The topological center of the spectrum of some distal algebras

Abstract

The topological center of the spectrum of the Weyl algebra W, i.e. the norm closure of the algebra generated by the set of functions $\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i\in\mathbb {N}\}$, is characterized in a recent paper by Jabbari and Namioka (Ellis group and the topological center of the flow generated by the map $n\mapsto \lambda^{n^{k}}$, to appear in Milan J. Math.). By the techniques essentially used in the cited paper, the topological center of the spectrum of the subalgebra W_k, the norm closure of the algebra generated by the set of functions $\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i=0,1,2,\ldots,k\}$, will be characterized, for all k∈ℕ. Also an example of a non-minimal dynamical system, with the enveloping semigroup Σ, for which the set of all continuous elements of Σ is not equal to the topological center of Σ, is given.