Abstract
In the previous two chapters we introduced the concept of a topological dynamical system and discussed certain basic notions such as minimality, recurrence, and transitivity. However, a deeper study requires a change of perspective: Instead of the state space transformation \( \varphi: K \rightarrow K \) we now consider its
Keywords
- Koopman Operator
- Topological Dynamical System
- Peripheral Point Spectrum
- Gelfand Space
- Gelfand-Naimark Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
European authors sometimes use the nomenclature Čech–Stone compactification.
- 2.
Named after Carl Neumann (1832–1925).
Bibliography
I. Gelfand and A. Kolmogorov [1939] On rings of continuous functions on topological spaces., C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 22 (1939), 11–15.
I. Gelfand and M. Neumark [1943] On the imbedding of normed rings into the ring of operators in Hilbert space, Rec. Math. [Mat. Sbornik] N.S. 12(54) (1943), 197–213.
A. M. Gleason [1967] A characterization of maximal ideals, J. Analyse Math. 19 (1967), 171–172.
J.-P. Kahane and W. Żelazko [1968] A characterization of maximal ideals in commutative Banach algebras, Studia Math. 29 (1968), 339–343.
B. O. Koopman [1931] Hamiltonian systems and transformation in Hilbert space, Proc. Nat. Acad. Sci. U.S.A. 17 (1931), 315–318.
[1932c] Zur Operatorenmethode in der klassischen Mechanik, Ann. Math. (2) 33 (1932), no. 3, 587–642.
W. Żelazko [1968] A characterization of multiplicative linear functionals in complex Banach algebras, Studia Math. 30 (1968), 83–85.
[1973] Banach Algebras, Elsevier Publishing Co., Amsterdam-London-New York; PWN–Polish Scientific Publishers, Warsaw, 1973. Translated from the Polish by M. E. Kuczma.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Tanja Eisner, Bálint Farkas, Markus Haase, and Rainer Nagel
About this chapter
Cite this chapter
Eisner, T., Farkas, B., Haase, M., Nagel, R. (2015). The C ∗-Algebra C(K) and the Koopman Operator. In: Operator Theoretic Aspects of Ergodic Theory. Graduate Texts in Mathematics, vol 272. Springer, Cham. https://doi.org/10.1007/978-3-319-16898-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-16898-2_4
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16897-5
Online ISBN: 978-3-319-16898-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)