Department of Mathematical Sciences

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Browsing Department of Mathematical Sciences by browse.metadata.advisor "Benjamin, Ronalda"

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    Domination and generalized domination in ordered Banach algebras
    (Stellenbosch : Stellenbosch University, 2024-03) Rabearivony, Andriamahazosoa Dimbinantenaina; Mouton, Sonja; Benjamin, Ronalda; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: Let A be a complex unital Banach algebra. An algebra cone in A is a non-empty subset C of A containing the unit of A and which is closed under addition, multiplication and non-negative scalar multiplication. Any algebra cone C in A corresponds to a partial ordering in A satisfying C = {a ∈ A : 0 ≤ a}. The pair (A, C) is called an ordered Banach algebra (OBA). In this dissertation, we investigate the domination (resp. generalized dom- ination) problems in OBAs, stated as follows: given an OBA (A, C) and two elements a, b ∈ A such that 0 ≤ a ≤ b (resp. ±a ≤ b), under which conditions do we have that a property of b is inherited by a? In 2014, Mouton and Muzundu investigated the domination problem for ergodic elements, where an element a ∈ A is called ergodic if the sequence (Ln 1 aᵏ) is convergent. While their theorem is an outstanding extension k=1 n of an operator theoretic result, the assumption of weak monotonicity of the spectral radius in the quotient algebra is quite strong, limiting its applicability to the regular operators on a Dedekind complete Banach lattice. Our first main result in this dissertation (Theorem 2.3.10) provides a partial solution to this problem in the form of an ergodic domination-type theorem without assuming weak monotonicity of the spectral radius in the quotient algebra. Furthermore, since their result was only for the domination problem, we extend it, as well as most of the other existing domination results, to generalized domination results. (See, in particular, Theorems 3.10.5, 3.12.6 and 3.13.8 regarding ergodicity.) These two contributions provide (partial) answers to two open questions in the survey paper [42] of Mouton and Raubenheimer.