Derivations on Some Algebras of Measurable Operators Affiliated with Real W*-algebras of Type I

Abdugafur Rakhimov; Ulugbek Karimov

doi:doi:10.11648/j.ijamtp.20261201.12

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Derivations on Some Algebras of Measurable Operators Affiliated with Real W*-algebras of Type I

Abdugafur Rakhimov^*

, Ulugbek Karimov

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 12, Issue 1)

Received: 12 December 2025 Accepted: 12 January 2026 Published: 27 January 2026

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Abstract

It is well known that every derivation on a von Neumann algebra is inner, which reflects the strong rigidity of these algebras. In contrast, for general C*-algebras there may exist non-inner derivations, indicating a more complicated and diverse algebraic structure. This fundamental difference has stimulated extensive research on derivations on various classes of operator algebras. In recent years, increasing attention has been paid to derivations defined on algebras of unbounded operators, in particular on algebras of measurable, locally measurable, and τ-measurable operators associated with von Neumann algebras. Such algebras arise naturally within the framework of noncommutative integration theory and provide a rich setting for extending classical results from the theory of bounded operators. In particular, a complete description of derivations on these algebras has been established in a number of works when they are associated with type I von Neumann algebras, demonstrating that under appropriate assumptions the derivations possess strong regularity properties and admit explicit representations. The present article is devoted to the development of a real analogue of the results described above. More precisely, derivations on algebras of measurable, locally measurable, and τ-measurable operators associated with real type I von Neumann algebras are investigated. By carefully adapting the methods from the complex case and taking into account the specific algebraic and topological features of real operator algebras, a complete characterization of all derivations on the algebras under consideration is obtained. These results generalize known theorems for complex von Neumann algebras to the real setting and contribute to a deeper understanding of derivations on algebras of unbounded operators associated with real operator algebras.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 12, Issue 1)
DOI	10.11648/j.ijamtp.20261201.12
Page(s)	28-33
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Derivations, Algebra of Measurable Operators, Locally Measurable Operator, -measurable Operator, Von Neumann Algebras

1. Introduction

Derivations on operator algebras on a Hilbert space have been widely studied. In particular, it is known that all derivations on a von Neumann algebra are inner, while on a C*-algebra there are non-inner derivations. Recently, considerable attention has been devoted to the study of derivations on algebras of unbounded operators. In papers

[1-5]

a full description of all derivations on algebras of measurable, locally measurable, and

-measurable operators affiliated with von Neumann algebra of type I is given. Papers

[6-9]

study derivations on algebras of measurable and locally measurable operators affiliated with von Neumann algebras, focusing on their continuity and structural properties under natural measure-type topologies. In the present paper, a real analogue of these results is considered. More precisely, descriptions of derivations on some algebras of measurable, locally measurable, and

-measurable operators affiliated with real von Neumann algebra of type I are obtained.

2. Derivations on Some Algebras of Measurable Operators

2.1. Preliminaries

Let

be a *-subalgebra of

of all bounded linear operators on complex Hilbert space

. The set

is called the commutant of the *-algebra

. The set

is called the center of the algebra

. A *-subalgebra

with the property

is called W*-algebra, where

is the double commutant of

. It is equivalent to that, *-subalgebra

is weakly closed and consists identity (i.e. 1

). W*-algebras are also known as von Neumann Algebras. A W*-algebra

is called a factor if its center is trivial, i.e.

coincides with

is called discrete, or of type I, if for every nonzero central projection

there exists a nonzero abelian projection

such that

is called of type

is a nonfinite (infinite) algebra of type I.

