Projects / Programmes
Mappings on algebras and stability
| 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 |
Algebra, homomorphism, antihomomorphism, Jordan homomorphism, approximate homomorphism, isometry, approximate isometry, linear preserver, derivation.
Researchers (4)
Organisations (1)
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.
