Approximate Noncommutative Dynamics and Spectral Entropy on Non-Separable Banach Spaces
Abstract
Juan Alberto Molina Garcia
This article develops a general framework for analysing approximate noncommutative dynamics on non-separable Banach spaces, extending classical ideas from operator theory, ergodic theory and noncommutative analysis beyond the restrictions of separability, countability and measurable functional calculus.
We introduce the notion of an ε-noncommutative dynamical scheme, a family of bounded linear operators (T_t )_(t∈R) acting on a non-separable Banach space Xand satisfying only approximate multiplicativity,
â?¥T_(t+s)-T_t T_sâ?¥≤ε(t,s)
where the error function εobeys temperedness and asymptotic consistency. This notion enables the study of operator families whose algebraic structure or continuity properties fail to meet classical semigroup axioms, as typically occurs in non-separable L^∞ (μ), l^∞, and C(K) spaces with non-compact K.
To quantify long-term dynamical complexity, we introduce a new invariant, the generalized spectral entropy h_spec (T), constructed from approximate spectral partitions, ε-resolvent structures, and quasi-spectral profiles adapted to the absence of σ-compact spectral decomposition. We prove existence, stability under perturbations, and a variational characterization of this entropy for large classes of operator families, including weakly compact, fragmentable, Radon–Nikodým, almost-commuting, and extended Riesz–Schauder classes. Applications include: Conditions ensuring the emergence of positive spectral entropy in non-separable Banach algebras. Structural decomposition of approximate semigroups acting on non-separable function spaces. A noncommutative variational principle linking h_specwith asymptotic growth of approximate resolvents.
Our results provide the first systematic attempt to extend spectral entropy and noncommutative dynamical analysis to the non-separable setting, offering a unified viewpoint and novel tools for future developments in operator dynamics, Banach space theory and noncommutative ergodic analysis.