Jordan homomorphisms of structural matrix algebras

doctoral thesis

• Title (english): Jordanovi homomorfizmi strukturnih matričnih algebri
• Author: Mateo Tomašević
• Mentor: Ilja Gogić (mentor)
• Mentor: Peter Šemrl https://orcid.org/0000-0002-0899-9379 (komentor)
• Committee member: Ljiljana Arambašić (predsjednik povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Committee member: Rajna Rajić (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Committee member: Peter Šemrl https://orcid.org/0000-0002-0899-9379 (član povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2025, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

Jordan homomorphisms between two associative algebras are linear maps which preserve squares. This dissertation studies Jordan homomorphisms between structural matrix algebras (SMAs), which are unital subalgebras of \(M_n\) (the algebra of all complex square matrices of order \(n\)) spanned by some set of matrix units. The first objective of the dissertation is to completely describe the form of all Jordan embeddings between two SMAs using concepts introduced by Coelho’s description of SMA automorphisms. Secondly, continuing the work of Molnar and Šemrl on rank-one preservers of the algebra of uppertriangular matrices \(\mathcal{T}_n\), it considers several natural linear preserver problems on SMAs and puts them into the broader context of Jordan embeddings on SMAs. The third part of the dissertation contains a full extension of the well-known nonlinear preserver results of Petek and Šemrl (on \(M_n\) and \( \mathcal{T}_n \)) in the context of SMAs. More precisely, a characterization of SMAs \(\mathcal{A} \subseteq M_n\) is obtained with the property that all injective continuous commutativity and spectrum preserving map \(\mathcal{A} \to M_n\) are necessarily Jordan embeddings.

Abstract (english)

Jordanovi homomorfizmi između dviju asocijativnih algebri su linearna preslikavanja koja čuvaju kvadrate. U ovoj disertaciji se proučavaju Jordanovi homomorfizmi između strukturnih matričnih algebri (SMA), tj. unitalnih podalgebri od \(M_n\) (algebre svih kompleksnih kvadratnih matrica reda \(n\)) razapetih nekim skupom matričnih jedinica. Prvi cilj disertacije je u potpunosti opisati formu svih Jordanovih ulaganja između dviju SMA koristeći koncepte uvedene u Coelhinom opisu automorfizama SMA. Nastavljajući rad Molnara i Šemrla na preserverima ranga jedan algebre gornjetrokutastih matrica \(\mathcal{T}_n\), u drugom dijelu disertacije se promatra nekoliko prirodnih linearnih preservera na SMA i stavlja ih se u širi kontekst Jordanovih ulaganja na SMA. Treći dio disertacije sadrži potpuno proširenje dobro poznatog preserverskog rezultata Petek i Šemrla u kontekstu SMA. Preiznije, dobivena je karakterizacija SMA \(\mathcal{A} \subseteq M_n\) takvih da su sva injektivna neprekidna preslikavanja \(\mathcal{A} \to M_n\) koja čuvaju komutativnost i spektar nužno Jordanova ulaganja.

• Keywords: Jordan homomorphism; structural matrix algebra; rank preserver; spectrum preserver; commutativity preserver
• Keywords (english): Jordanov homomorfizam; strukturna matrična algebra; preserver ranga; preserver spektra; preserver komutativnosti
• Language: english
• URN:NBN: urn:nbn:hr:217:219656
• Study programme: Title: Doctoral study; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica u području prirodnih znanosti (doktor/doktorica u području prirodnih znanosti)
• Type of resource: Text
• Extent: iv, 100 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2025-01-31 13:37:04