N1-0061 Maps on matrices, quantum structures, and Minkowski space

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Research project is (co) funded by the Slovenian Research Agency.

UL Member: Faculty of Mathematics and Physics
Code: N1-0061
Project: Maps on matrices, quantum structures, and Minkowski space
Period: 1.6.2017 - 31.5.2020   
Range per year: 1,82 FTE, category B
Head: Peter Šemrl
Research activity: Natural sciences and mathematics
Research Organisations: link on SICRIS
Researchers: link on SICRIS
Citations for bibliographic records: link on SICRIS

Project description:

We shall study two classical long standing open preserver problems. Although both can beformulated in an elementary way, a variety of tools from diverse areas of mathematics such as linear algebra, functional analysis, geometry, complex analysis and algebraic topology has been used in attempts to solve them.

The first problem we will deal with is to find the optimal versions of all four Hua's fundamental theorems of geometry of matrices. Our belief that this goal can be achieved is based on a recent work of PI (Mem. Amer. Math. Soc. \bf 232 \rm (2014), 74pp.) followed by two papers of PI with two coauthors (approx. 140 pages altogether) where the optimal version of the fundamental theorem of geometry of rectangular matrices and its modification for matrices over a special class of division rings were proved and the optimality was demonstrated by counterexamples.

Based on known results we foresee applications in several areas of mathematics and physics. Within the project we will limit our interest to applications in the theory of general preservers on various operators spaces, and in particular, in the study of symmetries on quantum structures.

There is quite a surprising connection between Hua's theorems and Alexandrov's basic result in chronogeometry charaterizing Poincare similarities of Minkowski space as bijective maps preserving light cones in both directions. It is expected that the techniques we will develop will help us find the optimal version of Alexandrov's theorem.

The second problem we will deal with is the famous Kaplansky's problem of characterizing bijective linear invertibility preservers on semisimple Banach algebras.