A functional

is called a trace on

for all

;

for every

and

(with the convention

);

for every

and every unitary operator

A trace

is called finite if

for every

; semifinite if

for every

; faithful if from

with

it follows that

; normal if from

with

it follows that

A linear subspace

is said to be affiliated with algebra

(notation:

) if

for every unitary operator

is a closed linear subspace of

and

denotes the orthogonal projection onto

, then

if and only if

, where

is complete lattice of all projections from

A linear subspace

is called strongly dense in

with respect to

and there exists a sequence of projections

such that

is a finite projection for each

In this case one says that the strongly dense linear subspace

is determined by the sequence of projections

From the condition

it follows directly that every strongly dense linear subspace is dense in

A linear operator

with domain

, acting in the Hilbert space

, is called affiliated with

(notation:

) if

for every unitary

, that is,

and

for all

It is easy to see that a bounded linear operator

is affiliated with

if and only if

. A closed linear operator

acting in a Hilbert space

is called measurable with respect to algebra

and its domain

is strongly dense in

We denote by

the set of all operators that are measurable with respect to algebra

. Clearly,

A closed linear operator

acting in a Hilbert space

is called locally measurable with respect to algebra

and there exists a sequence of central projections

such that

for all

We denote by

the set of all linear operators that are locally measurable with respect to algebra

. It is clear that

, and

is a factor.

2.2. Derivations on the Algebra

Let

be an algebra over the complex numbers. A linear operator

is called a derivation if it satisfies the Leibniz rule

for all

. Every element

defines a derivation

. Such derivations

are called inner derivations.

If the element

that defines the derivation

belongs to a larger algebra

that contains

, then

is called a spatial derivation.

In the special case when

is commutative, all inner derivations are zero, so they are trivial. One of the main problems in the theory of derivations is to determine when derivations are automatically inner or spatial, and whether non-inner derivations exist, especially nontrivial derivations on commutative algebras.

Recall that, real *-subalgebra

is called real W*-algebra, if it is weakly closed, 1

, and

. Real W*-algebras are also known as real von Neumann Algebras. Let

be a real von Neumann algebra and

. We consider the algebra

of all measurable operators affiliated with a real von Neumann algebra

It is proved that (see

[10]

)

, analogically it is proven that

. Below, we give a full description of derivations on the algebra

of all locally measurable operators affiliated with a type I real von Neumann algebra

It is clear that if a derivation

is inner, then it is

-linear, that is,

for all

, where

is the center of

. The following result shows that the opposite statement is also true.

Theorem 3.1. Let

be a type I real von Neumann algebra with the center

. Then every

-linear derivation

on the algebra

is inner.

Proof. Since

, the derivation

can be extended by

-linearity to derivation

where

is a center of

. Since every derivation on

is inner (see

[2]

, Theorem 2.1), then derivation

is inner, i.e. there is an element

such that

. Then we have

Since

and

, then

. Hence

, i.e.

is inner. The theorem is proven.

Theorem 3.2. If

is a type

real von Neumann algebra, then any derivation on the algebra

is inner.

The proof of the theorem 3.2 is carried out similarly to the proof of Theorem 3.1.

Suppose that

isa commutative algebra and suppose that

is the algebra of

matrices over

. If

, are the matrix units in

, then every element

has the form

Let

be a derivation. Setting

(1)

we obtain a well-defined linear operator

on the algebra

. Moreover

is a derivation on the algebra

and its restriction onto the center of the algebra

coincides with the given

Now let

be a homogeneous real von Neumann algebra of type

, with the center

. Since

is the center of

, it follows that

and

. Let

be a derivation and

be a derivation on the algebra

defined by (1). Extend

by setting

(2)

where

is the extension of

, defined by

. It is clear that

is a derivation on

, and

is a derivation on

The next lemma explains the form of derivations on the algebra of measurable operators in homogeneous type

, real von Neumann algebras.

Lemma 3.3. Let

be a homogeneous real von Neumann algebra of type

. Each derivation

on the algebra

has a unique representation as a sum

in this representation,

is an inner derivation given by an element

, and

is a derivation of type (1) coming from a derivation

on the center

Proof. The proof of the this lemma and lemma 3.3 in

[2]

is very similar. Here, we give a sketch of the proof. Let

be an arbitrary derivation on the algebra

. Now consider the restriction of this derivation

onto the center

of this algebra, and let

be the derivation on the algebra

constructed as in (1). Put

. Then for any

we have

, i.e.

is identically zero on

. Therefore

-linear and by Theorem 3.1 we obtain that

is an inner derivation, and thus

for an appropriate

. Therefore

, then

. Since

is identically zero on the center of the algebra

, this implies that

is also identically zero on the center of

. This means that

, and therefore

, i.e. the decomposition of

is unique. The proof is complete.

