Abstract
For a semigroup \(\mathcal {S}\) of Markov operators on a space of continuous functions, we use \(\mathcal {S}\)-invariant ideals to describe qualitative properties of \(\mathcal {S}\) such as mean ergodicity and the structure of its fixed space. For this purpose we focus on primitive\(\mathcal {S}\)-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 \(\mathcal {S}\).
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Acknowledgements
The author wants to thank Roland Derndinger, Nikolai Edeko, Ulrich Groh and Rainer Nagel for ideas, suggestions and inspiring discussions. The author is also very grateful to the referee for their suggestions. Their thorough reports have led to major improvements of the article. In particular, their valuable advice vastly helped to improve the abstract, the introduction and the readability of the article overall.
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Kreidler, H. The primitive spectrum of a semigroup of Markov operators. Positivity 24, 287–312 (2020). https://doi.org/10.1007/s11117-019-00678-0
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DOI: https://doi.org/10.1007/s11117-019-00678-0