Mathematics Ph.D. Dissertations

Title

A Constructive Approach to the Universality Criterion for Semigroups

Date of Award

2017

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

First Advisor

Kit Chan (Advisor)

Second Advisor

John Laird (Other)

Third Advisor

Juan Bes (Committee Member)

Fourth Advisor

Alex Izzo (Committee Member)

Abstract

We prove a generalization of the well-known Universality Criterion in the setting of continuous homomorphisms acting on a separable, complete, metrizable topological semigroup by constructing a particular universal element. Consequently, we simplify the proof of a 1955 classical result of Heins on the existence of universal Blaschke products on the unit ball of bounded analytic functions on the open unit disk, and the proofs of a number of other related results, by unifying them in the semigroup setting. Motivated by our generalization, we provide new applications such as the homomorphisms of conjugation on the unit ball of the operator algebra of c0 and ℓp, where 1≤ p<∞, and also the homomorphisms of composition on the closed unit ball of L∞[0,1] with the weak-star topology, as well as on the semigroup of all measurable integrable functions bounded in sup-norm by 1.

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