Kazan State University
Scientific seminar
Operator algebras and their applications
Supervised by Prof. A.N.Sherstnev.
The seminar arised within Research Institute of Mechanics and Mathematics named after N.G. Chebotarev in the late 60ths. Beginning 1974, the seminar relocated to the Chair of Mathematical Analysis, where it is now going on to be active. Let us dwell upon the major directions of the research.
1. The Noncommutative Measure and Integration Theory
Because of the progress in the theory of von Neumann algebras and the expansion of its applications during the late 60ths -- early 70ths (that was stimulated by the fruitful Tomita-Takesaki theory and the theory of normal weights), extending the noncommutative integration theory by I. Segal (1953) has become urgent to normal weights being noncentral analogs to integrals by unbounded measures on the class of bounded functions. This attracted members of the seminar in the 70ths. For the first time, a solution to the above problem was obtained at our seminar in 1972-1978 [52,54,48]. This integration scheme was essentially advanced during the subsequent 2 or 3 years. The principal idea, on which the construction of a noncommutative L 1 associated with a faithful normal semifinite weight on a von Neumann algebra M, acting in a Hilbert space H was based, is that of its representation as a space of integrable bilinear forms on the dense in H lineal of weight intrinsically related to the weight [52,54]. The well-developed machinery of bilinear forms has also turned out to be extremely fruitful (as we will see further) for solving other important problems of the general measure theory on projections in a von Neumann algebra. Let us state the main results obtained when solving this problem.
1) A theory of L1-spaces associated with a faithful normal semifinite weight has been built. Their representation as a space of integrable bilinear forms on the lineal of weight has been suggested. A dual description has been obtained for L1 showing it to be isometrically isomorphic to the predual of the von Neumann algebra M. The isomorphism naturally preserves the natural order structures in L1 and the predual [52,54,48].
2) A scale of spaces Lp, associated with a locally measurable weight on a semifinite von Neumann algebra has been constructed. These spaces have been described in terms of locally measurable operators and bilinear forms [45]. Real spaces have been constructed and represented via integrals of real functions by generalized decompositions of identity [39]. Conditions for a real Borel function f, ensuring that the corresponding operator function f(A) is continuous in the topology of convergence in measure in the spaces Lp associated with a faithful normal semifinite trace on a von Neumann algebra has been found [40]. Constructing and studying Lp-spaces associated with a state on a von Neumann algebra M has also been implemented in [11] by applying the scale Lp,a (a belongs to [0,1]) introduced in [46]. In this approach, the spaceLp,a for each fixed a in the complex interpolation scale constructed by the Banach couple (L1,a,M) serves to be an analog to the space Lp. The spaces Lp,a form a complex interpolation scale for every fixed p. A fairly general scheme for constructing Lp-type spaces has been suggested within an approach to the axiomatics of quantum mechanics that describes states of the system as elements in a convex set [41].
A theory of conditional expectation on the space L1 of integrable bilinear forms has been developed. A relation of this concept to the usual one of a conditional expectation on von Neumann algebras has been stated [47,48(II)].
3) The main results in the integration theory with respect to a trace and the theory of trace inequalities in von Neumann algebras have been extended to the spaces in spectral duality of E. Alfsen and F. Schultz [42-44].
New characterizations of traces among weights on von Newmann algedras have been obtained [65]. A non-commutative analog of A.V. Bukhvalov and G.Ya Lozanovskiy theory on ``near-compactness'' properties of convex sets in Banach function spaces has been suggested [56], [57]. This has been applied to the study of weak sequential completeness of factor spaces [66].
4) A general method for constructing noncommutative F-normed ideal spaces (in particular, Orlicz spaces) associated with a semiadditive measure on the projections of a von Neumann algebra has been suggested. Their topological and order properties has been investigated [6]. The Kothe duality theory of rearrangement invariant spaces of measurable operators has been extended over the normed ideal spaces [7]. A new majorization property of products of measurable operators has been established [8].
The properties of the topology of convergence in measure on semifinite von Neumann algebras has been investigated in [60, 61]. For a wide class of C*-algebras representation of elements as finite sums of products of projections has been established in [62, 63]. The properties of the set of semi-orthogonal projections has been investigated in [64].
The successful activity of the seminar has promoted its growth as an authority among the mathematical community. It was Kazan State University who, together with Moscow and St. Petersburg State Universities, organized and held Summer Mathematical Schools on Noncommutative Probability Theory in 1971 and 1978 [5,51]. A subject issue (1982, no. 8) of "Izv. Vuzov. Matematika" was dedicated to Noncommutative Integration and related topics in Probability Theory and Mathematical Physics.
Another direction of research at the seminar is the structure of measures on the orthoprojections in a von Neumann algebra. A fundamental result by A. Gleason (1957) on countably additive measures on the closed subspaces of a separable Hilbert space (together with the possibility to apply that result in constructing nice axiomatic models for quantum mechanics) stimulated interest to the so-called linearity problem. That was of the description of measures on orthoprojections and was formulated to be an extension problem of a measure on projections in a von Neumann algebra to a linear functional. At our seminar, it was M.S.Matveichuk who succeeded in solving this problem [19,20] and had published the solution before investigators abroad did. (E. Christensen (1982) and F. Yeadon (1983-1984) also obtained the result.)
Developing an exhaustive measure theory on projections implies (as in the classical "commutative" case) the study of not necessarily bounded measures. In the classical theory, an unbounded measure is either a measure on a sigma-ring of sets taking nonnegative values or a measure on a sigma-algebra of sets with values in the completed real half-line. These approaches are substantially equivalent. A sigma-orthoadditive mapping from Mpr (the set of all orthoprojections in a von Neumann algebra M) to the completed real half-line is a natural analog to the latter definition for measures on the projections in a von Neumann algebra M. In the noncommutative case, the first definition transforms to a sigma-orthoadditive mapping on an "ideal" m in Mpr to the nonnegative half-line. But, these approaches are now not equivalent, as an unbounded measure on an ideal need not extend to a measure on all projections [50,55]. The study of unbounded measures within the latter approach has turned out to be successful. The problem of describing the unbounded measures has completely been s
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Giới thiệu với các bạn về trường mình. Prof. A.N.Sherstnev là ông cụ già hướng dẫn của mình ở tổ Giải tích và giải tích hàm.
Nếu những bậc cao thủ muốn trao đổi với thầy thì cứ liên hệ trực tiếp!
Bài báo được lấy tại trang web của trường TH Kazan.
http://www.kcn.ru/ta...iimm/sem_04.htm
Kazan State University
Scientific seminar
Operator algebras and their applications
Supervised by Prof. A.N.Sherstnev.
The seminar arised within Research Institute of Mechanics and Mathematics named after N.G. Chebotarev in the late 60ths. Beginning 1974, the seminar relocated to the Chair of Mathematical Analysis, where it is now going on to be active. Let us dwell upon the major directions of the research.
