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Introduction to Operator Algebras (ItOA)10 ECTS (englische Bezeichnung: Introduction to Operator Algebras)
(Prüfungsordnungsmodul: Introduction to Operator Algebras)
Modulverantwortliche/r: Kang Li Lehrende:
Kang Li
Startsemester: |
WS 2021/2022 | Dauer: |
1 Semester | Turnus: |
unregelmäßig |
Präsenzzeit: |
90 Std. | Eigenstudium: |
210 Std. | Sprache: |
Englisch |
Lehrveranstaltungen:
Empfohlene Voraussetzungen:
Es wird empfohlen, folgende Module zu absolvieren, bevor dieses Modul belegt wird:
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:
- Mathematik (Master of Science)
(Po-Vers. 2019w | NatFak | Mathematik (Master of Science) | Gesamtkonto | Studienrichtung Algebra und Geometrie | Introduction to Operator Algebras)
- Mathematik (Master of Science)
(Po-Vers. 2019w | NatFak | Mathematik (Master of Science) | Gesamtkonto | Studienrichtung Analysis und Stochastik | Introduction to Operator Algebras)
Dieses Modul ist daneben auch in den Studienfächern "Wirtschaftsmathematik (Master of Science)" verwendbar. Details
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
- Termin: 22.02.2022, 10:00 Uhr, Ort: Übungsraum Ü5, Cauerstr. 11, Erlangen
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