Remark 3.4. Under the conditions of the Lemma 3.3 we can extend the derivation

. By Lemma 3.3 (see

[2]

has the form

, where

with

, and

is the derivation defined in (2). From the proof of the Lemma 3.3, it is easy to see that

, and for

with

we obtain

, i.e. we have

Now let

be any finite real von Neumann algebra of type I with the center

. There exists a family of central projections from

which we denote

, with

such that the algebra

is *-isomorphic to the C*-product of real von Neumann algebras

of type

respectively,

, i.e.

Analogically to Proposition 1.1 (see

[2]

), one can prove that

Let

be a derivation on

, and

be its restriction onto the center

. Since

maps each

into itself,

generates a derivation

for each

Let

be the derivation on the matrix algebra

defined as in (1).

Put

(3)

It is easy to see that the map

is a derivation on

Lemma 3.3 now leads to the following result:

Lemma 3.5. Let

be a finite real von Neumann algebra of type I. Each derivation

on the algebra

has a unique representation as a sum

where

is an inner derivation implemented by an element

, and

is a derivation given as (3).

We shall now consider derivations on the algebra

of locally measurable operators with respect to an arbitrary type I real von Neumann algebra

Suppose that

is a type I real von Neumann algebra. There exists a central projection

such that

is a finite real von Neumann algebra;

is a real von Neumann algebra of type

Take a derivation

and suppose that

is its restriction onto its center

By Theorem 3.2

is inner and thus we have

, i.e.

Let

be the derivation on

defined as in (3) and consider its extension

which is defined as

(4)

The following theorem gives the general form of derivations on the algebra

By combining Lemma 3.5 with Theorem 3.2, we obtain the following result.

Theorem 3.6. Let

be a type I real von Neumann algebra. Each derivation

has a unique representation as a sum

where

is an inner derivation implemented by an element

, and

is a derivation of the form (4), generated by a derivation

on the center of

2.3. Derivations on the Algebra S(R)

In this section, we present a description of derivations on the algebra

of measurable operators associated with a type I real von Neumann algebra

Let

be an arbitrary subalgebra of

which contains

Take a derivation

and let us show that

can be extended to a derivation

on the whole

Since

is of type I, for an arbitrary element

there exists a sequence

of mutually orthogonal central projections with

and

for all

. Set

(5)

Since every derivation

is identically zero on central projections of

the equality (5) yields a well-defined derivation

which coincides with

More specifically, from

-linearity of

, implies

-linearity of

, and by Theorem 3.1 the derivation

is inner on

. Therefore

is a spatial derivation on

, i.e. there exists an element

such that

Hence, we arrive at the following result.

Theorem 4.1. Let

be a type I real von Neumann algebra with the center

, and let

be an arbitrary subalgebra in

containing

. Then any

-linear derivation

is spatial and implemented by an element of

Corollary 4.2. Let

be a type I real von Neumann algebra with the center

and let

be a

-linear derivation on

. Then

is spatial and implemented by an element of

Lemma 4.3. If

is a type

real von Neumann algebra, then every derivation

has the form

, for an appropriate

Proof. Derivation

extends to derivation

. By Lemma 3.4 (see

[2]

) derivation

is inner, i.e. there is an element

, such that

. Then we have

. Since

and

, then

. Hence

, i.e.

is inner. The Lemma is proven.

Analogically, we prove the next lemma.

Lemma 4.4. Let

be a type I real von Neumann algebra with the center

. Then every

-linear deriva tion

on the algebra

is inner. In particular, if

is of type

then every derivation on

is inner.