1. The Noncommutative Measure and Integration Theory
Because of the progress in the theory of von Neumann algebras and the expansion of its applications during the late 60ths -- early 70ths (that was stimulated by the fruitful Tomita-Takesaki theory and the theory of normal weights), extending the noncommutative integration theory by I. Segal (1953) has become urgent to normal weights being noncentral analogs to integrals by unbounded measures on the class of bounded functions. This attracted members of the seminar in the 70ths. For the first time, a solution to the above problem was obtained at our seminar in 1972-1978 [52,54,48]. This integration scheme was essentially advanced during the subsequent 2 or 3 years. The principal idea, on which the construction of a noncommutative L 1 associated with a faithful normal semifinite weight on a von Neumann algebra M, acting in a Hilbert space H was based, is that of its representation as a space of integrable bilinear forms on the dense in H lineal of weight intrinsically related to the weight [52,54]. The well-developed machinery of bilinear forms has also turned out to be extremely fruitful (as we will see further) for solving other important problems of the general measure theory on projections in a von Neumann algebra. Let us state the main results obtained when solving this problem.
1) A theory of L1-spaces associated with a faithful normal semifinite weight has been built. Their representation as a space of integrable bilinear forms on the lineal of weight has been suggested. A dual description has been obtained for L1 showing it to be isometrically isomorphic to the predual of the von Neumann algebra M. The isomorphism naturally preserves the natural order structures in L1 and the predual [52,54,48].
2) A scale of spaces Lp, associated with a locally measurable weight on a semifinite von Neumann algebra has been constructed. These spaces have been described in terms of locally measurable operators and bilinear forms [45]. Real spaces have been constructed and represented via integrals of real functions by generalized decompositions of identity [39]. Conditions for a real Borel function f, ensuring that the corresponding operator function f(A) is continuous in the topology of convergence in measure in the spaces Lp associated with a faithful normal semifinite trace on a von Neumann algebra has been found [40]. Constructing and studying Lp-spaces associated with a state on a von Neumann algebra M has also been implemented in [11] by applying the scale Lp,a (a belongs to [0,1]) introduced in [46]. In this approach, the spaceLp,a for each fixed a in the complex interpolation scale constructed by the Banach couple (L1,a,M) serves to be an analog to the space Lp. The spaces Lp,a form a complex interpolation scale for every fixed p. A fairly general scheme for constructing Lp-type spaces has been suggested within an approach to the axiomatics of quantum mechanics that describes states of the system as elements in a convex set [41].
A theory of conditional expectation on the space L1 of integrable bilinear forms has been developed. A relation of this concept to the usual one of a conditional expectation on von Neumann algebras has been stated [47,48(II)].
3) The main results in the integration theory with respect to a trace and the theory of trace inequalities in von Neumann algebras have been extended to the spaces in spectral duality of E. Alfsen and F. Schultz [42-44].
New characterizations of traces among weights on von Newmann algedras have been obtained [65]. A non-commutative analog of A.V. Bukhvalov and G.Ya Lozanovskiy theory on ``near-compactness'' properties of convex sets in Banach function spaces has been suggested [56], [57]. This has been applied to the study of weak sequential completeness of factor spaces [66].
4) A general method for constructing noncommutative F-normed ideal spaces (in particular, Orlicz spaces) associated with a semiadditive measure on the projections of a von Neumann algebra has been suggested. Their topological and order properties has been investigated [6]. The Kothe duality theory of rearrangement invariant spaces of measurable operators has been extended over the normed ideal spaces [7]. A new majorization property of products of measurable operators has been established [8].
The properties of the topology of convergence in measure on semifinite von Neumann algebras has been investigated in [60, 61]. For a wide class of C*-algebras representation of elements as finite sums of products of projections has been established in [62, 63]. The properties of the set of semi-orthogonal projections has been investigated in [64].
The successful activity of the seminar has promoted its growth as an authority among the mathematical community. It was Kazan State University who, together with Moscow and St. Petersburg State Universities, organized and held Summer Mathematical Schools on Noncommutative Probability Theory in 1971 and 1978 [5,51]. A subject issue (1982, no. 8) of "Izv. Vuzov. Matematika" was dedicated to Noncommutative Integration and related topics in Probability Theory and Mathematical Physics.
Another direction of research at the seminar is the structure of measures on the orthoprojections in a von Neumann algebra. A fundamental result by A. Gleason (1957) on countably additive measures on the closed subspaces of a separable Hilbert space (together with the possibility to apply that result in constructing nice axiomatic models for quantum mechanics) stimulated interest to the so-called linearity problem. That was of the description of measures on orthoprojections and was formulated to be an extension problem of a measure on projections in a von Neumann algebra to a linear functional. At our seminar, it was M.S.Matveichuk who succeeded in solving this problem [19,20] and had published the solution before investigators abroad did. (E. Christensen (1982) and F. Yeadon (1983-1984) also obtained the result.)
Developing an exhaustive measure theory on projections implies (as in the classical "commutative" case) the study of not necessarily bounded measures. In the classical theory, an unbounded measure is either a measure on a sigma-ring of sets taking nonnegative values or a measure on a sigma-algebra of sets with values in the completed real half-line. These approaches are substantially equivalent. A sigma-orthoadditive mapping from Mpr (the set of all orthoprojections in a von Neumann algebra M) to the completed real half-line is a natural analog to the latter definition for measures on the projections in a von Neumann algebra M. In the noncommutative case, the first definition transforms to a sigma-orthoadditive mapping on an "ideal" m in Mpr to the nonnegative half-line. But, these approaches are now not equivalent, as an unbounded measure on an ideal need not extend to a measure on all projections [50,55]. The study of unbounded measures within the latter approach has turned out to be successful. The problem of describing the unbounded measures has completely been solved for the algebra of all bounded operators on a separable Hilbert space [15,13]. This has been accomplished by applying methods of the bilinear forms theory. (For the first time, those methods were utilized in the study of measures on ideals [50]. Later on, these were essentially refined by applying a number of structural theorems by B. Simon (1978).) This result is a generalization of the aforementioned classical Gleason theorem to unbounded measures. The one has allowed for introducing an L1-space for a wide class of unbounded measures on projections and representing the space as a predual of a suitable von Neumann algebra [16].
The extension problem of an unbounded measure on projections to a weight (or the linearity problem for unbounded measures) has also naturally arised. For all the sufficient conditions obtained of extending a measure to a weight, the problem has turned out to negatively answer for arbitrary finitely additive measures. An example of an unbounded semifinite finitely additive measure on the projections in a von Neumann algebra with no direct summand of type I_2 nonextendable to a weight has been given [18,58].
Unbounded analogs to vector orthoadditive measures on orthoprojections in a von Neumann algebra taking values in a Hilbert space have been introduced and studied [14,17]. Also, order properties of orthogonal vector fields and their relationships with topological properties are studied. It turns out, in particular, that the order property of an orthogonal vector field being normal substantially differs from the corresponding scalar analog. A characterization of a broad class of orthogonal vector fields in the class of linear mappings is obtained [59].
