Distal Functions and Unique Ergodicity

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°°(S) that contains the constants and is closed under the complex conjugation and left translation by elements of 5. Also it is proved that [7] for any k e N and any lei the function /: Z -> C defined by f(h) = ea" is distal on Z . Now let W be the norm closure of the algebra generated by the set of functions {«i»/" :k€N, XeVL), which will be called the Weyl algebra. According to the facts mentioned above, all members of the Weyl Algebra are distal functions on Z. In this paper, we will show that any element of W is uniquely ergodic (Theorem 2.13) and that the sel W does not exhaust all the distal functions on 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. 1. Preliminaries Let S be a semigroup and X = l°° (S) be the algebra of all bounded complex functions on S, equiped with the topology of pointwise convergence on S. Then (S, l°°(S)) forms a flow, where the action of S on X = l°°(S) is defined by (s,fi)^sfi=Rsfi, where (sf)(t) = (Rsfi)(t) = fits) for all fi G X and s, t G S. A member / G l°°(S) is called a distal function on S if it is, distal point relative to the flow (S, l°°(S)). For this flow and its distal functions we have the following results [7, 8]: 1.1. Theorem. Let S be a semigroup and let Z(X) be the enveloping semigroup of the flow (S,l°°(S)). Then l(X) is compact. Furthermore, each og!,(X) is linear, multiplicative, and preserves the complex conjugation and the constant functions. Also \\af\\ < \\fi\\ holds for all fiGl°°(S). Received by the editors June 20, 1988 and, in revised form, January 31, 1989. Presented to the Society April 12, 1986 in Indianapolis, Indiana. 1980 Mathematics Subject Classification (1985 Revision). Primary 54H20; Secondary 28D99. ©1991 American Mathematical Society 0002-9947/91 $1.00+ $.25 per page