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Projects / Programmes source: ARIS

Mappings on algebras and stability

Research activity

Code Science Field Subfield
1.01.01  Natural sciences and mathematics  Mathematics  Analysis 

Code Science Field
P140  Natural sciences and mathematics  Series, Fourier analysis, functional analysis 
Keywords
Algebra, homomorphism, antihomomorphism, Jordan homomorphism, approximate homomorphism, isometry, approximate isometry, linear preserver, derivation.
Evaluation (rules)
source: COBISS
Researchers (4)
no. Code Name and surname Research area Role Period No. of publicationsNo. of publications
1.  00158  PhD Zvonimir Bohte  Mathematics  Researcher  1998 - 2000  292 
2.  01639  PhD Anton Cedilnik  Mathematics  Researcher  1998 - 2000  111 
3.  07082  PhD Gorazd Lešnjak  Mathematics  Researcher  2000  154 
4.  05953  PhD Peter Šemrl  Mathematics  Head  1998 - 2000  497 
Organisations (1)
no. Code Research organisation City Registration number No. of publicationsNo. of publications
1.  0101  Institute of Mathematics, Physics and Mechanics  Ljubljana  5055598000  20,258 
Abstract
Linear preservers are maps defined on algebras (matrix algebras, operator algebras,...) that leaves invariant certain functions defined on algebras (an example: spectrum preserving maps), certain subsets of the algebra (an example: nilpotent preserving maps), or certain relations on the algebra (an example: commutativity preserving maps). It turns out that in many cases such maps are Jordan automorphisms or that they differ from such maps in a multiplicative constant and an operator of a scalar type. We are mainly interested in linear maps preserving spectral properties (preserving invertibility, the spectral radius, nilpotents) and in linear maps preserving commutativity. We intend to contribute new results as well as new methods to the Kaplansky’s problem on characterizing linear bijective maps on semisimple Banach algebras that preserve invertibility. We believe that in the finite-dimensional case we will be able to characterize linear maps that preserve commutativity without the bijectivity assumption. Such maps are important as they can be considered as generalizations of Lie homomorphisms. We will also study derivations (or the compositum of derivations) that map the algebra into the socle, the set of algebraic elements or the set of all quasinilpotent elements. When studying such derivations the important tools are the theory of subharmonic functions and the theory of irreducible representations. Another topic is the study of non linear perturbations of isometries and algebraic homomorphisms. We believe that we will have to improve some results in the theory of functional inequalities in order to get the desired stability results for such mappings.
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