Conditional expectation and stochastic integrals in non-commutative Lp spaces

1991 ◽  
Vol 110 (2) ◽  
pp. 365-383 ◽  
Author(s):  
Stanisław Goldstein
Keyword(s):  
General Scheme ◽  
Stochastic Integrals ◽  
Norm Topology ◽  
Von Neumann ◽  
Measure Topology ◽  
Modular Automorphism ◽  
Faithful Normal State ◽  
Measurable Operators ◽  
Commutative Theory

The aim of the paper is to propose a general scheme for the consideration of non-commutative stochastic integrals. The role of a probability space is played by a couple (, φ0), where is a von Neumann algebra and φ0 is a faithful normal state on . Our processes live in the algebra of all measurable operators associated with the crossed product of by the modular automorphism group The algebra contains all the (Haagerup's) Lp spaces over . The measure topology of the algebra has the nice feature of inducing the Lp norm topology on the Lp spaces, which makes it particularly suitable for defining stochastic integrals. The commutative theory fits smoothly into the scheme, although there exists no canonical way of embedding the algebra of (commutative) random variables into . In fact, for any commutative stochastic process we have a family of different non-commutative stochastic processes corresponding to the process. This arbitrariness seems to be quite natural in the non-commutative context. An appropriate example can be found at the end of the paper (Section 6, C4).

scholarly journals LOCAL DERIVATIONS ON ALGEBRAS OF MEASURABLE OPERATORS

2011 ◽  
Vol 13 (04) ◽  
pp. 643-657 ◽  
Author(s):  
S. ALBEVERIO ◽  
SH. A. AYUPOV ◽  
K. K. KUDAYBERGENOV ◽  
B. O. NURJANOV
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Continuity Condition ◽  
Type I ◽  
Atomic Lattice ◽  
Von Neumann ◽  
Measurable Operators ◽  
Finite Trace ◽  
Necessary And Sufficient

The paper is devoted to local derivations on the algebra [Formula: see text] of τ-measurable operators affiliated with a von Neumann algebra [Formula: see text] and a faithful normal semi-finite trace τ. We prove that every local derivation on [Formula: see text] which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras, they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the continuity condition on local derivations in the above results is redundant. Finally we give necessary and sufficient conditions on a commutative von Neumann algebra [Formula: see text] for the algebra [Formula: see text] to admit local derivations which are not derivations.

DERIVATIONS ON ALGEBRAS OF MEASURABLE OPERATORS

2010 ◽  
Vol 13 (02) ◽  
pp. 305-337 ◽  
Author(s):  
SH. A. AYUPOV ◽  
K. K. KUDAYBERGENOV
Keyword(s):  
Measurable Function ◽  
Von Neumann Algebras ◽  
Special Section ◽  
Type I ◽  
Von Neumann ◽  
Abelian Case ◽  
Measurable Operators

The present paper is a survey of recent results concerning derivations on various algebras of measurable operators affiliated with von Neumann algebras. A complete description of derivation is obtained in the case of type I von Neumann algebras. A special section is devoted to the Abelian case, namely to the existence of nontrivial derivations on algebras of measurable function. Local derivations on the above algebras are also considered.

scholarly journals Embeddings of ℓp into Non-commutative Spaces

2003 ◽  
Vol 74 (3) ◽  
pp. 331-350 ◽  
Author(s):  
Narcisse Randrianantoanina
Keyword(s):  
Symmetric Space ◽  
Von Neumann Algebra ◽  
Lorentz Spaces ◽  
Natural Conditions ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Measurable Operators

AbstractLet ℳ be a semi-finite von Neumann algebra equipped with a faithful normal trace τ. We prove a Kadec-Pelczyński type dichotomy principle for subspaces of symmetric space of measurable operators of Rademacher type 2. We study subspace structures of non-commutative Lorentz spaces Lp, q, (ℳ, τ), extending some results of Carothers and Dilworth to the non-commutative settings. In particular, we show that, under natural conditions on indices, ℓp cannot be embedded into Lp, q (ℳ, τ). As applications, we prove that for 0 < p < ∞ with p ≠ 2, ℓp cannot be strongly embedded into Lp(ℳ, τ). This provides a non-commutative extension of a result of Kalton for 0 < p < 1 and a result of Rosenthal for 1 ≦ p < 2 on Lp [0, 1].

