Abstract
Starting with a complex commutative semi-simple regular Banach algebra A and an automorphism σ of A, we form the crossed product of A by the integers, where the latter act on A via iterations of σ. The automorphism induces a topological dynamical system on the character space Δ(A) of A in a natural way. We prove an equivalence between the property that every nonzero ideal in the crossed product has non-zero intersection with the subalgebra A, maximal commutativity of A in the crossed product, and density of the non-periodic points of the induced system on the character space. We also prove that every non-trivial ideal in the crossed product always intersects the commutant of A non-trivially. Furthermore, under the assumption that A is unital and such that Δ(A) consists of infinitely many points, we show equivalence between simplicity of the crossed product and minimality of the induced system, and between primeness of the crossed product and topological transitivity of the system.
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© 2008 Birkhäuser Verlag Basel/Switzerland
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Svensson, C., Silvestrov, S., de Jeu, M. (2008). Connections Between Dynamical Systems and Crossed Products of Banach Algebras by ℤ. In: Janas, J., Kurasov, P., Naboko, S., Laptev, A., Stolz, G. (eds) Methods of Spectral Analysis in Mathematical Physics. Operator Theory: Advances and Applications, vol 186. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8755-6_19
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DOI: https://doi.org/10.1007/978-3-7643-8755-6_19
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8754-9
Online ISBN: 978-3-7643-8755-6
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