Due to Covid-19 students should refer to the module Blackboard pages for assessment details
|Assessment length / details
|2 Hours (Written Examination)
|2 Hours (Written Examination)
On successful completion of this module students should be able to:
Give examples of associative algebras and illustrate the basic theory.
Define basic notions from algebras of operators and recognise their occurrence and relevance.
Solve specific problems from algebras of operators.
Combining the reconstruction of a fascinating historical tale with the development of more and more refined technical tools the topic of this module provides an opportunity for undergraduate students to follow the emergence of Abstract Algebra along one of its most central lines and to learn to appreciate the unifying power of modern mathematics.
The theory of associative algebras grows historically out of the study of hypercomplex number systems in the 19th century and occupies a central position in modern Abstract Algebra. We start with the historic background, introduce a multitude of examples and then develop carefully the basics of this theory up to some classifying structure theorems. Following the strategy in the book of Farenick (see Essential Reading) we mainly study the finite-dimensional case, thus keeping everything accessible with tools from Linear Algebra. Focusing on algebras of linear operators we concentrate on those examples which are most relevant for applications outside Abstract Algebra. With the introduction of von Neumann algebras and C*-algebras towards the end of the course we finally see some important mathematical developments of the 20th century emerge from these origins which still play an important role in present day research.
Basic definitions and concepts. Historical examples: hypercomplex number systems, quaternions. More examples: group algebras, algebras of linear operators.
Theory of Associative Algebras
Division Algebras, invariant subspaces, representations and ideals, simple and semisimple algebras, structure theorems.
Von Neumann Algebras and C*-Algebras
Introduction of the notion of abstract operator algebras. The role of the involution in algebras of linear operators. Normed algebras. Von Neumann and C*-algebras. How to connect Algebra and Analysis.
|Application of Number
|Throughout the module.
|Students will be expected to submit written worksheet solutions
|Improving own Learning and Performance
|Feedback via tutorials
|Use of spreadsheets. Aberlearn Blackboard.
|Personal Development and Career planning
|Students will be exposed to an area of application that they have not previously encountered and they will be encouraged to combine previously learned theories towards a broader appreciation of modern mathematics.
|All situations considered are problem-based to a greater or lesser degree.
|Students will be encouraged to consult various books and journals for examples of application.
|Subject Specific Skills
|Using techniques from Abstract Algebra to solve problems.
This module is at CQFW Level 6