Finally, combining Lemmas 3.5 and 4.4, we obtain the main result of this section.

Theorem 4.5. Let

be a type I real von Neumann algebra. Then every derivation

on the algebra

has a unique representation as a sum

where

is inner and implemented by an element

, and

is the derivation of the form (3) generated by a derivation

on the center of

Let

be a faithful normal semifinite trace on a von Neumann algebra

. A linear subspace

is called

-dense if

and for every

there exists a projection

such that

and

From Proposition 3.15 (see

[5]

) it follows that every

-dense subspace

is strongly dense.

A closed linear operator

acting in

is called

-measurable with respect to algebra

and

-dense in

We denote by

the set of all

-measurable operators. Clearly,

. And

if and only if

and

P(T) is

-dense in

Now, we consider a general form of derivations on the algebra

-measurable operators affiliated with a type I real von Neumann algebra

and a faithful normal semifinite trace

. Similarly to the proof of Theorem 2.1 (see

[2]

) it is proved the following theorem.

Theorem 4.6. Let

be a type I real von Neumann algebra with the center

and a faithful normal semi-finite trace. Then every

-linear derivation

on the algebra

is inner. As a special case, if

is of type

, then every derivation on

is inner.

In a similar way to (3), and formulae (12) from

[2]

we can define a derivation

. Now, similarly to Lemma 3.5 one can prove the following.

Lemma 4.7. Let

be a finite real von Neumann algebra of type I with a faithful normal semi-finite trace

. Each derivation

on the algebra

can be uniquely represented in the form

where

is an inner derivation implemented by an element

, and

is a derivation given as (5).

Finally, Theorem 4.6 and Lemma 4.7 imply the following result.

Theorem 4.8. Let

be a type I real von Neumann algebra with a faithful normal semi-finite trace

. Then every derivation

on the algebra

can be uniquely represented in the form

where

is an inner derivation implemented by an element

, and

is the derivation from lemma 4.7.

If we consider the measure topology

on the algebra

(see Section 1), then it is clear that every non-zero derivation of the form

is discontinuous in

. Therefore, the above Theorem 4.8 implies the following.

Corollary 4.9. Let

be a type I real von Neumann algebra with a faithful normal semi-finite trace

. A derivation

on the algebra

is inner if and only if it is continuous in the measure topology.

3. Materials and Methods

The article employs methods from the theory of bounded and unbounded operator algebras. In proving the main results, the method of passing to the enveloping complex algebra is also used. In doing so, the results obtained in the complex case are applied.

4. Results

Descriptions of derivations on certain algebras of measurable, locally measurable, and

-measurable operators, affiliated with real von Neumann algebra of type I, are obtained.

5. Discussion

Derivations on operator algebras on a Hilbert space have been widely studied. More specifically, it is known that all derivations on a von Neumann algebra are inner, while on a C*-algebra there are non-inner derivations. Recently, considerable attention has been devoted to the study of derivations on algebras of unbounded operators. In 2008-2013, a complete description of all derivations on algebras of measurable, locally measurable, and

-measurable operators affiliated with von Neumann algebra of type I is obtained. Here, a real analogue of these results is considered. That is, descriptions of derivations on some algebras of measurable, locally measurable, and

-measurable operators affiliated with real von Neumann algebra of type I are obtained.

6. Conclusions

A real analogue of the study of derivations on the algebra of unbounded operators is considered. In particularly, descriptions of derivations on certain algebras of measurable, locally measurable, and

-measurable operators, affiliated with real von Neumann algebra of type I, are obtained.

