C*-algebre kompaktnih operatora

master's thesis

• Title (english): C*-algebras of compact operators
• Author: Luka Junaković
• Mentor: Ilja Gogić (mentor)
• Committee member: Zvonko Iljazović (predsjednik povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-12-01, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilj ovog rada bio je prezentirati osnovnu teoriju C*-algebri s naglaskom na C*- algebre kompaktnih operatora. Nakon što se obrade glavni rezultati teorije Banachovih algebri, započinjemo s promatranjem jedne posebne klase Banachovih algebri, tzv. C*-algebri koje su po definiciji Banachove *-algebre čiji svaki element zadovoljava ||a*a|| = \(||a||^2\). Dokazan je Komutativni Gelfand-Naimarkov teorem koji ukratko kaže da svaku komutativnu C*-algebru možemo realizirati kao algebru neprekidnih funkcija na nekom lokalno kompaktnom Hausdorffovom prostoru koje iščezavaju u beskonačnosti. Nakon toga se razvija teorija reprezentacija C*-algebri što kulminira poznatim (nekomutativnim) Gelfand-Naimark teoremom koji kaže da se svaka C*-algebra može realizirati kao zatvorena, samoadjunigrana podalgebra algebre ograničenih operatora na nekom Hilbertovom prostoru. Nakon toga, naglasak je stavljen na analizu C*-algebri kompaktnih operatora, gdje u potpunosti opisujemo reprezentacije C*-algebri kompaktnih operatora te pomoću tog opisa dolazimo do njihovog strukturnog teorema.

Abstract (english)

The goal of this thesis was to present the basic theory of C*-algebras with emphasis on C*-algebras of compact operators. After the main results in the theory of Banach algebras are covered, we proceed with the study of so called C*-algebras, which are defined as complete *-algebras whose every element satisfies ||a*a|| = |\(||a||^2\). Commutative Gelfand-Naimark theorem is proven which, in short, states that every commutative C*-algebra can be realized as an algebra of continuous functions on some locally compact Hausdorff space which vanish at infinity. Next, we begin with the development of the representation theory of C*-algebras which culminates with the famous (noncommutative) Gelfand-Naimark theorem which states that every C*-algebra can be realized as a closed, self-adjoint, subalgebra of algebra of bounded operators on some Hilbert space. After that, the emphasis is given to the study of C*-algebras of compact operators, where we completely describe the representations of C*-algebras of compact operators and thus we deduce their structural theorem.

• Keywords: teorije Banachovih algebri; Banachove *-algebre; komutativni Gelfand-Naimarkov teorem; kompaktni Hausdorffov prostor; Hilbertov prostor
• Keywords (english): theory of Banach algebras; Banach C*-algebras; commutative Gelfand-Naimark theorem; compact Hausdorff space; Hilbert space
• Language: croatian
• URN:NBN: urn:nbn:hr:217:979432
• Study programme: Title: Pure Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-12-16 12:08:11