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Transactions of the American Mathematical Society

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Distal functions and unique ergodicity


Author: Ebrahim Salehi
Journal: Trans. Amer. Math. Soc. 323 (1991), 703-713
MSC: Primary 43A60; Secondary 54H20
DOI: https://doi.org/10.1090/S0002-9947-1991-0986700-2
MathSciNet review: 986700
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Abstract: A. Knapp [5] has shown that the set, $ D(S)$, of all distal functions on a group $ S$ is a norm closed subalgebra of $ {l^\infty }(S)$ that contains the constants and is closed under the complex conjugation and left translation by elements of $ S$. Also it is proved that [7] for any $ k \in \mathbb{N}$ and any $ \lambda \in \mathbb{R}$ the function $ f:\mathbb{Z} \to \mathbb{C}$ defined by $ f(n) = {e^{i\lambda {n^k}}}$ is distal on $ \mathbb{Z}$. Now let $ {\mathbf{W}}$ be the norm closure of the algebra generated by the set of functions

$\displaystyle \{ n \mapsto {e^{i\lambda {n^k}}}:k \in \mathbb{N},\;\lambda \in \mathbb{R}\} ,$

which will be called the Weyl algebra. According to the facts mentioned above, all members of the Weyl Algebra are distal functions on $ \mathbb{Z}$. In this paper, we will show that any element of $ {\mathbf{W}}$ is uniquely ergodic (Theorem 2.13) and that the set $ {\mathbf{W}}$ does not exhaust all the distal functions on $ \mathbb{Z}$ (Theorem 2.14). The latter will answer the question that has been asked (to the best of my knowledge) by P. Milnes [6].

The term Weyl algebra is suggested by S. Glasner. I would like to express my warmest gratitude to S. Glasner for his helpful advise, and to my advisor Professor Namioka for his enormous helps and contributions.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-0986700-2
Article copyright: © Copyright 1991 American Mathematical Society

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