On bilinear maps on matrices with applications to commutativity preservers

Matej Brešar (Author), Peter Šemrl (Author)

Abstract

Naj bo ▫$M_n$▫ algebra vseh ▫$n \times n$▫ matrik nad komutativnim enotskim kolobarjem ▫$\mathcal{C}$▫, and naj bo ▫$\mathcal{L}$▫ modul nad ▫$\mathcal{C}$▫. Podane so različne karakterizacije bilinearnih preslikav ▫$\{\,.\,,\,.\,\}: M_n \times M_n \to \mathcal{L}$▫ z lastnostjo, da je ▫$\{x,y\} = 0$▫, kadarkoli ▫$x$▫ in ▫$y$▫ komutirata. Kot glavno aplikacijo dobimo dokončno rešitev problema opisa (ne nujno bijektivnih) linearnih ohranjevalcev komutativnosti iz ▫$M_n$▫ v ▫$M_n$▫ za primer, ko je ▫$\mathcal{C}$▫ poljubno polje; še več, enak opis velja v vsaki končno razsežni centralni enostavni algebri.

Keywords

matematika;matrična algebra;bilinearna preslikava;ohranjevalec komutativnosti;funkcijska identiteta;neasociativni produkt;centralna enostavna algebra;mathematics;matrix algebra;central simple algebra;functional identity;nonassociative product;Lie-admissible algebra;commutativity preserving map;

Data

Language:	English
Year of publishing:	2006
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	512.643
COBISS:	13984857
ISSN:	0021-8693
Views:	36
Downloads:	20
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	O blinearnih preslikavah na matrikah in aplikacije v teoriji ohranjevalcev komutativnosti
Secondary abstract:	Let ▫$M_n$▫ be the algebra of all ▫$n \times n$▫ matrices over a commutative unital ring ▫$\mathcal{C}$▫, and let ▫$\mathcal{L}$▫ be a ▫$\mathcal{C}$▫-module. Various characterizations of bilinear maps ▫$\{\,.\,,\,.\,\}: M_n \times M_n \to \mathcal{L}$▫ with the property that ▫$\{x,y\} = 0$▫ whenever ▫$x$▫ any ▫$y$▫ commute are given. As the main application of this result we obtain the definitive solution of the problem of describing (not necessarily bijective) commutativity preserving linear maps from ▫$M_n$▫ into ▫$M_n$▫ for the case where ▫$\mathcal{C}$▫ is an arbitrary field; moreover, this description is valid in every finite dimensional central simple algebra.
Secondary keywords:	matematika;matrična algebra;bilinearna preslikava;ohranjevalec komutativnosti;funkcijska identiteta;neasociativni produkt;centralna enostavna algebra;
URN:	URN:SI:UM:
Type (COBISS):	Not categorized
Pages:	str. 803-837
Volume:	ǂVol. ǂ301
Issue:	ǂno ǂ2
Chronology:	2006
ID:	1472729