Properties of geometric Peirce decompositions of facially symmetric spaces | Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika – Tomsk State University Journal of Mathematics and Mechanics. 2025. № 96. DOI: 10.17223/19988621/96/2

Properties of geometric Peirce decompositions of facially symmetric spaces

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.

Download file

Keywords

weakly and strongly facially symmetric spaces, symmetric face, geometric tripotent, geometric triangle and quadrilateral, geometric Peirce projectors

Authors

Name	Organization	E-mail
Ibragimov Mukhtar M.	Karakalpak State University named after Berdakh	m.ibragimov1909@gmail.com
Arziev Allabay D.	V.I. Romanovsky Institute of Mathematics, Uzbekistan Academy of Sciences	allabayarziev@inbox.ru

Всего: 2

References

Боголюбов Н.Н., Логунов А.А., Тодоров Н.Т. Основы аксиоматического подхода в квантовой теории поля. М.: Наука, 1969.

Браттели У., Робинсон Д. Операторные алгебры и квантовая статистическая механика. М.: Мир, 1982.

Эмх. Ж. Алгебраические методы в статистической механике и квантовой теории поля. М.: Мир, 2009.

Jordan P., von Neumann J., Wigner E. On an algebraic generalization of the quantum mechanical formalism // Ann. of Math. 1934. V. 35 (1). P. 29-64. doi: 10.2307/1968117.

Сигал А. Математические проблемы релятивистской физики. М.: Мир, 1968.

Аюпов Ш.А., Ядгоров Н.Ж. Спектральные выпуклые множества в конечномерных пространствах // Известия АН УзССР. Сер. физ.-мат. наук. 1989. № 3. С. 3-7.

Аюпов Ш.А., Ядгаров Н.Ж. Геометрия пространства состояний модулярных йордановых алгебр // Известия Российской академии наук. Сер. математическая. 1993. Т. 57 (6). С. 199-211.

Аюпов Ш.А., Ядгаров Н.Ж. Свойства спектральных выпуклых множеств // Доклады АН УзССР. 1989. № 7. С. 3-4.

Ajupov Sh., Iochum B., Yadgorov N. Symmetry versus facial homogeneity for self-dual cones // Linear Algebra and its Applications. 1990. V. 142. P. 83-89. doi: 10.1016/0024-3795(90)90257-D.

Аюпов Ш.А., Иокум В., Ядгоров Н.Ж. Геометрия пространств состояний конечномерных йордановых алгебр // Известия АН УзССР. Сер. физ.-мат. наук. 1990. № 3. С. 19-22.

Alfsen E.M., Hanche-Olsen H., Shultz F.W. Space spaces of C*-algebras // Acta Math. 1980. V. 144. P. 267-305. doi: 10.1007/BF02392126.

Alfsen E.M., Shultz F.W. On non-commutative spectral theory and Jordan algebras // Proc. London Math. Soc. 1979. V. s3-38 (3). P. 497-516. doi: 10.1112/plms/s3-38.3.497.

Alfsen E.M., Shultz F.W. State Spaces of Jordan algebras // Acta Math. 1978. V. 140. P. 155190. doi: 10.1007/BF02392307.

Araki H. On the characterization of the state space of quantum mechanics // Commun. Math. Physics. 1980. V. 75. P. 1-24. doi: 10.1007/BF01962588.

Iochum В., Shultz F. W. Normal state spaces of Jordan and von Neumann algebras // Journal of Functional Analysis. 1983. V. 50 (3). P. 317-328. doi: 10.1016/0022-1236(83)90008-3.

Friedman Y., Russo B. A geometric spectral theorem // The Quarterly Journal of Mathematics. 1986. V. 37 (3). P. 263-277. doi: 10.1093/qmath/37.3.263.

Friedman Y., Russo B. Affine structure of facially symmetric spaces // Mathematical Proc. of the Cambridge Philosophical Society. 1989. V. 106 (1). P. 107-124. doi: 10.1017/ S030500410006802X.

Friedman Y., Russo B. Classification of atomic facially symmetric spaces // Canadian Journal of Mathematics. 1993. V. 45 (1). P. 33-87. doi: 10.4153/CJM-1993-004-0.

Friedman Y., Russo B. Geometry of the Dual ball of the Spin Factor // Proc. Lon. Math. Soc. 1992. V. s3-65 (1). P. 142-174. doi: 10.1112/plms/s3-65.1.142.

Friedman Y., Russo B. Some affine geometric aspects of operator algebras // Pacific Journal of Mathematics. 1989. V. 137 (1). P. 123-144. doi: 10.2140/pjm.1989.137.123.

Ibragimov M. Geometric properties of geometric tripotents and split faces in neutral SFS-space // Science and Education in Karakalpakstan. 2023. № 2/2. P. 43-49.

Ибрагимов М.М., Кудайбергенов К.К. Геометрическое описание Li-пространств // Известия вузов. Математика. 2013. № 9. С. 21-27.

Ибрагимов М.М., Кудайбергенов К.К., Тлеумуратов С.Ж., Сейпуллаев Ж.Х. Геометрическое описание предсопряженного пространства к атомической коммутативной алгебре фон Неймана // Математические заметки. 2018. № 93 (5). С. 728-735. doi: 10.4213/mzm9314.

Ibragimov M.M., Tleumuratov S.J., Seypullaev J.X. Some geometric properties of a strongly facially symmetric space // Methods of functional analysis and topology. 2005. V. 11 (3). P. 234-238 ().

Yadgorov N.Dj., Ibragimov M.M., Kudaybergenov K.K. Geometric characterization of Li-spaces // Studia Mathematika. 2013. V. 219. P. 97-107. doi: 10.4064/sm219-2-1.

Seypullaev J.X. Finite strongly facially symmetric spaces // Uzb. Math. Journal. 2020. № 4. P. 140-148. doi: 10.29229/uzmj.2020-4-15.

Friedman Y., Russo B. Structure of the predual of a JBW*-triple // Journal fur die Reine und Angewandte Mathematik. 1985. Bd. 356. S. 67-89. doi: 10.1515/crll.1985.356.67.

Dang T., Friedmann Y. Classification of JBW*-triple factors and applications // Mathematica Scandinavica. 1987. V. 61. P. 292-330. doi: 10.7146/math.scand.a-12206.