MATHEMATICS
Properties of geometric Peirce decompositions of facially symmetric spaces
M. M. Ibragimova,
A. D. Arzievb a Karakalpak State University named after Berdakh, Nukus, Uzbekistan
b V.I. Romanovsky Institute of Mathematics, Uzbekistan Academy of Sciences,
Tashkent, Uzbekistan
Abstract:
In this paper, we consider problems of the theory of facially symmetric spaces which was introduced in the 1980s by Y. Friedman and B. Russo as a geometrical model of quantum mechanics. These spaces are determined based on the study of the structure of the predual space of the
$JBW^*$-triple, guided by geometric introductions to the measurement process in the set of observables in quantum mechanical systems. The main example of facially symmetric spaces is the Banach space whose dual space is a
$JBW^*$-triple. The main goal of this project was the geometric characterization of Banach spaces admitting an algebraic structure. More precisely, facially symmetric spaces provide the corresponding structure, where the problem of characterization of the unit ball of a predual space of a
$JBW^*$-triple is studied, describing important properties of a convex set in geometric terms such as orthogonality, projective unit, normed face, symmetric face, generalized (or geometric) tripotent and generalized (or geometric) Peirce projectors, etc.
One of the key concepts in facially symmetric spaces is the concept of geometric tripotent. In this paper, we study the relationship between the notions corresponding to different geometric tripotents on facially symmetric spaces and, on this basis, we study properties of geometric Peirce decompositions. More precisely, it is proved that geometric Peirce projectors corresponding to a certain class of geometric tripotents coincide. It is also shown that the geometric Peirce subspace corresponding to the minimal geometric tripotent is linearly isometric to the Hilbert space.
Keywords:
weakly and strongly facially symmetric spaces, symmetric face, geometric tripotent, geometric triangle and quadrilateral, geometric Peirce projectors.
UDC:
517.98
MSC: 46B20,
46E30 Received: 08.07.2024
DOI:
10.17223/19988621/96/2
Fulltext:
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