Spectral Theory for Frölicher Algebras via Locally Convex and Convenient Structures

This article develops a framework for extending spectral theory to Frölicher algebras, which define smoothness via curves and functionals rather than topological or bornological structures. Motivated by classical spectral theory in locally convex algebras and its smooth extension in convenient algebras, we compare and connect these three categories through their notions of smoothness and character spectra. Beginning with the Gelfand theory for locally convex algebras, we generalize spectral constructions to convenient algebras via smooth homomorphisms, and further to Frölicher algebras, defining the spectrum in terms of Frölicher-smooth homomorphisms. We construct a smooth Gelfand transform and show that every convenient algebra admits a canonical Frölicher structure, yielding a faithful functor, though not an essentially surjective one. We introduce the concept of Frölicher bialgebras and sketch possible extensions of spectral analysis in this context. This work offers a unified smooth spectral framework, bridging topological, convenient, and Frölicher settings, and provides new tools for infinite-dimensional geometry, global analysis, and noncommutative smooth structures.