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, , of all distal functions on a group is a norm closed subalgebra of that contains the constants and is closed under the complex conjugation and left translation by elements of . Also it is proved that [7] for any and any the function defined by is distal on . Now let be the norm closure of the algebra generated by the set of functions

*Weyl algebra*. According to the facts mentioned above, all members of the Weyl Algebra are distal functions on . In this paper, we will show that any element of is uniquely ergodic (Theorem 2.13) and that the set does not exhaust all the distal functions on (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.

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DOI:
https://doi.org/10.1090/S0002-9947-1991-0986700-2

Article copyright:
© Copyright 1991
American Mathematical Society