Differences of idempotents in C∗-algebras and the quantum Hall effect. II. Unbounded idempotents
https://doi.org/10.26907/2949-3919.2023.4.35-48
Abstract
Let a von Neumann algebra M of operators act on a Hilbert space H, I be the unit of M, τ be a faithful semifinite normal trace on M. Let S(M, τ ) be the ∗-algebra of all τ -measurable operators and L1(M, τ ) be the Banach space of all τ -integrable operators, P, Q ∈ S(M, τ ) be idempotents. If P − Q ∈ L1(M, τ ) then τ (P − Q) ∈ R. In particular, if A = A3 ∈ L1(M, τ ), then τ (A) ∈ R. If P −Q ∈ L1(M, τ ) and PQ ∈ M, then for all n ∈ N we have (P −Q)2n+1 ∈ L1(M, τ ) and τ ((P − Q)2n+1) = τ (P − Q) ∈ R. If A ∈ L2(M, τ ) and U ∈ M is an isometry, then ∥UA − A∥22 ≤ 2∥(I − U )AA∗∥1.
Keywords
Hilbert space,
von Neumann algebra,
normal trace,
measurable operator,
idempotent,
tripotent,
quantum Hall effect
Supporting agencies: The work is performed under the development program of Volga Region Mathematical Center (agreement no. 075-02-2023-944).
About the Authors
A. M. Bikchentaev
Казанский (Приволжский) федеральный университет
Russian Federation
Airat Midkhatovich Bikchentaev,
Volga Region Mathematical Center,18 Kremlyovskaya str., Kazan 420008, Russia
Russian Federation
Airat Midkhatovich Bikchentaev,
Volga Region Mathematical Center,18 Kremlyovskaya str., Kazan 420008, Russia
Mahmoud Khadour
Kazan Federal University
Russian Federation
Russian Federation
Mahmoud Khadour
18 Kremlyovskaya str., Kazan 420008, RussiaReferences
1. J. Avron, R. Seiler, B. Simon, The index of a pair of projections, J. Funct. Anal. 120 (1), 220–237 (1994). DOI: https://doi.org/10.1006/jfan.1994.1031
2. N.J. Kalton, A note on pairs of projections, Bol. Soc. Mat. Mexicana (3) 3 (2), 309–311 (1997). DOI: https://doi.org/10.1007/978-3-319-18796-9_8
3. A.M. Bikchentaev, Differences of idempotents in C∗-algebras, Siberian Math. J. 58 (2), 183–189 (2017). DOI: https://link.springer.com/article/10.1134/S003744661702001X
4. J. Bellissard, A. van Elst, H. Schulz-Baldes, The noncommutative geometry of the quantum Hall effect. Topology and physics, J. Math. Phys. 35 (10), 5373–5451 (1994). DOI: https://doi.org/10.1063/1.530758
5. F. Gesztesy (coordinating Editor), From Mathematical Physics to Analysis: a walk in Barry Simon’s Mathematical Garden, II, Notices Amer. Math. Soc. 63 (8), 878–889 (2016). DOI: http://doi.org/10.1090/noti1412
6. A.M. Bikchentaev, Differences of idempotents in C∗-algebras and the quantum Hall effect, Theoret. and Math. Phys. 195 (1), 557–562 (2018). DOI: https://link.springer.com/article/10.1134/S0040577918040074
7. A.M. Bikchentaev, Concerning the theory of τ-measurable operators affiliated to a semfinite von Neumann algebra, Math. Notes 98 (3), 382–391 (2015). DOI: https://doi.org/10.1134/S0001434615090035
8. M. Takesaki, Theory of operator algebras. I. Encyclopaedia Math. Sci. 124. Operator Algebras and Non-commutative Geometry, 5. Springer-Verlag, Berlin, 2002. DOI: https://doi.org/10.1007/978-1-4612-6188-9
9. M. Takesaki, Theory of operator algebras. II. Encyclopaedia Math. Sci. 125. Operator Algebras and Non-commutative Geometry, 6. Springer-Verlag, Berlin, 2003. DOI: https://doi.org/10.1007/978-3-662-10451-4
10. A.M. Bikchentaev, The algebra of thin measurable operators is directly finite, Constr. Math. Anal. 6 (1), 1–5 (2023). DOI: https://doi.org/10.33205/cma.1181495
11. A.M. Bikchentaev, Ideal spaces of measurable operators affiliated to a semfinite von Neumann algebra, Siberian Math. J. 59 (2), 243–251 (2018). DOI: https://doi.org/10.1134/S0037446618020064
12. A.M. Bikchentaev, On a property of Lp spaces on semifinite von Neumann algebras, Math. Notes 64 (2), 159–163 (1998). DOI: https://doi.org/10.1007/BF02310299
13. A.M. Bikchentaev, Renormalizations of measurable operator ideal spaces affiliated to a semfinite von Neumann algebra, Ufa Math. J. 11 (3), 3–10 (2019). DOI: https://doi.org/10.13108/2019-11-3-3
14. S. G. Krein, Ju. I. Petunin, E. M. Semenov, Interpolation of Linear Operators, Translations of Mathematical Monographs 54, AMS, Providence R.I., 1982.
15. L.G. Brown, H. Kosaki, Jensen’s inequality in semifinite von Neumann algebra, J. Operator Theory 23 (1), 3–19 (1990). URL: https://www.theta.ro/jot/archive/1990-023-001/1990-023-001-001.html
16. A.M. Bikchentaev, On idempotent τ-measurable operators affiliated to a von Neumann algebra, Math. Notes 100 (3-4), 515–525 (2016). DOI: https://doi.org/10.1134/S0001434616090224
17. A.M. Bikchentaev, R.S. Yakushev, Representation of tripotents and representations via tripotents, Linear Algebra Appl. 435 (9), 2156–2165 (2011). DOI: https://doi.org/10.1016/j.laa.2011.04.003
18. A.M. Bikchentaev, Kh. Fawwaz, Differences and commutators of idempotents in C∗- algebras, Russian Math. (Iz. VUZ) 65 (8), 13–22 (2021).DOI: https://link.springer.com/article/10.3103/S1066369X21080028
19. A.N. Sherstnev, Methods of bilinear forms in non-commutative measure and integral theory, Fizmatlit, Moskva, 2008 [in Russian].
20. G. Pisier, Q. Xu, Non-commutative Lp-spaces, Handbook of the geometry of Banach spaces V. 2., P. 1459–1517. North-Holland, Amsterdam, 2003. DOI: https://doi.org/10.1016/S1874-5849(03)80041-4
21. M. Choda, Characterization of approximately inner automorphisms, Proc. Amer. Math. Soc. 84 (2), 231–234 (1982). DOI: https://doi.org/10.1090/S0002-9939-1982-0637174-2
Review
For citations:
Bikchentaev A.M., Khadour M. Differences of idempotents in C∗-algebras and the quantum Hall effect. II. Unbounded idempotents. Mathematics and Theoretical Computer Science. 2023;1(4):35-48. (In Russ.) https://doi.org/10.26907/2949-3919.2023.4.35-48
Views:
182
Email this article 