A.M. Bikchentaev*, S.R. Nasyrov**, E.A. Turilova***
Kazan Federal University, Kazan, 420008 Russia
E-mail: *Airat.Bikchentaev@kpfu.ru, **snasyrov@kpfu.ru, ***Ekaterina.Turilova@kpfu.ru
Received June 6, 2018
Abstract
The paper describes the biography and the main scientific achievements of Professor Anatolij Nikolaevich Sherstnev.
The scientific interests of A.N. Sherstnev are associated with probabilistic metric and normed spaces, as well as with the non-commutative theory of measure and integral. He was the first to suggest that bilinear forms should be used for solving important problems of the general theory of measure on projectors of von Neumann algebras. The fruitful ideas allowed him and his followers to advance the general theory of non-commutative integration in operator algebras in various directions. In particular, the scales of Lp-spaces of integrable bilinear forms were constructed, the theory of conditional expectation in the indicated spaces was developed, possibility of generalizations of the main constructions to the case of Jordan algebras was investigated. Furthermore, some important problems in construction of unbounded measures on projectors of von Neumann algebras were solved.
A.N. Sherstnev published over 100 research works, including the well-known monographs and textbooks. In 1974, he organized the interuniversity scientific seminar "Operator Algebras and Applications''. In 1973–1998, he was the head of the Department of Mathematical Analysis at Kazan State University.
References
1. Segal I.E. A non-commutative extension of abstract integration. Ann. Math., 1953, vol. 57, no. 3, pp. 401–457. doi: 10.2307/1969729.
2. Sherstnev A.N. States on von Neumann algebras. Funct. Anal. Its Appl., 1974, vol. 8, no. 3, pp. 89–90. doi: 10.1007/BF01075708.
3. Sherstnev A.N. Every smooth weight is an l-weight. Izv. Vyssh. Uchebn. Zaved. Mat., 1977, no. 8, pp. 88–91. (In Russian)
4. Trunov N.V., Sherstnev A.N. On the general theory of integration with respect to a weight in algebras of operators. I. Izv. Vyssh. Uchebn. Zaved. Mat., 1978, no. 7, pp. 79–88. (In Russian)
5. Trunov N.V., Sherstnev A.N. On the general theory of integration with respect to a weight in algebras of operators. II. Izv. Vyssh. Uchebn. Zaved. Mat., 1978, no. 12, pp. 88–98. (In Russian)
6. Sherstnev A.N. On the general theory of the measure and integral in von Neumann algebras. Izv. Vyssh. Uchebn. Zaved. Mat., 1982, no. 8, pp. 20–35. (In Russian)
7. Trunov N.V., Sherstnev A.N. Introduction to the theory of noncommutative integration. J. Sov. Math., vol. 27, no. 6, pp. 1504–1523. doi: 10.1007/BF01103856.
8. Sherstnev A.N. Metody bilineinykh form v nekommunikativnoi teorii mery i integrala [Methods of Bilinear Forms in Noncommutative Measure and Integral Theory]. Moscow, Fizmatlit, 2008. 264 p. (In Russian)
9. Shweizer B., Sklar A. Probabilistic Metric Spaces. New York, Amsterdam, Oxford, North-Holland, 1983. 275 p.
10. Dvurecenskij A. Gleason's Theorem and Its Applications. Dordrecht, Kluwer Acad. Publ., 1993. 331 p.
11. Hamhalter J. Quantum Measure Theory. Dordrecht, Kluwer Acad. Publ., 2003. 410 p.
12. Sherstnev A.N. Boolean logics. Uchenye Zapiski Kazanskogo Universiteta, 1968, vol. 128, no. 2, pp. 48–62. (In Russian)
13. Lugovaya G.D., Sherstnev A.N. The Gleason theorem for unbounded measures. Izv. Vyssh. Uchebn. Zaved. Mat., 1980, no. 12, pp. 30–32. (In Russian)
14. Lugovaya G.D., Sherstnev A.N. On the extension problem for unbounded measures on projections to weights. Dokl. Akad. Nauk, 1999, vol. 365, no. 2, pp. 165–166. (In Russian)
15. Dorofeev S.V., Sherstnev A.N. Functions of frame type and their applications. Izv. Vyssh. Uchebn. Zaved. Mat., 1990, no. 4, pp. 23–29. (In Russian)
16. Sherstnev A.N., Turilova E.A. Classes of subspaces affiliated with a von Neumann algebra. Russ. J. Math. Phys., 1999, vol. 6, no. 4, pp. 426–434.
17. Sherstnev A.N. An analog of the Hahn-Banach theorem for commutative semigroups. Russ. J. Math. Phys., 2002, vol. 9, no. 2, pp. 198–201.
18. Stolyarov A.I., Tikhonov O.E., Sherstnev A.N. Characterization of normal traces on von Neumann algebras by inequalities for the modulus. Math. Notes, 2002, vol. 72, nos. 3–4, pp. 411–416. doi: 10.1023/A:1020559623287.
19. Sherstnev A.N. Two-dimensional reductions of the cone of positive diagonal operators in 
. Linear Algebra Appl., 2003, vol. 370, pp. 375–378. doi: 10.1016/S0024-3795(03)00480-4.
20. Bikchentaev A.M., Sherstnev A.N. Projective convex combinations in C*-algebras with the unitary factorization property. Math. Notes, 2004, vol. 76, nos. 3–4, pp. 578–581. doi: 10.1023/B:MATN.0000043487.24773.04.
21. Sherstnev A.N., Tikhonov O.E. On characterization of integrable sesquilinear forms. Math. Slovaca, 2012, vol. 62, no. 6, pp. 1167–1172.
22. Sherstnev A.N. Measures on projections in a W*-algebra of type I2. Funct. Anal. Its Appl., 2013, vol. 47, no. 4, pp. 302–314. doi: 10.1007/s10688-013-0037-5.
23. Lugovaya G.D., Sherstnev A.N. Description of orthogonal vector fields over W*-algebra of type I2. Russ. Math., 2015, vol. 59, no. 4, pp. 28–37. doi: 10.3103/S1066369X15040040.
24. Lugovaya G.D., Sherstnev A.N. Funktsional'nyi analiz: Spetsial'nye kursy [Functional Analysis: Special Courses]. Moscow, Izd. Gruppa URSS, 2013. 256 p. (In Russian)
25. Sherstnev A.N. Matematicheskii i funktsional'nyi analiz: Konspekt lektsii [Mathematical and Functional Analysis: Lecture Notes]. Moscow, Izd. Gruppa URSS, 2018. 376 p. (In Russian)
For citation: Bikchentaev A.M., Nasyrov S.R., Turilova E.A. Anatolij Nikolaevich Sherstnev (on 80th birthday anniversary). Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 3, pp. 590–598. (In Russian)
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