Modifying the structure of associative algebras : doctoral thesis

english

doktorska dela

2015

algebras with trace   Banach algebras   commuting maps   free analysis   functional identities   length of a vector space   Lie maps   linear preservers   matrix invariants   noncommutative polynomials   noncommutative resolutions   Nullstellensatz   quasi-identities   spectrum

Univerza v Ljubljani, Fakulteta za matematiko in fiziko
(Obvezni izvod spletne publikacije)

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.

12.05.2023