Abbreviations

B(H)	Algebra of all Bounded Linear Operators Acting on a Complex Hilbert Space H
P(M)	Complete Lattice of all Projections from M

Author Contributions

Abdugafur Rakhimov: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Supervision, Validation, Writing – original draft, Writing – review & editing

Ulugbek Karimov: Formal Analysis, Investigation, Methodology, Validation, Writing – original draft, Writing – review & editing

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Albeverio S., Ayupov Sh. A., Kudaybergenov K. K., Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra, Siberian Adv. Math. 2008. No 18. 8694. https://doi.org/10.1007/s11196-008-9091-x
[2]	Albeverio S., Ayupov Sh. A., Kudaybergenov K. K., Structure of derivations on various algebras of measurable oper?ators, J. Funct. Anal. 2009, 252. 2917-2943. https://doi.org/10.1016/j.jfa.2008.11.003
[3]	S. Albeverio, Sh. A. Ayupov, K. K. Kudaybergenov. Local Derivations on Algebras of Measurable Operators. Comm. Contemp. Math., 2011. Vol. 13, No. 4 (2011) 1-15. https://doi.org/10.1142/S0219199711004270
[4]	Ayupov Sh. A., Kudaybergenov K. K., Innerness of continuous derivations on algebras of measurable operators affili?ated with finite von Neumann algebras. Journal of Mathematical Analysis and Applications. 408/1, (2013), 256-267. https://doi.org/10.1016/j.jmaa.2013.07.031
[5]	Ayupov Sh. A, Rakhimov A.A, Usmanov Sh. M. Jordan, Real and Lie Structures in Operator Algebras, Kluw.Acad.Pub., MAIA, 1997, 418, 235p. https://doi.org/10.1007/978-94-015-8605-4
[6]	A. F. Ber, V. I. Chilin, F. A. Sukochev. Continuity of derivations in algebras of locally measurable operators. Integr. Equ. Oper. Theory / Proc. Lond. Math. Soc. 2013. https://doi.org/10.48550/arXiv.1108.5427
[7]	A. F. Ber, V. I. Chilin, F. A. Sukochev. Innerness of continuous derivations on algebras of locally measurable operators. Proc. London Math. Soc., 2014. https://doi.org/10.48550/arXiv.1302.4883
[8]	A. F. Ber, B. de Pagter, F. A. Sukochev. Some remarks on derivations in algebras of measurable operators. Math. Notes, 2010. https://doi.org/10.48550/arXiv.0907.1195
[9]	Muratov M. A., Chilin V. I., Topological algebras of measurable and locally measurable operators, CMFD, 2016, 61. 115-163. https://doi.org/10.1007/s10958-019-04320-y
[10]	Karimov U.Sh., Algebra of measurable operators affiliated with finite real W*-algebras and its derivations. Bull.Math.Inst. 2025. Vol 8. Issue 6. 161-165.

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APA Style

Rakhimov, A., Karimov, U. (2026). Derivations on Some Algebras of Measurable Operators Affiliated with Real W*-algebras of Type I. International Journal of Applied Mathematics and Theoretical Physics, 12(1), 28-33. https://doi.org/10.11648/j.ijamtp.20261201.12

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ACS Style

Rakhimov, A.; Karimov, U. Derivations on Some Algebras of Measurable Operators Affiliated with Real W*-algebras of Type I. Int. J. Appl. Math. Theor. Phys. 2026, 12(1), 28-33. doi: 10.11648/j.ijamtp.20261201.12

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AMA Style

Rakhimov A, Karimov U. Derivations on Some Algebras of Measurable Operators Affiliated with Real W*-algebras of Type I. Int J Appl Math Theor Phys. 2026;12(1):28-33. doi: 10.11648/j.ijamtp.20261201.12

