Celorrio Ramirez, M.Eugenia and Dales, G. and Choi, Yemon
(2023)
*Arens Regularity of Banach Function Algebras and Decomposable Blaschke Products whose Degree is a Power of 2.*
PhD thesis, Lancaster University.

## Abstract

This thesis presents three pieces of work. Within the first two thirds of the thesis, we study Arens regularity of Banach algebras. We first study Arens regularity of weighted semigroup algebras that arise from totally ordered semilattices. This is a natural continuation of [24], where they focus on studying Arens regularity of the unweighted case. We provide a sufficient condition for when a weighted semigroup algebra is not strongly Arens irregular and a characterization of Arens regularity of the weighted semigroup algebra. We then focus on three specific totally ordered semilattices, the natural numbers with the minimum operation, the natural numbers with the maximum operation and the integers with the maximum operation to obtain stronger results than those obtained for a generic totally ordered semilattice. Later on, we focus on two different Banach sequence algebras, the James p-algebra and the Feinstein algebra. Amongst other properties, we prove that the Feinstein algebra is Arens regular, which provides a second example of an Arens regular natural Banach sequence algebra that is not an ideal in its bidual, the first one being the remarkable example obtained in [7]. We study whether the James p-algebra is a BSE algebra with a BSE norm, for 1 In the final part of the thesis, we focus on Blaschke products. We study the decomposability of a finite Blaschke product B of degree a potency of 2 into n degree-2 Blaschke products, examining the connections between Blaschke products, the elliptical range theorem, Poncelet theorem, and the monodromy group. We show that if the numerical range of the compression of the shift operator of B a Blaschke product of degree n is an ellipse, then B can be written as a composition of lower-degree Blaschke products that correspond to a factorization of the integer n. We also show that a Blaschke product of degree 2^n with an elliptical Blaschke curve has at most n distinct critical values, and we use this to examine the monodromy group associated with a regularized Blaschke product B. We prove that if B can be decomposed into n degree-2 Blaschke products, then the monodromy group associated with B is the wreath product of n cyclic groups of order 2. Lastly, we study the group of invariants of a Blaschke product B of order 2^n when B is a composition of n Blaschke products of order 2.