Methods of Complex Function Theory in Some Problems of Analysis: KMS States and Corona Problem

This thesis surveys a pair of topics which both depend on holomorphic functions and Banach Algebras. Firstly, the prerequisite background knowledge common to both such as holomorphic functions, Banach spaces and algebras, and module theory is provided. Secondly, KMS states arising on Cuntz-Krieger algebras are described. A Cuntz-Krieger graph algebra A is the universal C^* algebra satisfying certain defining relations between its partial isometries p_v which are derived from the directed graph. It becomes a C^* dynamical system when equipped with a gauge action a_t defined on partial isometries by a_t(p_v) = e^{it}p_v. Its KMS states can now be studied. (The KMS condition arises in physics in which it is a local equilibrium condition for the states of the operator algebra generated by local observables with the action of conjugation by the (time) evolution operator U_t = e^{it H}.) Examples including KMS states on matrix algebras and the generalization of Cuntz-Krieger algebras to Cuntz-Pimsner algebras are provided. Thirdly, algebras of bounded holomorphic functions are discussed. The maximal ideal space for an algebra of bounded holomorphic functions on a Riemann surface R is described. (In particular, this holds for domains in the complex plane C.) By the correspondence between maximal ideals and algebra homomorphism to $\mathbb{C}$, the maximal ideal space may be equipped with its induced weak* topology which is known as the Gelfand topology. A couple of interesting problems are the corona problem and the complement problem. The corona problem concerns whether the point evaluations (functionals induced by the canonical map R \to H^(R)*) are weak-star dense in the algebra of continous functions on the maximal ideal space and the complement problem concerns whether a nonsquare matrix in the algebra H^(R) can be augmented to become a unipotent matrix of the algebra. Wolff's proof of the corona theorem for the open unit disk D is given.

Name thesis:degree_name

Master of Science (MSc)

Discipline thesis:degree_discipline

Mathematics and Statistics

Grantor dc:publisher.institution

Graduate Studies

Year dc:date.issued

2015

Author dc:creator

• Broemeling, Luke Paul

Advisors dc:contributor.advisor

• Brudnyi, Alexander
• Brenken, Berndt

• Mathematics

Statement dc:rights

• University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.

Language dc:language.iso

eng

DOI

http://dx.doi.org/10.11575/PRISM/26896

OAI identifier oai:identifier

oai:ucalgary.scholaris.ca:11023/2294