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@article{10.11648/j.ijamtp.20261201.12,
  author = {Abdugafur Rakhimov and Ulugbek Karimov},
  title = {Derivations on Some Algebras of Measurable Operators Affiliated with Real W*-algebras of Type I},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {12},
  number = {1},
  pages = {28-33},
  doi = {10.11648/j.ijamtp.20261201.12},
  url = {https://doi.org/10.11648/j.ijamtp.20261201.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20261201.12},
  abstract = {It is well known that every derivation on a von Neumann algebra is inner, which reflects the strong rigidity of these algebras. In contrast, for general C*-algebras there may exist non-inner derivations, indicating a more complicated and diverse algebraic structure. This fundamental difference has stimulated extensive research on derivations on various classes of operator algebras. In recent years, increasing attention has been paid to derivations defined on algebras of unbounded operators, in particular on algebras of measurable, locally measurable, and τ-measurable operators associated with von Neumann algebras. Such algebras arise naturally within the framework of noncommutative integration theory and provide a rich setting for extending classical results from the theory of bounded operators. In particular, a complete description of derivations on these algebras has been established in a number of works when they are associated with type I von Neumann algebras, demonstrating that under appropriate assumptions the derivations possess strong regularity properties and admit explicit representations. The present article is devoted to the development of a real analogue of the results described above. More precisely, derivations on algebras of measurable, locally measurable, and τ-measurable operators associated with real type I von Neumann algebras are investigated. By carefully adapting the methods from the complex case and taking into account the specific algebraic and topological features of real operator algebras, a complete characterization of all derivations on the algebras under consideration is obtained. These results generalize known theorems for complex von Neumann algebras to the real setting and contribute to a deeper understanding of derivations on algebras of unbounded operators associated with real operator algebras.},
 year = {2026}
}

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TY  - JOUR
T1  - Derivations on Some Algebras of Measurable Operators Affiliated with Real W*-algebras of Type I
AU  - Abdugafur Rakhimov
AU  - Ulugbek Karimov
Y1  - 2026/01/27
PY  - 2026
N1  - https://doi.org/10.11648/j.ijamtp.20261201.12
DO  - 10.11648/j.ijamtp.20261201.12
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 28
EP  - 33
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20261201.12
AB  - It is well known that every derivation on a von Neumann algebra is inner, which reflects the strong rigidity of these algebras. In contrast, for general C*-algebras there may exist non-inner derivations, indicating a more complicated and diverse algebraic structure. This fundamental difference has stimulated extensive research on derivations on various classes of operator algebras. In recent years, increasing attention has been paid to derivations defined on algebras of unbounded operators, in particular on algebras of measurable, locally measurable, and τ-measurable operators associated with von Neumann algebras. Such algebras arise naturally within the framework of noncommutative integration theory and provide a rich setting for extending classical results from the theory of bounded operators. In particular, a complete description of derivations on these algebras has been established in a number of works when they are associated with type I von Neumann algebras, demonstrating that under appropriate assumptions the derivations possess strong regularity properties and admit explicit representations. The present article is devoted to the development of a real analogue of the results described above. More precisely, derivations on algebras of measurable, locally measurable, and τ-measurable operators associated with real type I von Neumann algebras are investigated. By carefully adapting the methods from the complex case and taking into account the specific algebraic and topological features of real operator algebras, a complete characterization of all derivations on the algebras under consideration is obtained. These results generalize known theorems for complex von Neumann algebras to the real setting and contribute to a deeper understanding of derivations on algebras of unbounded operators associated with real operator algebras.
VL  - 12
IS  - 1
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Author Information

Abdugafur Rakhimov

Department of Algebra and Functional Analysis, National University of Uzbekistan, Tashkent, Uzbekistan

Research Fields: Functional analysis, Theory of operator algebras, Theory of unbounded operators, Topology, Fuzzy topology

Contact Email

http://orcid.org/0000-0002-2675-1510
Ulugbek Karimov

Department of Algebra and Functional Analysis, National University of Uzbekistan, Tashkent, Uzbekistan

Research Fields: Functional analysis, Theory of operator algebras, Theory of unbounded operators, Topology.

