Semi-orthogonal projections in unital C∗-algebras
https://doi.org/10.26907/2949-3919.2025.2.4-18
Abstract
Let Asem = {A ∈ A : Re A = A∗A} be the set of all semiorthogonal projections of the unital C∗-algebra A, I be the identity of A. The formula U = 2A − I (A ∈ Asem) defines a bijection between the set Asem and the set of all isometries from A. For any natural number n ≥ 2, there exists a non-commutative polynomial of degree n that yields a semi-orthogonal projection when substituted for an arbitrary set A1, . . . , An ∈ Asem. Each element A ∈ Asem is hyponormal and lies in the unit ball of the C∗-algebra A. If A ∈ Asem, then A2 is hyponormal. If A, A2 ∈ Asem, then A is a projection. If A ∈ Asem and A = An for some n ∈ N, n ≥ 2, then A is a normal element, and A is a projection for n = 2.
Supporting agencies: The work is performed under the development program of Volga Region Mathematical Center (agreement no. 075-02-2025-1725/1).
About the Author
A. M. Bikchentaev
Kazan Federal University
Russian Federation
Russian Federation
Airat Midkhatovich Bikchentaev
Volga Region Mathematical Center,
18 Kremlyovskaya str., Kazan 420008
References
1. [1] J. Gross, G. Trenkler, S.-O. Troschke, On semi-orthogonality and a special class of matrices, Linear Algebra Appl. 289 (1–3), 169–182 (1999). URL: https://www.sciencedirect.com/science/article/pii/S0024379597100027
2. [2] A.M. Bikchentaev, Semi-orthogonal projectors in a Hilbert space, in: At the turn of the 20th–21st centuries, 108–114, Kazanskoe Matematicheskoe Obshchestvo, Kazan’, 2003 [in Russian].
3. [3] A.M. Bikchentaev, Ideal F-norms on C∗-algebras. II, Russian Math. (Iz. VUZ) 63 (3), 78–82 (2019). DOI: https://doi.org/10.3103/S1066369X19030071
4. [4] A.M. Bikchentaev, Rearrangements of tripotents and diff ences of isometries in semifi von Neumann algebras, Lobachevskii J. Math. 40 (10), 1450–1454 (2019). DOI: https://doi.org/10.1134/S1995080219100068
5. [5] P.R. Halmos, A Hilbert space problem book, Graduate Texts in Math., vol. 19. Springer, New York, 1982. DOI: https://doi.org/10.1007/978-1-4684-9330-6
6. [6] B. Blackadar, Operator algebras, theory of C∗-algebras and von Neumann algebras, Encyclopaedia of mathematical sciences 122. Operator algebras and non-commutative geometry 3. Springer-Verlag, Berlin, 2006. DOI: https://doi.org/10.1007/3-540-28517-2
7. [7] G.J. Murphy, C∗-algebras and operator theory. Academic Press, Inc., Boston, MA, 1990.
8. [8] M. Takesaki, Theory of operator algebras. I, Encyclopaedia of mathematical sciences 124. Operator algebras and non-commutative geometry 5. Springer-Verlag, Berlin, 2002. URL: https://link.springer.com/book/9783540422488
9. [9] U. Haagerup, R.V. Kadison, G.K. Pedersen, Means of unitary operators, revisited, Math Scand. 100 (2), 193–197 (2007). DOI: https://doi.org/10.7146/math.scand.a-15021
10. [10] R.A. Horn, C.R. Johnson, Matrix analysis. Second edition. Cambridge University Press, Cambridge, 2013. DOI: https://doi.org/10.1017/CBO9781139020411
11. [11] A.M. Bikchentaev, On operator monotone and operator convex functions, Russian Math. (Iz. VUZ) 60 (5), 61–65 (2016). DOI: https://doi.org/10.3103/S1066369X16050054
12. [12] H. Radjavi, P. Rosenthal, Invariant subspaces, Second edition. Dover Publications, Inc., Mineola, NY, 2003.
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For citations:
Bikchentaev A.M. Semi-orthogonal projections in unital C∗-algebras. Mathematics and Theoretical Computer Science. 2025;3(2):4-18. (In Russ.) https://doi.org/10.26907/2949-3919.2025.2.4-18
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