J1-2454 Isomorphisms, isometries, and preservers

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

UL Member: Faculty of Mathematics and Physics

Code: J1-2454

Project: Isomorphisms, isometries, and preservers

Period: 1. 9. 2020 - 31. 8. 2023

Range per year: 0,7 FTE, category: C

Head: Peter Šemrl

Research activity: Natural sciences and mathematics

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Project description:

We will deal with four problems that are closely related. It has been known which operator intervals are order-isomorphic and for each pair of order-isomorphic operator intervals the general form of order-isomorphisms between the two intervals is known. An order-isomorphism is a bijective map between two operator intervals which preserves order in both directions. Can we drop the bijectivity assumption and still obtain some reasonable result? More precisely, we want to describe the general form of maps from an operator interval into the set of all bounded linear self-adjoint operators preserving order in both directions. There is no hope to get a reasonable result in the infinite-dimensional case. Hence, we will restrict to the finite-dimensional case. We believe that this problem is closely related to the problem of a maximal possible extension of an order-isomorphism between two operator intervals. One of the main tools when studying order-preserving maps on selfadjoint operators/matrices is the fundamental theorem of geometry of hermitian matrices which describes the general form of bijective adjacency preserving maps on Hermitian matrices. All other problems that we will consider are also closely related to the study of adjacency preservers.

The structural problem for isometries of Grassmann spaces on Hilbert spaces has been recently solved in full generality. Here, the Grassmann space is identified with the set of all projections of a given fixed rank and the distance is induced by the operator norm. We will study isometries of Grassmann spaces with respect to other norms.

Fundamental theorem of chronogeometry describes the general form of bijective maps on Minkowski space that preserve coherency in both directions. Our goal is to find the optimal version of this classical result. We would like to describe the general form of maps on Minkowski space that preserve coherency in one direction only (no injectivity or surjectivity is assumed). So far, we have succeeded to solve this problem under the additional assumption of continuity.

Similarly, we plan to obtain the optimal version of the fundamental theorem of geometry of Grassmann spaces. This theorem describes the general form of bijective maps on a Grassmann space preserving adjacency in both directions. We would like to remove the bijectivity condition and replace the assumption of preserving adjacency in both directions by a weaker assumption of preserving adjacency in one direction only. Moreover, we would like to consider adjacency preservers between two different Grassmann spaces. The fundamental theorem of geometry of Grassmann spaces can be reduced to the fundamental theorem of geometry of matrices. We have recently obtained the optimal version of the fundamental theorem of geometry of matrices and we expect that the techniques we have developed will help us to solve the problem.