Operator Algebra

Cod y Modiwl: MA38310
Teitl y Modiwl: Operator Algebra
Blwyddyn Academaidd: 2026/2027
Semester: Semester 2
Cyd-gysylltydd y Modiwl: Dr Gwion Evans
Rhestr Ddarllen: Gweld ar Aspire
Rhagofynion: MA20310
Rhagofynion: MA30210 neu MT30210
Staff Eraill sy'n Cyfrannu: Dr Gwion Evans

Dulliau Asesu

Math o Asesiad	Manylion Asesiad	Cyfran
Arholiad Semester	2 Awr (Written Examination)	100%
Arholiad Ailsefyll	2 Awr (Written Examination)	100%

Canlyniadau Dysgu

Wedi cwblhau'r modiwl dylai'r myfyrwyr fedru:

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.

Disgrifiad cryno

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.

Nod

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.

Cynnwys

Associative Algebras, Historical Background and Examples

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.

Sgiliau Modiwl

Math o Sgiliau	Manylion Sgiliau
Cyfathrebu	Students will be expected to submit written worksheet solutions
Datblygu personol a chynllunio gyrfa	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.
Datrys Problemau	All situations considered are problem-based to a greater or lesser degree.
Gwella dysgu a pherfformiad ei hun	Feedback via tutorials
Rhifedd	Throughout the module.
Sgiliau pwnc penodol	Using techniques from Abstract Algebra to solve problems.
Sgiliau ymchwil	Students will be encouraged to consult various books and journals for examples of application.
Technoleg Gwybodaeth	Use of spreadsheets. Aberlearn Blackboard.

Nodau

Mae'r modiwl hwn yn cydymffurfio a FfCChC Lefel 6