiliar with the subject, and then highlight the design decisions in our formalization which proved to be important for usability. Our exposition is aimed at a general mathematical audience and provides a glimpse into the world of formalization by laying bare the discovery process.