Dynamical Systems and Commutants in Non-Commutative Algebras

Abstract: This thesis work is about commutativity which is a very important topic in Mathematics, Physics, Engineering and many other fields. In Mathematics, it is well known that matrix multiplication (or composition of linear operators on a finite dimensional vector space) is not always commutative. Commuting matrices or more general linear or non-linear operators play an essential role in Mathematics and its applications in Physics and Engineering. Many important relations in Mathematics, Physics and Engineering are represented by operators satisfying a number of commutation relations. Such commutation relations are key in areas such as representation theory, dynamical systems, spectral theory, quantum mechanics, wavelet analysis and many others.In Chapter 2 of this thesis we treat commutativity of monomials of operators satisfying certain commutation relations in relation to one-dimensional dynamical systems. We derive explicit conditions for commutativity of the said monomials in relation to the existence of periodic points of certain onedimensional dynamical systems.In Chapter 3, we treat the crossed product algebra for the algebra of piecewise constant functions on given set and describe the commutant of this algebra of functions which happens to be the maximal commutative subalgebra of the crossed product containing this algebra.In Chapters 4 and 5, we give a characterization of the commutant for the algebra of piecewise constant functions on the real line, by comparing commutants for a non-decreasing sequence of algebras.In Chapter 6 we give a description of the centralizer of the coefficient algebra in the Ore extension of the algebra of functions on a countable set with finite support.