In the early 70ths, the study of an analog to the Gleason theorem for not necessarily positive signed measures on orthoprojections was begun at the seminar [53]. It was established that the assertion as the Gleason theorem was valid for signed measures on B(H), the algebra of all bounded operators on a Hilbert space H (dim H > 2) provided that the restriction of a signed measure to the one-dimensional orthoprojections was bounded (which was equivalent to the boundedness of the signed measure). More generally, the description of completely additive signed measures on a von Neumann algebra M, was immediately related to their boundedness. Let m be a completely additive signed measure. Is it bounded?
It was Gleason (1957) who observed that this question answers in the negative for the algebra B(H) with finite-dimensional H. Only in 1990, we affirmatively answered the question for B(H) with infinite-dimensional H [10]. Later on, S.V. Dorofeev [9] proved that every signed measure is bounded (and thus is a linear combination of measures) in case the von Neumann algebra M has no direct summand of type I_n (n is finite).
Classes of orthoclosed and splitting subspaces of a unitary space that are affiliated to a von Neumann algebra are considered. A topological definition of measures on the above classes of subspaces is given and results on the relationships of these measures to measures on orthoprojections of the von Neumann algebra are presented [67].
2. Orthomodular Posets
The orthoprojections in a commutative von Neumann algebra naturally form a Boolean algebra. If the von Neumann algebra in question is noncommutative, we come to a more general structure that may naturally be viewed as an analog to the set of possible propositions about a suitable quantum-mechanical system. An abstract "Boolean" analog to this structure is called an orthomodular poset (abbr., OMP) or a quantum logic. The study of these was initiated by G. Birkhoff and J. von Neumann in the 30ths.
In Kazan, the study of OMPs was begun at our seminar in the late 60ths [49]. In that paper, the dimension theory by Loomis of orthomodular lattices was extended over the arbitrary OMPs. That approach was developed in [12]. The advanced research in OMPs has systematically been held by P.G. Ovchinnikov beginning the 80ths. He has suggested a new axiomatics for OMPs based on their combinatorial characterization [27], has described the automorphisms of the poset of skew projections on a Hilbert space [28], has proved a well-known Ptak-Pulmannova conjecture on the Jauch-Piron property [29], has extended the Birkhoff theorem on doubly stochastic matrices to hypergraphs [30], has given the first known example of an atomistic nonorthoatomistic OMP [31], has found an exact topological analog to the concept of an orthoposet [32], and has developed a general theory of measures and signed measures on finite concrete logics [33, 34]. F.F. Sultanbekov has exhaustively studied a number of particular finite OMPs and concrete logics [35,36] and has successfully applied a computer in the study of finite concrete logics [37]. In doing that, he has developed a series of specific routines [37,38].
Along with studying general OMPs, the search of analogs to the Gleason theorem for some interesting particular ones such as OMPs of skew projections on Banach spaces or vector spaces over "unusual" fields was held [23-25,68].
3. Probability Measures on Banach Spaces
The problem of (sigma)-additivity of a cylindrical probability is one of the most important and difficult in the theory of probability distributions on infinite-dimensional Banach spaces. That problem is closely related to trajectories of the most important stochastic processes. The classical theorems by N. Wiener and A.N. Kolmogorov were the first to be obtained in this field. The most efficient criteria of the (sigma)-additivity are topological ones that are infinite-dimensional analogs to the Bochner theorem. The description problem of those Banach spaces admitting an analog to the Bochner theorem has been motivated by the basic research by V.V. Sazonov (1958), R.A. Minlos (1959), and Yu.V. Prokhorov (1960).
In solving this and related problems, D.Kh. Mushtari has developed a topological technique for studying the weak compactness and the (sigma)-additivity of cylindrical probabilities in Banach spaces. This technique has allowed for the complete description of those Banach spaces admitting a topological solution to the problem. He has extended a number of major results over the topological or metric linear spaces. A number of probability characterizations of the Frechet nuclear spaces have also been obtained. Stable probabilities on Banach spaces have been studied [2,21,22,26].
REFERENCES
a) Monographs and Surveys
1. Measures on Projections and Orthomodular Posets (Dedicated to A. N. Sherstnev on the occasion of his 60th Anniversary). -- Kazan Math. Soc., (1998) 118 pp. 2. Mushtari D. H., Probabilities and Topologies on Linear Spaces. -- Kazan Mathematics Foundation, 1996. 248 p. 3.Trunov N. V. and Sherstnev A. N., Introduction to the noncommutative integration theory. -- Sovrem. problemy math. Noveysh. dostizheniya, vol. 27 (Itogi nauki i techniki. VINITI AN USSR), M., 1985, 167-190. (Russian) 4. Sherstnev A. N., On certain problems in the theory of unbounded measures on projections. -- Atti Sem. Mat. Fis. Univ. Modena, 1994, v. 42, 357-366.
