TOPICS IN ADVANCED ANALYSIS B
Linear Algebra, a knowledge of the basic tools of Functional Analysis, Lebesgue integration theory and the fundamentals of Spectral Theory.
Verification of learning will consist of two parts:
1) A short written thesis about a chosen topic to be exposed to the class as a lecture.
2) A traditional oral exam, during which the student will have to show that she/he has acquired the basic notions and the proofs of the main theorems.
The purpose of the teaching is to let the student acquire, via both abstract results and concrete examples, the basic concepts from the theory of locally compact groups and their representations. The course aims also to introduce applications of Operator Algebra Theory, Harmonic Analysis and Representation Theory. At the end of the course we expect that:
1) The student has acquired the fundamental notions pertaining to the above mentioned fields
2) On the base of the proofs illustrated during the lectures, the student is able to make by him/herself reasonings of medium complexity that lead him/her to deduce abstract properties of the above mentioned objects;
3) The student is able to investigate the main properties of the objects alluded to above in concrete situations.
Banach Algebras: Basic Concepts
Nonunital Banach Algebras
The Spectral Theorem
Spectral Theory of ∗-Representations
Von Neumann Algebras
Representations of a Group and Its Group Algebra
Functions of Positive Type
Analysis on Locally Compact Abelian Groups
The Dual Group
The Fourier Transform
The Pontrjagin Duality Theorem
Representations of Locally Compact Abelian Groups
Closed Ideals in L1(G)
Analysis on Compact Groups
Representations of Compact Groups
The Peter-Weyl Theorem
Fourier Analysis on Compact Groups
The Inducing Construction
The Frobenius Reciprocity Theorem
Pseudomeasures and Induction in Stages
Systems of Imprimitivity
The Imprimitivity Theorem
Introduction to the Mackey Machine
Gerald B. Folland A Course in Abstract Harmonic Analysis, 2nd Edition, CRC Press, 2015