Lokalno kompaktne grupe

master's thesis

• Title (english): Locally compact groups
• Author: Nikola Šalgaj
• Mentor: Marcela Hanzer (mentor)
• Committee member: Marcela Hanzer (predsjednik povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Committee member: Marko Radulović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu bavimo se lokalno kompaktnim grupama kao najopćenitijim matematičkim objektima koji imaju lijevu (desnu) Haarovu mjeru, tj. Radonovu mjeru invarijantnu s obzirom na lijeve (desne) translacije. Nakon preliminarnih sadržaja sakupljenih u prvom poglavlju, u drugom poglavlju razmatramo osnovna svojstva topoloških i lokalno kompaktnih grupa te kao glavni rezultat dokazujemo egzistenciju i jedinstvenost lijeve (desne) Haarove mjere na proizvoljnoj lokalno kompaktnoj grupi. Iduće poglavlje bavi se algebrom mjera M(G) i prostorom \(L^1(G)\), gdje je G neka lokalno kompaktna grupa, i pokazujemo kako s uvedenom konvolucijom oba prostora posjeduju strukturu Banachove ∗-algebre. Naposljetku, u četvrtom poglavlju predstavljamo dva teorema o egzistenciji i jedinstvenosti invarijatnih i kvazi-invarijantnih mjera na homogenim prostorima, što su prostori snabdjeveni djelovanjem neke lokalno kompaktne grupe.

Abstract (english)

In this paper, we study the locally compact groups as the most general mathematical objects which have left (right) Haar measure, i.e. left (right) translation-invariant Radon measure. After the preliminaries assembled in Chapter 1, in Chapter 2 we examine basic features of topological and locally compact groups and as a main result we prove the existence and uniqueness of left (right) Haar measure on a locally compact group. Chapter 3 deals with the measure algebra M(G) and space \(L^1(G)\), where G is some locally compact group, and following the introduction of convolution, we show that both spaces have the structure of Banach ∗-algebra. Finally, in Chapter 4 we present two theorems about the existence and uniqueness of invariant and quasi-invariant measures on the homogeneous spaces, which are spaces equipped with an action of some locally compact group.

• Keywords: lijeva (desna) Haarova mjera; Radonova mjera; algebra mjera; Banachova ∗-algebra
• Keywords (english): left (right) Haar measure; Radon measure; measure algebra; Banach ∗-algebra
• Language: croatian
• URN:NBN: urn:nbn:hr:217:774773
• 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: 2022-11-08 13:37:44