b) Papers (1968-1999)
5. Begovatov E. A., Minlos R. A. and Sherstnev A. N., A report on Summer mathematical School ``Non-commutative Theory of Probability". -- Uspehi mat. nauk, 1978, vol. 34, no. 5, 242-246. (Russian) 6. Bikchentaev A. M., On noncommutative function spaces. --- Amer. Math. Soc. Transl. (2), 1992, v. 154, 179-187. 7. Bikchentaev A. M., On a property Lp-spaces in semifinite von Neumann algebras. --- Mat. Zametki, 1998, vol. 64, no. 2, 185-190. (Russian) 8. Bikchentaev A. M., Majorization for products of measurable operators. -- Inter. J. Theor. Physics, 1998, v. 37, no. 1, 571-576. 9. Dorofeev S., On the problem of boundedness of a signed measure on projections of a von Neumann algebra. -- J. Funct. Anal., 1992, v. 103, no. 1, 33-40. 10. Dorofeev S. and Sherstnev A.N., Frame-type funcions and their applications. -- Izv. vuzov. Matematika, 1990, no. 4, 23-29 (Russian); see also [1], 2-13. 11. Zolotarev A.A., Lp-spaces wiyh respect to a state in von Neumann algebra and the interpolation. -- Izv. vuzov. Matematika, 1982, no. 8, 36-43. (Russian) 12. Kalinin V.V., Orthomodular posets with dimensionality. -- Algebra and Logics, 1976, vol. 15, 535-557. (Russian) 13. Lugovaya G.D., Bilinear forms determining measures on projections. --- Izv. vuzov. Matematika, 1983, no. 2, 88. (Russian) 14. Lugovaya G.D., On unbounded orthogonal measures on projections on a Hilbert space. -- Izv. vuzov. Matematika, 1993, no. 12, 43-46. (Russian) 15. Lugovaya G.D. and Sherstnev A.N., On Gleason's theorem for unbounded measures. -- Izv. vuzov. Matematika, 1980, no. 12, 30-32. (Russian) 16. Lugovaya G.D. and Sherstnev A.N., A model of L1-space with respect to an unbounded measure on projections. -- Izv. vuzov. Matematika, 1984, no. 12, 35-42. (Russian) 17. Lugovaya G., Sherstnev A., On extending an unbounded orthogonal vector measure in a semifinite von Neumann algebra to a weight. -- Demonstratio Math., 1997, v. 30, no. 3, 591-598. 18. Lugovaya G.D. and Sherstnev A.N., On the extension problem of unbounded measures on projections to a weight. -- Doklady AN, 1999, vol. 365, no. 2, 165-166. (Russian) 19. Matvejchuk M.S., A theorem on states on quantum logics, I. -- Teoret. i Mat. Fiz., 1980, vol. 45, 2, 244-250; II. -- ibid., 1981, vol. 48, 3, 261-265. (Russian) 20. Matvejchuk M.S., Description of finite measures on semifinite algebras. -- Funct. Anal. i Prilozhen., 1981, vol. 15, no. 3, 41-53. (Russian) 21. Mushtari D. H., Certain general questions of the theory of probability measures in linear spaces. -- Teoriya Veroyatn. i Primen., 1973, vol. 18, no. 1, 66 - 77. (Russian) 22. Mushtari D. H., Sur l'existence d'une topologie du type Sazonov sur un espace de Banach. -- Semin. Maurey - Schwartz, 1975-76, expose 17, 11 p. 23. Mushtari D. H., Logics of projections in Banach spaces. -- Izv. vuzov. Matematika, 1989, 8, 59-70. (Russian) 24. Mushtari D. H., Orthoadditive functions on idempotents. -- In [1] 38-83. 25. Mushtari D. H. and Matvejchuk M.S., Charges on the logic of skew projections. --- Dokl. Akad. Nauk SSSR, 1985, vol. 283, 43-46. (Russian) 26. Mushtari D. H. and Chuprunov A.N., Sufficient topologies and norms. -- Teoriya Veroyatn. i Primen., 1983, vol. 28, no. 4, 700 - 714. (Russian) 27. Ovchinnikov P. G., Another combinatorial characterization of orthomodular posets. -- In [1], 87-94. 28. Ovchinnikov P. G., Automorphisms of the poset of skew projections. -- J. Funct. Anal., 1993, v. 115, 184-189. 29. Ovchinnikov P. G., Countable Jauch-Piron logics. -- Int. J. Theor. Phys., 1993, v. 32, 885-890. 30. Ovchinnikov P. G., A theorem on pure states on a hypergraph. --- Math. Nachr., 1994, v. 167, 275-278. 31. Ovchinnikov P. G., On alternative orthomodular posets. -- Demonstratio Math., 1994, v. 27, 89-93. 32. Ovchinnikov P. G., Exact topological analogs to orthoposets. -- Proc. Amer. Math. Soc., 1997, v. 125, 2839-2841. 33. Ovchinnikov P. G., Measures on finite concrete logics. --- Proc. Amer. Math. Soc., 1999, v. 127, 1957-1966. 34. Ovchinnikov P. G., Sultanbekov F. F., Finite concrete logics: their structure and measures on them. -- Int. J. Theor. Phys., 1998, v. 37, 147-153. 35. Sultanbekov F. F., Set logics and their representations. --- Int. J. Theor. Phys., 1993, v. 32, 2177-2186. 36. Sultanbekov F. F., Operation associated to partitions and best extension of signed measures on set logics. -- Int. J. Theor. Phys., 1995, v. 34, 1735-1739. 37. Sultanbekov F. F., Solving problems on finite concrete logics with the help of a PC. -- Int. J. Theor. Phys., 1998, v. 37, 199-202. 38. Sultanbekov F. F., Routines for investigation of concrete logics. --- In [1], 95-113. 39. Tikhonov O. E., Lp-spaces with respect to a weight on a von Neumann algebra. -- Izv. vuzov. Matematika, 1982, 8, 76-78. (Russian) 40. Tikhonov O. E., The continuity of operator functions in topologies associated with a trace on a von Neumann algebra. -- Izv. vuzov. Matematika, 1987, 1, 77-79. (Russian) 41. Tikhonov O. E., Banach spaces, associated with a state space, and the information function. -- In: Konstr. teoriya funktsiy i funkts. analiz (Kazan), 1990, no. 7, 67-90. (Russian) 42. Tikhonov O. E., Spectral decomposition with respect to a weight in a base-norm space. -- Izv. vuzov. Matematika, 1991, no. 1, 73-80. (Russian) 43. Tikhonov O. E., Spectral theory for base-norm spaces. -- In: Konstr. teoriya funktsiy i funkts. analiz (Kazan), 1992, no. 8, 76-91. (Russian) 44. Tikhonov O. E., Trace inequalities for spaces in spectral duality. -- Studia Mathematica, 1993, v. 104, no. 1, 99-110. 45. Trunov N.V., Lp-spaces, associated with a weight on a semifinite von Neumann algebra. -- In: Konstr. teoriya funktsiy i funkts. analiz (Kazan), 1981, no. 3, 88-92. (Russian) 46. Trunov N.V., On the theory of noncommutative spaces L1 and L2. - In: Konstr. teoriya funktsiy i funkts. analiz (Kazan), 1983, no. 4, 96-105. (Russian) 47. Trunov N.V. and Sherstnev A.N., Conditional expectations in a scheme of noncommutative probability theory. -- Trans. VIII Prague Conf. on Inform. Theory, Stat. Decis. Funct. Random Proc., vol. B, Prague, 1978, 287-299. (Russian) 48. Trunov N.V. and Sherstnev A.N., On the general noncommutative integration theory with respect to a weight in operator algebras, I. -- Izv. vuzov. Matematika, 1978, no. 7, 79-88; II. -- Izv. vuzov. Matematika, 1978, no. 12, 88-99. (Russian) 49. Sherstnev A.N., On boolean logics. -- Uchen. Zapiski KGU, 1968, vol. 128, 48-62. (Russian) 50. Sherstnev A.N., On the representation of measures on orthoprojections on a Hilbert space by means of bilinear forms. -- Izv. vuzov. Matematika, 1970, 9, 90-97. (Russian) 51. Sherstnev A.N., Summer Mathematical School on Noncoomutative Probability Theory. -- Uspehi mat. nauk, 1972, vol. 27, no. 1, 287-288. (Russian) 52. Sherstnev A.N., On the general theory of states on von Neumann algebras. -- Funct. Anal. i Prilozhen., 1974, vol. 8, no. 3, 89-90. (Russian) 53. Sherstnev A.N., On the notion of a charge in a noncommutative scheme of measure theory. -- In: Veroyatn. metody i Kibernetika (Kazan), 1974, no. 10-11, 68-72. (Russian) 54. Sherstnev A.N., Every smooth weight is a l-weight. -- Izv. vuzov. Matematika, 1977, 8, 88-91. (Russian) 55. Sherstnev A.N., Measures on ideals of orthoprojectors in von Neumann algebra. -- Izv. vuzov. Matematika, 1978, no. 10, 108-109. (Russian) 56. Skvortsova G.Sh. and Tikhonov O.E., Convex sets in noncommutative L1-spaces that are closed in the topology of local convergence in measure, -- Izv. vuzov. Matematika, 1998, 8, 48-55. (Russian). English translation in Russian Math. (Iz. VUZ), 1998, vol. 42, no. 8, 46-52 (1999). 57. Tikhonov O.E., On the Lozanovskiy class of condensing operators and its applications to the non-commutative integration, -- Israel Math. Conf. Proc., 1999, vol. 13, 203-207.
