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92. Sourour, Ahmed Ramzi(3-VCTR-MS): The Lie structure of certain algebras of operators. (English summary). - Operator theory and Banach algebras (Rabat, 1999), 133--141, Theta, Bucharest, 2003Matej Brešar, 2004, review, book review, critique Published in DKUM: 10.07.2015; Views: 410; Downloads: 47
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97. 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: 44
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98. On bilinear maps on matrices with applications to commutativity preserversMatej Brešar, Peter Šemrl, 2006, original scientific article 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.
Keywords: 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: 105
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99. 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: 34
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100. 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: 55
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