Introduction to Operator Algebras (ItOA)10 ECTS
(englische Bezeichnung: Introduction to Operator Algebras)
(Prüfungsordnungsmodul: Introduction to Operator Algebras)

Lectures:

• • Introduction to Operator Algebras
    (Vorlesung, 4 SWS, Kang Li)

Empfohlene Voraussetzungen:

It is recommended to finish the following modules before starting this module:

Funktionalanalysis I (SS 2021)

Inhalt:

<ul> <li>Banach algebras: Basic properties. <li>Gelfand's theory of commutative Banach algebras and C-algebras: a) Special elements such as unitary, self-adjoint, normal, positive elements and their spectrum); b) The continuous functional calculus for normal elements in a C-algebra; c) Gelfand-Naimark theorem. <li>C-algebras (States and representations, and GNS construction) <li>von Neumann algebras (Bicommutant theorem, Kaplansky density theorem, Borel functional calculus) </ul>

Lernziele und Kompetenzen:

After following this course the student <ul> <li> knows the notion of spectrum in several contexts; in simple cases, he/she can compute the spectrum, has acquired insight in the elementary theory of operator algebras, in particular C-algebras and von Neumann algebras, <li>can deal with functions of operators, <li>can illustrate the various concepts and results treated in this course with relevant examples, <li>has gained intuition about linear mappings between infinite- dimensional Hilbert spaces and is able to verify intuitive conjectures by giving either rigorous proofs or counterexamples, <li>is able to explore some problems, examples, applications or extensions related to the course, independently using the literature. </ul>

Literatur:

We will mainly use the book <ul><li>C-Algebras and Operator Theory, Academic Press, 1990,, Gerard J. Murphy </ul> Several good books to read: <ul> <li>K. Zhu: An introduction to Operator Algebras. (a concise introduction) <li>R.V. Kadison and J.R. Ringose: Fundamentals of the theory of operator algebras. Volumes 1 & 2. (This contains far more material than we will be able to cover.) <li>B. Blackadar: Operator algebras. Theory of C-algebras and von Neumann algebras. (Contains lots of material, but does not include detailed proofs for everything.) <li>K. Davidson: C-algebras by example. (Useful example-based approach. But be careful: some parts are known to have mistakes.) </ul>

Verwendbarkeit des Moduls / Einpassung in den Musterstudienplan:

1. Wirtschaftsmathematik (Master of Science)
   (Po-Vers. 2019w | NatFak | Wirtschaftsmathematik (Master of Science) | Gesamtkonto | Mathematische Wahlpflichtmodule | Studienrichtung Algebra und Geometrie | Introduction to Operator Algebras)
2. Wirtschaftsmathematik (Master of Science)
   (Po-Vers. 2019w | NatFak | Wirtschaftsmathematik (Master of Science) | Gesamtkonto | Mathematische Wahlpflichtmodule | Studienrichtung Analysis und Stochastik | Introduction to Operator Algebras)

Studien-/Prüfungsleistungen:

Introduction to Operator Algebras (Prüfungsnummer: 50791)

Prüfungsleistung, mündliche Prüfung, Dauer (in Minuten): 20, benotet, 10 ECTS

Anteil an der Berechnung der Modulnote: 100.0 %

Erstablegung: WS 2021/2022, 1. Wdh.: WS 2021/2022

1. Prüfer:

Kang Li

Termin: 22.02.2022, 10:00 Uhr, Ort: Übungsraum Ü5, Cauerstr. 11, Erlangen