c) Papers (since 2000)
58. Lugovaya G.D. and Sherstnev A.N., On the extension problem for unbounded measures on projections. -- Math. Slovaca, 2000, vol. 50, no. 4, 473-481. 59. Lugovaya G.D. and Sherstnev A.N., Order properties of orthogonal vector fields and a representation theorem. -- Russian Journ. of Math. Physics, 2001, vol. 8, no. 2, 172-179. 60. Bikchentaev A.M., Minimality of convergence in measure topologies on finite von Neumann algebras. -- Mat. Zametki, 2004, vol. 75, no. 3, 342-349. (Russian) 61. Bikchentaev A.M., The continuity of multiplication for two topologies associated with a semifinite trace on von Neumann algebra. -- Lobachevskii J. Math., 2004, vol. 14, 17-24. 62. Bikchentaev A.M., Representation of linear operators in a Hilbert space in the form of finite sums of products of projections. -- Doklady AN, 2003, vol. 393, no. 4, 444-447. (Russian) 63. Bikchentaev A.M. and Sherstnev A.N., On projection-convex combinations in C*-algebras with the unitary factorization property. -- Mat. Zametki, 2004, vol. 76, no. 6. (Russian) 64. Bikchentaev A.M., Semi-orthogonal projectors in a Hilbert space. -- In book: At the turn of the 20-th--21st centuries, Kazan Mat. Obs., Kazan, 2003, 108-114. (Russian) 65. Stolyarov A.I., Tikhonov O.E. and Sherstnev A.N., Characterization of normal traces on von Neumann algebras by inequalities for the modulus, Mat. Zametki, 2002, vol. 72, 448-454; (Russian). English translation: Math. Notes, 2002, vol. 72, 411-416. 66. Skvortsova G.Sh., On weak sequential completeness of factor spaces of the space of integrable operators, Izv. vuzov. Matematika, 2002, 9, 71-74. (Russian) 67. Sherstnev A.N. and Turilova E.A., Classes of subspaces affiliated to a von Neumann algebra, Russian Journ. of Math. Physics, 1999, vol. 6, no. 4, 426-434. 68. Mushtari D.H., Some remarks on measures on orthogonal rational projections and the rational sphere, Lobachevskii Journ. of Math., 2001, vol. 9., 47-53.
Last revised: September, 30, 2004.
E-mail: [email protected]
Phone: (8432)315160
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Kazan State University
Scientific seminar
Operator algebras and their applications
Supervised by Prof. A.N.Sherstnev.
The seminar arised within Research Institute of Mechanics and Mathematics named after N.G. Chebotarev in the late 60ths. Beginning 1974, the seminar relocated to the Chair of Mathematical Analysis, where it is now going on to be active. Let us dwell upon the major directions of the research.
1. The Noncommutative Measure and Integration Theory
Because of the progress in the theory of von Neumann algebras and the expansion of its applications during the late 60ths -- early 70ths (that was stimulated by the fruitful Tomita-Takesaki theory and the theory of normal weights), extending the noncommutative integration theory by I. Segal (1953) has become urgent to normal weights being noncentral analogs to integrals by unbounded measures on the class of bounded functions. This attracted members of the seminar in the 70ths. For the first time, a solution to the above problem was obtained at our seminar in 1972-1978 [52,54,48]. This integration scheme was essentially advanced during the subsequent 2 or 3 years. The principal idea, on which the construction of a noncommutative L 1 associated with a faithful normal semifinite weight on a von Neumann algebra M, acting in a Hilbert space H was based, is that of its representation as a space of integrable bilinear forms on the dense in H lineal of weight intrinsically related to the weight [52,54]. The well-developed machinery of bilinear forms has also turned out to be extremely fruitful (as we will see further) for solving other important problems of the general measure theory on projections in a von Neumann algebra. Let us state the main results obtained when solving this problem.
1) A theory of L1-spaces associated with a faithful normal semifinite weight has been built. Their representation as a space of integrable bilinear forms on the lineal of weight has been suggested. A dual description has been obtained for L1 showing it to be isometrically isomorphic to the predual of the von Neumann algebra M. The isomorphism naturally preserves the natural order structures in L1 and the predual [52,54,48].
2) A scale of spaces Lp, associated with a locally measurable weight on a semifinite von Neumann algebra has been constructed. These spaces have been described in terms of locally measurable operators and bilinear forms [45]. Real spaces have been constructed and represented via integrals of real functions by generalized decompositions of identity [39]. Conditions for a real Borel function f, ensuring that the corresponding operator function f(A) is continuous in the topology of convergence in measure in the spaces Lp associated with a faithful normal semifinite trace on a von Neumann algebra has been found [40]. Constructing and studying Lp-spaces associated with a state on a von Neumann algebra M has also been implemented in [11] by applying the scale Lp,a (a belongs to [0,1]) introduced in [46]. In this approach, the spaceLp,a for each fixed a in the complex interpolation scale constructed by the Banach couple (L1,a,M) serves to be an analog to the space Lp. The spaces Lp,a form a complex interpolation scale for every fixed p. A fairly general scheme for constructing Lp-type spaces has been suggested within an approach to the axiomatics of quantum mechanics that describes states of the system as elements in a convex set [41].
A theory of conditional expectation on the space L1 of integrable bilinear forms has been developed. A relation of this concept to the usual one of a conditional expectation on von Neumann algebras has been stated [47,48(II)].
3) The main results in the integration theory with respect to a trace and the theory of trace inequalities in von Neumann algebras have been extended to the spaces in spectral duality of E. Alfsen and F. Schultz [42-44].
New characterizations of traces among weights on von Newmann algedras have been obtained [65]. A non-commutative analog of A.V. Bukhvalov and G.Ya Lozanovskiy theory on ``near-compactness'' properties of convex sets in Banach function spaces has been suggested [56], [57]. This has been applied to the study of weak sequential completeness of factor spaces [66].
4) A general method for constructing noncommutative F-normed ideal spaces (in particular, Orlicz spaces) associated with a semiadditive measure on the projections of a von Neumann algebra has been suggested. Their topological and order properties has been investigated [6]. The Kothe duality theory of rearrangement invariant spaces of measurable operators has been extended over the normed ideal spaces [7]. A new majorization property of products of measurable operators has been established [8].