Contact Email

http://orcid.org/0009-0007-0992-0590

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Rakhimov, A., Karimov, U. (2026). Derivations on Some Algebras of Measurable Operators Affiliated with Real W*-algebras of Type I. International Journal of Applied Mathematics and Theoretical Physics, 12(1), 28-33. https://doi.org/10.11648/j.ijamtp.20261201.12

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Rakhimov, A.; Karimov, U. Derivations on Some Algebras of Measurable Operators Affiliated with Real W*-algebras of Type I. Int. J. Appl. Math. Theor. Phys. 2026, 12(1), 28-33. doi: 10.11648/j.ijamtp.20261201.12

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Rakhimov A, Karimov U. Derivations on Some Algebras of Measurable Operators Affiliated with Real W*-algebras of Type I. Int J Appl Math Theor Phys. 2026;12(1):28-33. doi: 10.11648/j.ijamtp.20261201.12

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@article{10.11648/j.ijamtp.20261201.12,
  author = {Abdugafur Rakhimov and Ulugbek Karimov},
  title = {Derivations on Some Algebras of Measurable Operators Affiliated with Real W*-algebras of Type I},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {12},
  number = {1},
  pages = {28-33},
  doi = {10.11648/j.ijamtp.20261201.12},
  url = {https://doi.org/10.11648/j.ijamtp.20261201.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20261201.12},
  abstract = {It is well known that every derivation on a von Neumann algebra is inner, which reflects the strong rigidity of these algebras. In contrast, for general C*-algebras there may exist non-inner derivations, indicating a more complicated and diverse algebraic structure. This fundamental difference has stimulated extensive research on derivations on various classes of operator algebras. In recent years, increasing attention has been paid to derivations defined on algebras of unbounded operators, in particular on algebras of measurable, locally measurable, and τ-measurable operators associated with von Neumann algebras. Such algebras arise naturally within the framework of noncommutative integration theory and provide a rich setting for extending classical results from the theory of bounded operators. In particular, a complete description of derivations on these algebras has been established in a number of works when they are associated with type I von Neumann algebras, demonstrating that under appropriate assumptions the derivations possess strong regularity properties and admit explicit representations. The present article is devoted to the development of a real analogue of the results described above. More precisely, derivations on algebras of measurable, locally measurable, and τ-measurable operators associated with real type I von Neumann algebras are investigated. By carefully adapting the methods from the complex case and taking into account the specific algebraic and topological features of real operator algebras, a complete characterization of all derivations on the algebras under consideration is obtained. These results generalize known theorems for complex von Neumann algebras to the real setting and contribute to a deeper understanding of derivations on algebras of unbounded operators associated with real operator algebras.},
 year = {2026}
}

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TY  - JOUR
T1  - Derivations on Some Algebras of Measurable Operators Affiliated with Real W*-algebras of Type I
AU  - Abdugafur Rakhimov
AU  - Ulugbek Karimov
Y1  - 2026/01/27
PY  - 2026
N1  - https://doi.org/10.11648/j.ijamtp.20261201.12
DO  - 10.11648/j.ijamtp.20261201.12
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 28
EP  - 33
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20261201.12
AB  - It is well known that every derivation on a von Neumann algebra is inner, which reflects the strong rigidity of these algebras. In contrast, for general C*-algebras there may exist non-inner derivations, indicating a more complicated and diverse algebraic structure. This fundamental difference has stimulated extensive research on derivations on various classes of operator algebras. In recent years, increasing attention has been paid to derivations defined on algebras of unbounded operators, in particular on algebras of measurable, locally measurable, and τ-measurable operators associated with von Neumann algebras. Such algebras arise naturally within the framework of noncommutative integration theory and provide a rich setting for extending classical results from the theory of bounded operators. In particular, a complete description of derivations on these algebras has been established in a number of works when they are associated with type I von Neumann algebras, demonstrating that under appropriate assumptions the derivations possess strong regularity properties and admit explicit representations. The present article is devoted to the development of a real analogue of the results described above. More precisely, derivations on algebras of measurable, locally measurable, and τ-measurable operators associated with real type I von Neumann algebras are investigated. By carefully adapting the methods from the complex case and taking into account the specific algebraic and topological features of real operator algebras, a complete characterization of all derivations on the algebras under consideration is obtained. These results generalize known theorems for complex von Neumann algebras to the real setting and contribute to a deeper understanding of derivations on algebras of unbounded operators associated with real operator algebras.
VL  - 12
IS  - 1
ER  -

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