Mathematics
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Department of Mathematics & Statistics

MATH4OA Operator Algebras

Second Semester
10 points
 

Operator algebras are mathematical objects with a rich algebraic and analytic structure modelled on the properties of bounded linear operators on Hilbert space. The subject was originally a branch of functional analysis, which is the kind of analysis appropriate for studying problems involving infinite-dimensional phenomena. Operator algebras first arose in attempts to help put quantum mechanics on a firm mathematical footing, but have since been used in a broad range of mathematical situations. They are now being actively studied by mathematicians all over the world.

At Otago we have a lively research group working in this area, which includes Astrid an Huef, Lisa Clark and Iain Raeburn. They are currently interested in operator algebras associated to directed graphs, groupoids (a generalisation of groups), and dynamical systems arising in number theory.

The theory of operator algebras is a well-established and powerful tool, and the object of this module is to give students access to this tool by covering the basic theory with an emphasis on how it is used. The two main theorems in the course, proved by Gelfand and Naimark in the mid-twentieth century, say that there are two main examples of operator algebras: algebras of continuous functions, and algebras of linear operators. The interplay between these two sets of examples gives the subject its power.

Prerequisites

Math 301 Hilbert spaces (available in S1 2013)

Lecturer

Iain Raeburn (Room 515, phone 479 5115, email: iraeburn@maths.otago.ac.nz)

Internal Assessment

2 written assignments

Final exam

There will be a two-hour exam.

Final mark

While we strive to keep details as accurate and up-to-date as possible, information given here should be regarded as provisional. Individual lecturers will confirm teaching and assessment methods.