The primitive spectrum of a semigroup of Markov operators
Henrik Kreidler

TL;DR
This paper investigates the structure of semigroups of Markov operators on continuous functions, using invariant ideals to analyze properties like ergodicity and fixed points, with applications to dynamical systems.
Contribution
It introduces a novel approach based on primitive S-ideals and topology to describe qualitative properties of Markov semigroups, inspired by C*-algebra theory.
Findings
Characterization of mean ergodicity via invariant ideals
Description of fixed space structure using primitive S-ideals
Identification of centers of attraction in dynamical systems
Abstract
For a semigroup S of Markov operators on a space of continuous functions, we use S-invariant ideals to describe qualitative properties of S such as mean ergodicity and the structure of its fixed space. For this purpose we focus on primitive S-ideals and endow the space of those ideals with an appropriate topology. This approach is inspired by the representation theory of C*-algebras and can be adapted to our dynamical setting. In the particularly important case of Koopman semigroups, we characterize the centers of attraction of the underlying dynamical system in terms of the invariant ideal structure of S.
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