scholarly journals Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra

2008 ◽  
Vol 18 (2) ◽  
pp. 86-94 ◽  
Author(s):  
S. Albeverio ◽  
Sh. A. Ayupov ◽  
K. K. Kudaybergenov
Keyword(s):  
Von Neumann Algebra ◽  
Type I ◽  
Von Neumann ◽  
Measurable Operators

scholarly journals ACCOUNTS INSURANCE

2020 ◽  
Vol 8 (6) ◽  
pp. 166-169
Author(s):  
K. Pavlenko
Keyword(s):  
Russian Federation ◽  
General Scheme ◽  
Scientific Research ◽  
Scientific Article ◽  
Accounts Receivable ◽  
Ongoing Process ◽  
Main Disadvantage ◽  
The Russian Federation

A scientific article is devoted to the analysis of the role of insurance in the management of receivables of enterprises of the economy of the Russian Federation. The relevance of scientific research is associated with the ongoing process of increasing the size of receivables from domestic organizations. The theoretical aspects and fundamentals of accounts receivable insurance are considered. The advantages by which insurance is one of the most relevant methods for managing receivables of enterprises are highlighted. The main negative fundamental factors that increase the degree of problems of managing receivables from domestic organizations are listed. The main disadvantage of insurance is described as a method of managing receivables. The features of the policy and general scheme of insurance of receivables of enterprises are considered.

scholarly journals Principles of incorporating data about the environmental condition of natural-historical parks of metropolises in the atlas information system (on the example of the Moskvoretsky park)

2021 ◽  
Vol 27 (3) ◽  
pp. 387-408
Author(s):  
Olga Markova
Keyword(s):  
Information System ◽  
Protected Areas ◽  
Cultural Landscape ◽  
Cultural Landscapes ◽  
General Scheme ◽  
North West ◽  
The North ◽  
Specially Protected Natural Areas ◽  
Protected Natural Areas

Within the framework of this research, the task of developing the principles of creating an atlas information system (AIS) for assessing the ecological state of specially protected natural areas of the Moscow metropolis was carried out. The research was carried out on the example of the Moskvoretsky park, the largest and most integrated natural-historical park in Moscow, located in the North-West and West administrative districts of the capital. Such parks have environmental, historical, cultural, educational and recreational significance as natural complexes and monuments of national history and culture that are especially valuable for the city. A general scheme for organizing data in the atlas information system about the protected areas of Moscow and specifically about the Moskvoretsky natural-historical park has been developed. All the various data are hierarchically linked. To obtain concentrated information, data from various sources were used and interconnected. The landscape and ecological structure of the Moskvoretsky park is considered. The historical and geographical processes that formed the appearance of the cultural landscapes of the natural-historical park—urbanization, hydrotechnical transformations, the construction of sports facilities, landscaping—are revealed. A map of the Moskvoretsky рark with all the protected areas that are part of its structure, other noteworthy objects, infrastructure, green areas was compiled. The role of this map is assumed as the basis for the further creation of an atlas information system and filling it with various data. A table for individual protected areas that are part of the structure of the Moskvoretsky рark, their types and main characteristics, features of natural and cultural heritage, cultural landscape, environmental problems, and conflicts of nature use are given has been compiled. The main environmental problem of the park is the expansion of urban construction, alienating the territory from the field of nature conservation. Prospects for the creation of AIS are associated with monitoring and solving environmental and cultural problems of a part of the metropolis that is part of the natural-historical park.

scholarly journals Koopman wavefunctions and Clebsch variables in Vlasov–Maxwell kinetic theory

2021 ◽  
Vol 87 (4) ◽  
Author(s):  
Cesare Tronci ◽  
Ilon Joseph
Keyword(s):  
Kinetic Theory ◽  
Classical Mechanics ◽  
Canonical Formalism ◽  
Canonical Structure ◽  
Hamiltonian Structures ◽  
Von Neumann ◽  
Gauge Transformations ◽  
Phase Dynamics ◽  
Space Formulation

Motivated by recent discussions on the possible role of quantum computation in plasma simulations, here, we present different approaches to Koopman's Hilbert-space formulation of classical mechanics in the context of Vlasov–Maxwell kinetic theory. The celebrated Koopman–von Neumann construction is provided with two different Hamiltonian structures: one is canonical and recovers the usual Clebsch representation of the Vlasov density, the other is non-canonical and appears to overcome certain issues emerging in the canonical formalism. Furthermore, the canonical structure is restored for a variant of the Koopman–von Neumann construction that carries a different phase dynamics. Going back to van Hove's prequantum theory, the corresponding Koopman–van Hove equation provides an alternative Clebsch representation which is then coupled to the electromagnetic fields. Finally, the role of gauge transformations in the new context is discussed in detail.

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