For each natural number k and each irrational member λ of the unit circle, it is proved that the shift-orbit closure X f of the function f (n) = λ n k is homeomorphic to a k-torus. Using this homeomorphism, we investigate the Ellis group and its topological center of the dynamical system (X f , U), where U is the shift operator on l ∞ (Z). Finally, it is shown that the topological center of the spectrum of the Weyl algebra is the image of Z in the spectrum .Mathematics Subject Classification (2010). 43A60.

Generalizing the concept of inner amenability for Lau algebras, we define and study the notion of ϕ-inner amenability of any Banach algebra A, where ϕ is a homomorphism from A onto C. Several characterizations of ϕ-inner amenable Banach algebras are given.

Let [Formula: see text] and [Formula: see text] be two Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with the left and right compatible action of [Formula: see text] on [Formula: see text]. Let [Formula: see text] be a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. We show that (super) amenability of [Formula: see text] implies (super) module amenability of [Formula: see text] and (super) amenability [Formula: see text]. We investigate biprojectivity and biflatness of [Formula: see text] in the some especial cases. We also give some results related to module biprojectivity and module biflatness of [Formula: see text], when [Formula: see text] is biprojective or biflat.

In this paper, a general construction of a skew-product dynamical system, for which the skew-product dynamical system studied by Hahn is a special case, is given. Then the ergodic and topological properties (of a special type) of our newly defined systems (called Milnes-type systems) are investigated. It is shown that the Milnes-type systems are actually natural extensions of dynamical systems corresponding to some special distal functions. Finally, the topological centre of Ellis groups of any skew-product dynamical system is calculated.2000 Mathematics subject classification: primary 43A60; secondary 37B05, 37A05.

Let A, B be Banach A-modules with compatible actions and M be a left Banach A-A-module and a right Banach B-A-module. In the current paper, we study module amenability, n-weak module amenability and module Arens regularity of the triangular Banach algebra T = A M B (as an T :=. We employ these results to prove that for an inverse semigroup S with subsemigroup E of idempotents, the triangular Banachis permanently weakly module amenable (as an T 0 = 1 (E) 1 (E) -module). As an example, we show that T 0 is T 0 -module Arens regular if and only if the maximal group homomorphic image G S of S is finite.