The properties of the topology of convergence in measure on semifinite von Neumann algebras has been investigated in [60, 61]. For a wide class of C*-algebras representation of elements as finite sums of products of projections has been established in [62, 63]. The properties of the set of semi-orthogonal projections has been investigated in [64].
The successful activity of the seminar has promoted its growth as an authority among the mathematical community. It was Kazan State University who, together with Moscow and St. Petersburg State Universities, organized and held Summer Mathematical Schools on Noncommutative Probability Theory in 1971 and 1978 [5,51]. A subject issue (1982, no. 8) of "Izv. Vuzov. Matematika" was dedicated to Noncommutative Integration and related topics in Probability Theory and Mathematical Physics.
Another direction of research at the seminar is the structure of measures on the orthoprojections in a von Neumann algebra. A fundamental result by A. Gleason (1957) on countably additive measures on the closed subspaces of a separable Hilbert space (together with the possibility to apply that result in constructing nice axiomatic models for quantum mechanics) stimulated interest to the so-called linearity problem. That was of the description of measures on orthoprojections and was formulated to be an extension problem of a measure on projections in a von Neumann algebra to a linear functional. At our seminar, it was M.S.Matveichuk who succeeded in solving this problem [19,20] and had published the solution before investigators abroad did. (E. Christensen (1982) and F. Yeadon (1983-1984) also obtained the result.)
Developing an exhaustive measure theory on projections implies (as in the classical "commutative" case) the study of not necessarily bounded measures. In the classical theory, an unbounded measure is either a measure on a sigma-ring of sets taking nonnegative values or a measure on a sigma-algebra of sets with values in the completed real half-line. These approaches are substantially equivalent. A sigma-orthoadditive mapping from Mpr (the set of all orthoprojections in a von Neumann algebra M) to the completed real half-line is a natural analog to the latter definition for measures on the projections in a von Neumann algebra M. In the noncommutative case, the first definition transforms to a sigma-orthoadditive mapping on an "ideal" m in Mpr to the nonnegative half-line. But, these approaches are now not equivalent, as an unbounded measure on an ideal need not extend to a measure on all projections [50,55]. The study of unbounded measures within the latter approach has turned out to be successful. The problem of describing the unbounded measures has completely been solved for the algebra of all bounded operators on a separable Hilbert space [15,13]. This has been accomplished by applying methods of the bilinear forms theory. (For the first time, those methods were utilized in the study of measures on ideals [50]. Later on, these were essentially refined by applying a number of structural theorems by B. Simon (1978).) This result is a generalization of the aforementioned classical Gleason theorem to unbounded measures. The one has allowed for introducing an L1-space for a wide class of unbounded measures on projections and representing the space as a predual of a suitable von Neumann algebra [16].
The extension problem of an unbounded measure on projections to a weight (or the linearity problem for unbounded measures) has also naturally arised. For all the sufficient conditions obtained of extending a measure to a weight, the problem has turned out to negatively answer for arbitrary finitely additive measures. An example of an unbounded semifinite finitely additive measure on the projections in a von Neumann algebra with no direct summand of type I_2 nonextendable to a weight has been given [18,58].
Unbounded analogs to vector orthoadditive measures on orthoprojections in a von Neumann algebra taking values in a Hilbert space have been introduced and studied [14,17]. Also, order properties of orthogonal vector fields and their relationships with topological properties are studied. It turns out, in particular, that the order property of an orthogonal vector field being normal substantially differs from the corresponding scalar analog. A characterization of a broad class of orthogonal vector fields in the class of linear mappings is obtained [59].
In the early 70ths, the study of an analog to the Gleason theorem for not necessarily positive signed measures on orthoprojections was begun at the seminar [53]. It was established that the assertion as the Gleason theorem was valid for signed measures on B(H), the algebra of all bounded operators on a Hilbert space H (dim H > 2) provided that the restriction of a signed measure to the one-dimensional orthoprojections was bounded (which was equivalent to the boundedness of the signed measure). More generally, the description of completely additive signed measures on a von Neumann algebra M, was immediately related to their boundedness. Let m be a completely additive signed measure. Is it bounded?
It was Gleason (1957) who observed that this question answers in the negative for the algebra B(H) with finite-dimensional H. Only in 1990, we affirmatively answered the question for B(H) with infinite-dimensional H [10]. Later on, S.V. Dorofeev [9] proved that every signed measure is bounded (and thus is a linear combination of measures) in case the von Neumann algebra M has no direct summand of type I_n (n is finite).
Classes of orthoclosed and splitting subspaces of a unitary space that are affiliated to a von Neumann algebra are considered. A topological definition of measures on the above classes of subspaces is given and results on the relationships of these measures to measures on orthoprojections of the von Neumann algebra are presented [67].
2. Orthomodular Posets
The orthoprojections in a commutative von Neumann algebra naturally form a Boolean algebra. If the von Neumann algebra in question is noncommutative, we come to a more general structure that may naturally be viewed as an analog to the set of possible propositions about a suitable quantum-mechanical system. An abstract "Boolean" analog to this structure is called an orthomodular poset (abbr., OMP) or a quantum logic. The study of these was initiated by G. Birkhoff and J. von Neumann in the 30ths.
In Kazan, the study of OMPs was begun at our seminar in the late 60ths [49]. In that paper, the dimension theory by Loomis of orthomodular lattices was extended over the arbitrary OMPs. That approach was developed in [12]. The advanced research in OMPs has systematically been held by P.G. Ovchinnikov beginning the 80ths. He has suggested a new axiomatics for OMPs based on their combinatorial characterization [27], has described the automorphisms of the poset of skew projections on a Hilbert space [28], has proved a well-known Ptak-Pulmannova conjecture on the Jauch-Piron property [29], has extended the Birkhoff theorem on doubly stochastic matrices to hypergraphs [30], has given the first known example of an atomistic nonorthoatomistic OMP [31], has found an exact topological analog to the concept of an orthoposet [32], and has developed a general theory of measures and signed measures on finite concrete logics [33, 34]. F.F. Sultanbekov has exhaustively studied a number of particular finite OMPs and concrete logics [35,36] and has successfully applied a computer in the study of finite concrete logics [37]. In doing that, he has developed a series of specific routines [37,38].
Along with studying general OMPs, the search of analogs to the Gleason theorem for some interesting particular ones such as OMPs of skew projections on Banach spaces or vector spaces over "unusual" fields was held [23-25,68].
3. Probability Measures on Banach Spaces
The problem of (sigma)-additivity of a cylindrical probability is one of the most important and difficult in the theory of probability distributions on infinite-dimensional Banach spaces. That problem is closely related to trajectories of the most important stochastic processes. The classical theorems by N. Wiener and A.N. Kolmogorov were the first to be obtained in this field. The most efficient criteria of the (sigma)-additivity are topological ones that are infinite-dimensional analogs to the Bochner theorem. The description problem of those Banach spaces admitting an analog to the Bochner theorem has been motivated by the basic research by V.V. Sazonov (1958), R.A. Minlos (1959), and Yu.V. Prokhorov (1960).
