doktorska disertacija
Nejc Širovnik (Author), Joso Vukman (Mentor), Maja Fošner (Co-mentor)

Abstract

Disertacija je sestavljena iz štirih delov. V prvem definiramo osnovne pojme, kot so prakolobar, polprakolobar in standardna operatorska algebra ter dokažemo znan rezultat, da je standardna operatorska algebra prakolobar. Nato spoznamo pojme klasični kolobar kvocientov, levi (desni, simetrični) Martindaleov kolobar kvocientov ter razširjen centroid, ki izhajajo iz teorije Martindaleovih kolobarjev kvocientov. Sledi vpeljava preslikav, kot so odvajanje, jordansko odvajanje, jordansko trojno odvajanje, posplošeno odvajanje, levi (desni) centralizator in levi (desni) jordanski centralizator ter predstavitev pomembnih rezultatov v zvezi z njimi. Prvi odmevnejši izrek tega področja sega v leto 1957, ko je Herstein dokazal, da je vsako jordansko odvajanje na prakolobarju brez elementov reda dva odvajanje. Njegov rezultat je leta 1975 na polprakolobarje brez elementov reda dva posplošil Cusack. M. Brešar je leta 1989 dokazal, da je vsako jordansko trojno odvajanje na polprakolobarju brez elementov reda dva odvajanje. Zalar je leta 1991 dokazal, da je vsak levi (desni) jordanski centralizator na polprakolobarju brez elementov reda dva levi (desni) centralizator. Chernoff je leta 1973 karakteriziral vsa linearna odvajanja na standardnih operatorskih algebrah. Na koncu prvega poglavja predstavimo še teorijo funkcijskih identitet (Brešar - Beidar - Chebotarjeva teorija), ki jo uporabimo pri rezultatih na prakolobarjih. V nadaljevanju predstavimo preslikave, ki zadoščajo določenim enakostim na standardnih operatorskih algebrah, prakolobarjih ter polprakolobarjih. V drugem poglavju obravnavamo aditivne preslikave v zvezi z odvajanji in jordanskimi odvajanji. Na standardnih operatorskih algebrah dokažemo vrsto rezultatov, ki motivacijo črpajo iz rezultatov in domnev Vukmana, Eremite in Kosi-Ulblove. S pomočjo teorije funkcijskih identitet na prakolobarjih dokažemo izrek, ki izhaja iz Vukmanove domneve. Sledi obravnava preslikav z določenimi lastnostmi na polprakolobarjih, ki ponekod vsebujejo enoto. Tretje poglavje posvetimo preslikavam, ki so povezane s centralizatorji. Predstavimo motivacijo za obravnavo dveh izrekov na standardnih operatorskih algebrah kompleksnega Hilbertovega prostora. V zadnjem poglavju se lotimo odvajanjem sorodnih preslikav na standardnih operatorskih algebrah, prakolobarjih in polprakolobarjih z enoto. Navdih za študij preslikav te vrste predstavljajo rezultati, ki jih predstavimo v prvem in drugem poglavju ter enakost, ki sta jo leta 2011 objavila M. Fošner in Vukman.

Keywords

prakolobarji;polprakolobarji;Banachov prostor;algebra omejenih linearnih operatorjev;standardna operatorska algebra;aditivna preslikava;odvajanje;jordansko odvajanje;jordansko trojno odvajanje;centralizator;involucija;funkcijska identiteta;omejeni lineareni operatorji;disertacije;

Data

Language: Slovenian
Year of publishing:
Typology: 2.08 - Doctoral Dissertation
Organization: UM FNM - Faculty of Natural Sciences and Mathematics
Publisher: N. Širovnik]
UDC: 512.552(043.3)
COBISS: 20527112 Link will open in a new window
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Other data

Secondary language: English
Secondary title: Derivations and related mappings on some structures in algebra and functional analysis
Secondary abstract: The dissertation consists of four parts. The first part introduces prime rings, semiprime rings and standard operator algebras. We present the proof of a known result, which states that every standard operator algebra is a prime ring. Later we define the terms classical ring of quotients, left (right, symmetrical) Martindale ring of quotients and extended centroid of a ring, which originate from the theory of Martindale rings of quotients. Afterwards follows an introduction of some specific additive mappings, such as derivation, Jordan derivation, Jordan triple derivation, generalized derivation, left (right) centralizer and left (right) Jordan centralizer. The first result regarding the topic was published in 1957, when Herstein proved that every Jordan derivation on a 2-torsion free prime ring is a derivation. His result was generalized on 2-torsion free semiprime rings by Cusack in 1975 and in 1989 M. Brešar proved that every Jordan triple derivation on a 2-torsion free semiprime ring is a derivation. In 1991 Zalar showed that every left (right) Jordan centralizer on a 2-torsion free semiprime ring is left (right) centralizer. In 1973 Chernoff characterized all linear derivations on standard operator algebras. At the end of the first chapter the reader is acquainted with the theory of functional identities (Brešar - Beidar - Chebotar theory). In the following chapters it is our aim to find the form of mappings that satisfy certain relations on standard operator algebras, prime rings and semiprime rings. The second chapter treats additive mappings that are associated with derivations and Jordan derivations. On standard algebras we prove a series of results, which are motivated by the work of Vukman, Eremita and Kosi-Ulbl. Using the sophisticated theory of functional identities we prove the result that derives from Vukman's conjecture. Further on we deal with mappings with specific properties on semiprime rings, which sometimes contain a unit. The third chapter is devoted to mappings connected with centralizers. We introduce the reader with the motivation for two theorems on standard operator algebras of a complex Hilbert space. The last part of the dissertation tackles the mappings related to derivations. We study these mappings on standard operator algebras, prime rings and unital semiprime rings. The inspiration for dealing with such mappings comes from the results from previous chapters and also from the relation introduced by M. Fošner and Vukman in 2011.
Secondary keywords: prime rings;semiprime rings;Banach space;algebra of bounded linear operators;standard operator algebra;additivie mapping;derivation;Jordan derivation;Jordan triple derivation;centralizer;involution;functional identity;bounded linear operators;dissertations;
URN: URN:SI:UM:
Type (COBISS): Dissertation
Thesis comment: Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages: VII, 125 str.
ID: 8729218