Banachove i C*-algebre

master's thesis

• Author: Dorotea Rajšel
• Mentor: Damir Bakić (mentor)
• Committee member: Damir Bakić (predsjednik povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-02-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu dan je kratki pregled teorije Banachovih i \(C*\)-algebri. Banachove algebre su potpune normirane algebre, a \(C*\)-algebre su Banachove algebre s involucijom \(a\mapsto a^*\) koje imaju svojstvo da za svaki njihov element \(a\) vrijedi \(\|a\|^2 = \|a^*a\|.\) Na Banachovim algebrama se promatraju multiplikativni linearni funkcionali te je pokazana veza između multiplikativnih linearnih funkcionala i maksimalnih ideala komutativne Banachove algebre. Također, iznešeni su rezultati o spektru elementa Banachove algebre. Definira se te se pokazuje egzistencija Geljfandove transformacije na komutativnim Banachovim algebrama. Pokazuje se da je to preslikavanje izometrički \(*\)-izomorfizam na komutativnim \(C*\)-algebrama pa se koristi za identifikaciju komutativne \(C*\)-algebre s algebrom neprekidnih funkcija koje iščezavaju u beskonačnosti. Uvodi se pojam hereditarne \(C*\)-algebre te je pokazana uska veza između te klase podalgebri i njenih zatvorenih lijevih ideala. Definiraju se pozitivni elementi \(C*\)-algebre te se dokazuje egzistencija i jedinstvenost pozitivnog korijena. Pokazuju se osnovna svojstva pozitivnih funkcionala na \(C*\)-algebrama. Konačno, dokazano je postojanje vjerne reprezentacije svake \(C*\)-algebre u Geljfand-Naimarkovom teoremu.

Abstract (english)

This thesis gives an overview of Banach and \(C*\)-algebras theory. Banach algebras are complete normed algebras, whereas \(C*\)-algebras are Banach algebras with involution \(a\mapsto a^*\) satisfying the condition \(\|a\|^2 = \|a^*a\|\) for every element \(a\). Multiplicative linear functionals on Banach algebras have interesting properties and an important relation with maximal ideals when the algebra is abelian. Additionaly, the thesis includes a brief study of spectral theory. Gelfand transformation is defined and its existence is shown on abelian Banach algebras. Moreover, on abelian \(C*\)-algebras the Gelfand transformation is an isometric \(*\)-isomorphism. Hence, it gives a way to identify abelian \(C*\)-algebras with algebras of continuous functions that vanish at infinity. After introducing hereditary \(C*\)-algebras, their relation with closed left ideals is shown. Positive elements of \(C*\)-algebras are defined and the existence and uniqueness of positive root of a positive element is proven. Also, the properties of positive functionals on \(C*\)- algebras are studied. Finally, the existence of a faithful representation for every \(C*\)-algebras is proven in the Gelfand-Naimark theorem.

• Keywords: Banachove algebra; C* -algebre; normirana algebra; Geljfandova transformacija; Geljfand-Naimarkov teorem
• Keywords (english): Banach algebras; C*-algebras; normed algebra; Gelfand transformation; Gelfand-Naimark theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:844239
• Study programme: Title: Applied 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: 2020-09-23 11:30:32