In solving this and related problems, D.Kh. Mushtari has developed a topological technique for studying the weak compactness and the (sigma)-additivity of cylindrical probabilities in Banach spaces. This technique has allowed for the complete description of those Banach spaces admitting a topological solution to the problem. He has extended a number of major results over the topological or metric linear spaces. A number of probability characterizations of the Frechet nuclear spaces have also been obtained. Stable probabilities on Banach spaces have been studied [2,21,22,26].
REFERENCES
a) Monographs and Surveys
1. Measures on Projections and Orthomodular Posets (Dedicated to A. N. Sherstnev on the occasion of his 60th Anniversary). -- Kazan Math. Soc., (1998) 118 pp. 2. Mushtari D. H., Probabilities and Topologies on Linear Spaces. -- Kazan Mathematics Foundation, 1996. 248 p. 3.Trunov N. V. and Sherstnev A. N., Introduction to the noncommutative integration theory. -- Sovrem. problemy math. Noveysh. dostizheniya, vol. 27 (Itogi nauki i techniki. VINITI AN USSR), M., 1985, 167-190. (Russian) 4. Sherstnev A. N., On certain problems in the theory of unbounded measures on projections. -- Atti Sem. Mat. Fis. Univ. Modena, 1994, v. 42, 357-366.
b) Papers (1968-1999)
5. Begovatov E. A., Minlos R. A. and Sherstnev A. N., A report on Summer mathematical School ``Non-commutative Theory of Probability". -- Uspehi mat. nauk, 1978, vol. 34, no. 5, 242-246. (Russian) 6. Bikchentaev A. M., On noncommutative function spaces. --- Amer. Math. Soc. Transl. (2), 1992, v. 154, 179-187. 7. Bikchentaev A. M., On a property Lp-spaces in semifinite von Neumann algebras. --- Mat. Zametki, 1998, vol. 64, no. 2, 185-190. (Russian) 8. Bikchentaev A. M., Majorization for products of measurable operators. -- Inter. J. Theor. Physics, 1998, v. 37, no. 1, 571-576. 9. Dorofeev S., On the problem of boundedness of a signed measure on projections of a von Neumann algebra. -- J. Funct. Anal., 1992, v. 103, no. 1, 33-40. 10. Dorofeev S. and Sherstnev A.N., Frame-type funcions and their applications. -- Izv. vuzov. Matematika, 1990, no. 4, 23-29 (Russian); see also [1], 2-13. 11. Zolotarev A.A., Lp-spaces wiyh respect to a state in von Neumann algebra and the interpolation. -- Izv. vuzov. Matematika, 1982, no. 8, 36-43. (Russian) 12. Kalinin V.V., Orthomodular posets with dimensionality. -- Algebra and Logics, 1976, vol. 15, 535-557. (Russian) 13. Lugovaya G.D., Bilinear forms determining measures on projections. --- Izv. vuzov. Matematika, 1983, no. 2, 88. (Russian) 14. Lugovaya G.D., On unbounded orthogonal measures on projections on a Hilbert space. -- Izv. vuzov. Matematika, 1993, no. 12, 43-46. (Russian) 15. Lugovaya G.D. and Sherstnev A.N., On Gleason's theorem for unbounded measures. -- Izv. vuzov. Matematika, 1980, no. 12, 30-32. (Russian) 16. Lugovaya G.D. and Sherstnev A.N., A model of L1-space with respect to an unbounded measure on projections. -- Izv. vuzov. Matematika, 1984, no. 12, 35-42. (Russian) 17. Lugovaya G., Sherstnev A., On extending an unbounded orthogonal vector measure in a semifinite von Neumann algebra to a weight. -- Demonstratio Math., 1997, v. 30, no. 3, 591-598. 18. Lugovaya G.D. and Sherstnev A.N., On the extension problem of unbounded measures on projections to a weight. -- Doklady AN, 1999, vol. 365, no. 2, 165-166. (Russian) 19. Matvejchuk M.S., A theorem on states on quantum logics, I. -- Teoret. i Mat. Fiz., 1980, vol. 45, 2, 244-250; II. -- ibid., 1981, vol. 48, 3, 261-265. (Russian) 20. Matvejchuk M.S., Description of finite measures on semifinite algebras. -- Funct. Anal. i Prilozhen., 1981, vol. 15, no. 3, 41-53. (Russian) 21. Mushtari D. H., Certain general questions of the theory of probability measures in linear spaces. -- Teoriya Veroyatn. i Primen., 1973, vol. 18, no. 1, 66 - 77. (Russian) 22. Mushtari D. H., Sur l'existence d'une topologie du type Sazonov sur un espace de Banach. -- Semin. Maurey - Schwartz, 1975-76, expose 17, 11 p. 23. Mushtari D. H., Logics of projections in Banach spaces. -- Izv. vuzov. Matematika, 1989, 8, 59-70. (Russian) 24. Mushtari D. H., Orthoadditive functions on idempotents. -- In [1] 38-83. 25. Mushtari D. H. and Matvejchuk M.S., Charges on the logic of skew projections. --- Dokl. Akad. Nauk SSSR, 1985, vol. 283, 43-46. (Russian) 26. Mushtari D. H. and Chuprunov A.N., Sufficient topologies and norms. -- Teoriya Veroyatn. i Primen., 1983, vol. 28, no. 4, 700 - 714. (Russian) 27. Ovchinnikov P. G., Another combinatorial characterization of orthomodular posets. -- In [1], 87-94. 28. Ovchinnikov P. G., Automorphisms of the poset of skew projections. -- J. Funct. Anal., 1993, v. 115, 184-189. 29. Ovchinnikov P. G., Countable Jauch-Piron logics. -- Int. J. Theor. Phys., 1993, v. 32, 885-890. 30. Ovchinnikov P. G., A theorem on pure states on a hypergraph. --- Math. Nachr., 1994, v. 167, 275-278. 31. Ovchinnikov P. G., On alternative orthomodular posets. -- Demonstratio Math., 1994, v. 27, 89-93. 32. Ovchinnikov P. G., Exact topological analogs to orthoposets. -- Proc. Amer. Math. Soc., 1997, v. 125, 2839-2841. 33. Ovchinnikov P. G., Measures on finite concrete logics. --- Proc. Amer. Math. Soc., 1999, v. 127, 1957-1966. 34. Ovchinnikov P. G., Sultanbekov F. F., Finite concrete logics: their structure and measures on them. -- Int. J. Theor. Phys., 1998, v. 37, 147-153. 35. Sultanbekov F. F., Set logics and their representations. --- Int. J. Theor. Phys., 1993, v. 32, 2177-2186. 36. Sultanbekov F. F., Operation associated to partitions and best extension of signed measures on set logics. -- Int. J. Theor. Phys., 1995, v. 34, 1735-1739. 37. Sultanbekov F. F., Solving problems on finite concrete logics with the help of a PC. -- Int. J. Theor. Phys., 1998, v. 37, 199-202. 