Banach Algebras

Timetable

Module aim (aims)

The aim of the course is to acquaint participants with the language and methods of the Banach Algebra Theory – part of Analysis where the methods and objects from Abstract Algebra find its applications.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

Basic course in Complex Analysis and Functional Analysis.

Syllabus

Week 1: Basic definitions and examples.
Week 2: Banach algebra homomorphisms.
Week 3: Banach algebra constructions: direct sum, unitization, quotient.
Week 4: The group of invertible elements: definition and properties.
Week 5: Spectrum of an element (local spectrum) and spectral radius.
Week 6: Gelfand-Mazur Theorem and Gelfand-Beurling Formula.
Week 7: Holomorphic Functional Calculus.
Week 8: Holomorphic Functional Calculus – continued.
Week 9: Multiplicative functionals.
Week 10: Commutative Banach Algebras: multiplicative functionals and maximal ideals.
Week 11: The Gelfand Transform: definition and examples.
Week 12: Spectrum of an algebra (global spectrum) and the radical of an algebra.
Week 13: Involutive algebras: definition and exapmles.
Week 14: C*-algebras: definition and examples.
Week 15: Commutative C*-algebras and the Gelfand-Naimark Theorem.

Reading list

F. F. Bonsall, J. Duncan, “Complete Normed Algebras”, Springer, 1973,
R. V. Kadison, J. R. Ringrose, “Fundamentals of the Theory of Operator Algebras”, vol. 1, AMS, 1997,
W. Rudin, “Functional Analysis”, McGraw-Hill, 1991.