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105. Functional identitiesMatej Brešar, M. A. Chebotar, Wallace S. Martindale, 2007, scientific monograph Abstract: The theory of functional identities (FIs) is a relatively new one - the first results were published at the beginning of the 1990s, and this is the first book on this subject. An FI can be informally described as an identical relation involving arbitrary elements in an associative ring together with arbitrary (unknown) functions. The goal of the general FI theory is to describe these functions, or, when this is not possible, to describe the structure of the ring admitting the FI in question. This abstract theory has turned out to be a powerful tool for solving a variety of problems in ring theory, Lie algebras, Jordan algebras, linear algebra, and operator theory. The book is divided into three parts. Part I is an introductory one. Part II is the core of the book. It gives a full account of the general FI theory, which is based on the concept of a d-free set; various constructions and concrete examples of d-free sets are given, and FIćs on d-free sets are thoroughly studied. Part III deals with applications. Its main purpose is to demonstrate how one can find FI's when considering different problems, and then effectively use the general theory exposed in Part II. Perhaps the most illuminating example of the applicability are solutions of long-standing Herstein's conjectures on Lie homomorphisms and Lie derivations - in the proofs practically the entire FI theory is used.
Keywords: funkcijske identitete, d-proste množice, kvazi polinomi, Liejeve preslikave, jordanske preslikave, linearni ohranjevalci
Published in DKUM: 10.07.2015; Views: 1169; Downloads: 33
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106. Characterizing homomorphisms, derivations and multipliers in rings with idempotentsMatej Brešar, 2007, original scientific article Abstract: V določenih kolobarjih z necentralnimi idempotenti karakteriziramo homomorfizme, odvajanja in multilplikatorje z njihovim delovanjem na elementih, ki zadoščajo določenim zvezam. Tako je npr. obravnavan pogoj, da aditivna preslikava ▫$h$▫ med kolobarjema ▫$mathcal{A}$▫ in ▫$mathcal{B}$▫ zadošča ▫$h(x)h(y)h(z)=0$▫ kadarkoli je ▫$xy=yz=0$▫. Kot aplikacijo dobimo nove rezultate o lokalnih odvajanjih in lokalnih multiplikatorjih. Med drugim dokažemo, da je vsako odvajanje na prakolobarju z netrivialnim idempotentom odvajanje.
Keywords: matematika, kolobar, idempotent, homomorfizem, odvajanje, multiplikator, mathematics, ring, idempotent, homomorphism, derivation, multiplier
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107. On bilinear maps on matrices with applications to commutativity preserversMatej Brešar, Peter Šemrl, 2006, original scientific article 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
Published in DKUM: 10.07.2015; Views: 1315; Downloads: 100
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108. Jordan derivations revisitedMatej Brešar, 2005, original scientific article Abstract: Naj bo ▫$d$▫ jordansko odvajanje iz algebre ▫$mathcal{A}$▫ v ▫$mathcal{A}$▫-bimodul ▫$mathcal{M}$▫. Naš glavni rezultat med drugim pove, da je zožitev ▫$d$▫ na ideal algebre ▫$mathcal{A}$▫ generiran z določenimi višjimi komutatorji v ▫$mathcal{A}$▫ odvajanje. S pomočjo te splošne ugotovitve se zatem ugotovi, da je ob predpostavki različnih dodatnih pogojev ▫$d$▫ nujno odvajanje na ▫$mathcal{A}$▫. Nadalje je prikazanih več primerov pravih jordanskih odvajanj, karakterizirane so ▫$C^ast$▫-algebre s pravimi aditivnimi jordanskimi odvajanji, in poiskana je zveza s sorodnimi problemi o jordanskih homomorfizmih in jordanskimi ▫$mathcal{A}$▫-modulskimi homomorfizmi.
Keywords: matematika, algebra, komutator, jordansko odvajanje, odvajanje, mathematics, algebra, communitator, derivation, Jordan derivation
Published in DKUM: 10.07.2015; Views: 1146; Downloads: 98
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109. Lee, Tsiu-Kwen: Derivations on noncommutative Banach algebras. (English). - [J] Stud. Math. 167, No.2, 153-160 (2005). [ISSN 0039-3223; ISSN 1730-6337]Matej Brešar, 2006, review, book review, critique Keywords: matematika, teorija operatorjev, odvajanje, kvazinilpotentnost, Banachova algebra, Jacobsonov radikal, prakolobar, polenostavni kolobarji, mathematics, operator theory, derivation, quasinilpotent, Banach algebra, Jacobson radical, prime ring, semisimple ring
Published in DKUM: 10.07.2015; Views: 1166; Downloads: 28
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110. Duggal, B.P.: On the range closure of an elementary operator. (English). - [J] Linear Algebra Appl. 402, 199-206 (2005). [ISSN 0024-3795]Matej Brešar, 2006, review, book review, critique Keywords: matematika, teorija operatorjev, Hilbertov prostor, elementarni operator, skrčenje, posplošeni skalarni operator, mathematics, operator theory, Hilbert space, elementary operator, contraction, generalized scalar operator, single-valued extension property, SVEP
Published in DKUM: 10.07.2015; Views: 1195; Downloads: 53
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