38. Sultanbekov F. F., Routines for investigation of concrete logics. --- In [1], 95-113. 39. Tikhonov O. E., Lp-spaces with respect to a weight on a von Neumann algebra. -- Izv. vuzov. Matematika, 1982, 8, 76-78. (Russian) 40. Tikhonov O. E., The continuity of operator functions in topologies associated with a trace on a von Neumann algebra. -- Izv. vuzov. Matematika, 1987, 1, 77-79. (Russian) 41. Tikhonov O. E., Banach spaces, associated with a state space, and the information function. -- In: Konstr. teoriya funktsiy i funkts. analiz (Kazan), 1990, no. 7, 67-90. (Russian) 42. Tikhonov O. E., Spectral decomposition with respect to a weight in a base-norm space. -- Izv. vuzov. Matematika, 1991, no. 1, 73-80. (Russian) 43. Tikhonov O. E., Spectral theory for base-norm spaces. -- In: Konstr. teoriya funktsiy i funkts. analiz (Kazan), 1992, no. 8, 76-91. (Russian) 44. Tikhonov O. E., Trace inequalities for spaces in spectral duality. -- Studia Mathematica, 1993, v. 104, no. 1, 99-110. 45. Trunov N.V., Lp-spaces, associated with a weight on a semifinite von Neumann algebra. -- In: Konstr. teoriya funktsiy i funkts. analiz (Kazan), 1981, no. 3, 88-92. (Russian) 46. Trunov N.V., On the theory of noncommutative spaces L1 and L2. - In: Konstr. teoriya funktsiy i funkts. analiz (Kazan), 1983, no. 4, 96-105. (Russian) 47. Trunov N.V. and Sherstnev A.N., Conditional expectations in a scheme of noncommutative probability theory. -- Trans. VIII Prague Conf. on Inform. Theory, Stat. Decis. Funct. Random Proc., vol. B, Prague, 1978, 287-299. (Russian) 48. Trunov N.V. and Sherstnev A.N., On the general noncommutative integration theory with respect to a weight in operator algebras, I. -- Izv. vuzov. Matematika, 1978, no. 7, 79-88; II. -- Izv. vuzov. Matematika, 1978, no. 12, 88-99. (Russian) 49. Sherstnev A.N., On boolean logics. -- Uchen. Zapiski KGU, 1968, vol. 128, 48-62. (Russian) 50. Sherstnev A.N., On the representation of measures on orthoprojections on a Hilbert space by means of bilinear forms. -- Izv. vuzov. Matematika, 1970, 9, 90-97. (Russian) 51. Sherstnev A.N., Summer Mathematical School on Noncoomutative Probability Theory. -- Uspehi mat. nauk, 1972, vol. 27, no. 1, 287-288. (Russian) 52. Sherstnev A.N., On the general theory of states on von Neumann algebras. -- Funct. Anal. i Prilozhen., 1974, vol. 8, no. 3, 89-90. (Russian) 53. Sherstnev A.N., On the notion of a charge in a noncommutative scheme of measure theory. -- In: Veroyatn. metody i Kibernetika (Kazan), 1974, no. 10-11, 68-72. (Russian) 54. Sherstnev A.N., Every smooth weight is a l-weight. -- Izv. vuzov. Matematika, 1977, 8, 88-91. (Russian) 55. Sherstnev A.N., Measures on ideals of orthoprojectors in von Neumann algebra. -- Izv. vuzov. Matematika, 1978, no. 10, 108-109. (Russian) 56. Skvortsova G.Sh. and Tikhonov O.E., Convex sets in noncommutative L1-spaces that are closed in the topology of local convergence in measure, -- Izv. vuzov. Matematika, 1998, 8, 48-55. (Russian). English translation in Russian Math. (Iz. VUZ), 1998, vol. 42, no. 8, 46-52 (1999). 57. Tikhonov O.E., On the Lozanovskiy class of condensing operators and its applications to the non-commutative integration, -- Israel Math. Conf. Proc., 1999, vol. 13, 203-207.
c) Papers (since 2000)
58. Lugovaya G.D. and Sherstnev A.N., On the extension problem for unbounded measures on projections. -- Math. Slovaca, 2000, vol. 50, no. 4, 473-481. 59. Lugovaya G.D. and Sherstnev A.N., Order properties of orthogonal vector fields and a representation theorem. -- Russian Journ. of Math. Physics, 2001, vol. 8, no. 2, 172-179. 60. Bikchentaev A.M., Minimality of convergence in measure topologies on finite von Neumann algebras. -- Mat. Zametki, 2004, vol. 75, no. 3, 342-349. (Russian) 61. Bikchentaev A.M., The continuity of multiplication for two topologies associated with a semifinite trace on von Neumann algebra. -- Lobachevskii J. Math., 2004, vol. 14, 17-24. 62. Bikchentaev A.M., Representation of linear operators in a Hilbert space in the form of finite sums of products of projections. -- Doklady AN, 2003, vol. 393, no. 4, 444-447. (Russian) 63. Bikchentaev A.M. and Sherstnev A.N., On projection-convex combinations in C*-algebras with the unitary factorization property. -- Mat. Zametki, 2004, vol. 76, no. 6. (Russian) 64. Bikchentaev A.M., Semi-orthogonal projectors in a Hilbert space. -- In book: At the turn of the 20-th--21st centuries, Kazan Mat. Obs., Kazan, 2003, 108-114. (Russian) 65. Stolyarov A.I., Tikhonov O.E. and Sherstnev A.N., Characterization of normal traces on von Neumann algebras by inequalities for the modulus, Mat. Zametki, 2002, vol. 72, 448-454; (Russian). English translation: Math. Notes, 2002, vol. 72, 411-416. 66. Skvortsova G.Sh., On weak sequential completeness of factor spaces of the space of integrable operators, Izv. vuzov. Matematika, 2002, 9, 71-74. (Russian) 67. Sherstnev A.N. and Turilova E.A., Classes of subspaces affiliated to a von Neumann algebra, Russian Journ. of Math. Physics, 1999, vol. 6, no. 4, 426-434. 68. Mushtari D.H., Some remarks on measures on orthogonal rational projections and the rational sphere, Lobachevskii Journ. of Math., 2001, vol. 9., 47-53.
Last revised: September, 30, 2004.
E-mail: [email protected]
Phone: (8432)315160
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"Конспект лекций по математическому анализу", Казань: Унипресс, 1998, 488 с. (3-е издание) ( .pdf, .ps, .zip )
Рекомендовано Министерством общего и профессионального образования Российской Федерации в качестве учебного пособия для студентов математических направлений и специальностей университетов
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