Modifying the structure of associative algebras : doctoral thesis

Špenko, Špela (author), Brešar, Matej (mentor)

The structure of associative algebras can be modified by changing the operation of multiplication. In studying connections between the initial and the obtained modified category, the theory of functional identities has emerged. In this thesis we first study a subclass of functional identities - quasi-identities, which have played a fundamental role in the theory. They appear as linear relations among the noncommutative polynomial functions on algebras of matrices. We prove that quasi-identities follow from the Cayley-Hamilton identity if one allows central denominators, while the Cayley-Hamilton identity does not exhaust all quasi-identities globally. However, when considered in the class of all functional identities, every functional identity is a consequence of the Cayley-Hamilton identity. The analysis depends heavily on the theory of generic matrix algebras and trace rings. These are universal objects in the category of algebras (resp. algebras with trace) satisfying all polynomial (resp. trace) identities of ▫$n \times n$▫ matrices. Thus, they can be seen as analogues of polynomial rings from a noncommutative geometry standpoint. We explore some of their geometric properties. We prove a tracial Nullstellensatz and study the image of noncommutative polynomials and some special noncommutative polynomial maps on matrices. Moreover, we consider homological properties of trace rings and construct (twisted) noncommutative crepant resolutions of singularities for their centers. We further apply the theory of identities on matrices and matrix invariants to free function theory. This enables a unified approach to an understanding of free maps and free maps with involution. In Banach algebras we modify the multiplicative structure via the spectral function. We determine elements through their spectral functions and identify derivations through the spectra of their values. We investigate the stability of commuting maps, Lie maps and derivations, and obtain metric versions of Posner%s theorems. We conclude by modifying the structure of ▫$C^*$▫-algebras and especially algebras of matrices by introducing a multilinear multiplication induced by a noncommutative polynomial.

• Language: English
• Keywords: functional identities, quasi-identities, algebras with trace, Nullstellensatz, noncommutative polynomials, matrix invariants, noncommutative resolutions, length of a vector space, free analysis, Banach algebras, ▫$C^*$▫-algebras, spectrum, commuting maps, Lie maps, linear preservers
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Place of publishing: Ljubljana
• Publisher: [Š. Špenko]
• Year: 2015
• Number of pages: 206 str.
• PID: 20.500.12556/RUL-95860
• UDC: 512.5/.6:517.98(043.3)
• COBISS.SI-ID: 17347673
• Publication date in RUL: 24.10.2017

Secondary language

• Language: Slovenian
• Title: Preoblikovane strukture asociativnih algeber

Abstract

Strukturo asociativnih algeber lahko preoblikujemo s spremembo operacije množenja. Iz študija povezav med prvotno in preoblikovano strukturo je vzniknila teorija funkcijskih identitet. V disertaciji najprej proučujemo podrazred funkcijskih identitet - kvazi-identitete. Na algebrah matrik se pojavijo kot linearne relacije na nekomutativnih polinomskih funkcijah. Pokažemo, da kvazi-identitete izhajajo iz Cayley-Hamiltonove identitete, če dopustimo centralne imenovalce, globalno pa ta identiteta ne zaobjame vseh kvazi-identitet. Nasprotno je vsaka funkcijska identiteta znotraj celega razreda funkcijskih identitet posledica Cayley-Hamiltonove identitete. Obravnava se močno nasloni na teorijo generičnih matričnih algeber in kolobarjev s sledjo. Generična matrična algebra in kolobar s sledjo sta univerzalna objekta v kategoriji algeber (oz. algeber s sledjo), ki zadoščajo vsem polinomskim identitetam (oz. identitetam s sledjo) ▫$n\times n$▫ matrik. Torej sta s pogledom nekomutativne geometrije podobna polinomskim kolobarjem. Raziskujemo njune geometrijske lastnosti. Poiščemo Nullstellensatz s sledjo in postojimo pred slikami nekomutativnih polinomov in posebnimi nekomutativnimi polinomskimi preslikavami. Poglobimo se še v homološko naravo kolobarjev s sledjo in zgradimo nekomutativne krepantne odprave singularnosti njihovih centrov. Teorijo identitet na matrikah in matričnih invariant prenesemo v okolje proste funkcijske teorije, kjer omogoči poenoten pristop k razumevanju prostih preslikav in prostih preslikav z involucijo. V Banachovih algebrah preoblikujemo strukturo preko spektralne funkcije. Elemente prepoznamo po njihovih spektralnih funkcijah, odvajanja pa istovetimo preko spektrov njihovih vrednosti. Proučujemo stabilnost komutirajočih in liejevih preslikav ter odvajanj, in podamo metrične različice Posnerjevih izrekov. Zaključimo s preoblekami ▫$C^*$▫-algeber in matričnih algeber z vpeljavo multilinearnega množenja, porojenega z nekomutativnim polinomom.

• Keywords: funkcijske identitete, kvazi-identitete, Cayley-Hamiltonov polinom, algebre s sledjo, izrek o ničlah, nekomutativni polinomi, matrične invariante, nekomutativne odprave singularnosti, dolžina vektorskega prostora, prosta analiza, Banachove algebre, ▫$C^*$▫-algebre, spekter, komutirajoče preslikave, Liejeve preslikave, linearni ohranjevalci, Asociativne algebre, Disertacije