Home > ajphysmat.uz > Vol. 1 > Iss. 2 (2019)

## Scientific Bulletin. Physical and Mathematical Research

#### Abstract

The present paper is devoted to 2-local derivation on associative and Jordan matrix rings. In 1997, P. Semrl introduced the notion of 2-local derivations and described 2-local derivations on the algebra ¬B(H) of all bounded linear operators on the infinite-dimensional separable Hilbert space H. A similar description for the finite-dimensional case appeared later in 2004. In the paper Y. Lin and T. Wong 2-local derivations have been described on matrix algebras over finite dimensional division rings. In 2012 Sh. Ayupov, K. Kudaybergenov suggested a new technique and have generalized the above mentioned results for abritrary Hilbert spaces. Namely they considered 2-local derivations on the algebra B(H) of all linear bounded operators on an arbitrary (no separability is assumed) Hilbert space H and proved that every 2-local derivation on B(H) is a derivation. Later Sh. Ayupov, K. Kudaybergenov and F. Arzikulov extended the above results and give a proof of the theorem for an arbitrary von Neumann algebra. In the given paper it is introduced and studied a concept of 2-local two sided multuplication on M_2 (R). A 2-local two sided multuplication is defined as follows:let ∆:M_2 R→M_2 R be a mapping. If for each pairs X ,Y∈ M_2 (R) of elements there exists a matrix A∈M_2 (R) such that ∆(x)=AXA, ∆(y)=AYA, then the mapping ∆ is called 2-local two-sided multiplication. It is proved that every 2-local two sided multuplication of M_2 (R) is a two sided multuplication, if all components of all matrices defining this 2-local two sided multuplication are positive. Namely, we prove that for any matrix x∈M_2 (R) there exists a matrix a∈M_2 (R) with positive entries such that φ(x)=axa. For this propose we prove the following lemma: let ∆ be a 2-local two sided multuplication on M_2 (R) defined by 22 matrices with posotive entries of the field of real numbers. Then there exists a matrix A∈M_2 (R) with positive entries such that, for all matrix units e_ij∈M_2 (R) ,i,j=1,2 it is valid ∆〖(e〗_ij)=Ae_ij A for the 2-local two sided multuplication ∆, that is ∆〖(e〗_11)=Ae_11 A, ∆〖(e〗_12)=Ae_12 A, ∆〖(e〗_21)=Ae_21 A,∆〖(e〗_22)=Ae_22 A. Similarly we define a concept of 2-local two sided multuplication on M_2 (R)⊗C[a,b] as follows: let ∆:M_2 R⊗C[a,b]→M_2 R⊗C[a,b] be a mapping. If for each pairs X ,Y∈ M_2 (R)⊗C[a,b] of elements there exists a matrix A∈M_2 (R)⊗C[a,b] such that ∆(x)=AXA, ∆(y)=AYA, then the mapping ∆ is called 2-local two-sided multiplication. In the present paper, it is proved that every 2-local two sided multuplication of M_2 (R)⊗C[a,b] is a two sided multuplication, if all components of all matrices defining this 2-local two sided multuplication are positive functions. Namely, we prove that for any matrix x∈M_2 (R)⊗C[a,b] there exists a matrix a∈M_2 (R) with positive function entries such that φ(x)=axa.

#### First Page

73

#### Last Page

78

#### References

1. Šemrl, P. (1997) Local automorphisms and derivations on B(H). Proceedings of the American Mathematical Society. 125(9). Pp. 2677-2680.

2. Kim S., Kim J. (2004) Local automorphisms and derivations on Mn. Proceedings of the American Mathematical Society. 132 1389-1392.

3. Lin Y., Wong T. (2006) A note on 2-local maps. Proceedings of the Edinburgh Mathematical Society. 49. Pp. 701-708.

4. Ayupov Sh., Kudaybergenov K. (2012) 2-local derivations and automorphisms on B(H). Journal of Mathematical Analysis and Applications. 395. Pp. 15-18.

5. Ayupov Sh., Arzikulov F. (2014) 2-Local derivations on semi-ﬁnite von Neumann algebras, Glasgow Mathematical Journal. 56. Pp. 9-12.

6. Ayupov Sh., Kudaybergenov K. (2015) 2-Local derivations on von Neumann algebras. Positivity. 19. Pp. 445-455.

7. Ayupov S., Arzikulov F. (2017) 2-Local derivations on associative and Jordan matrix rings over commutative rings. Linear Algebra and its Applications. 522. Pp. 28-50.

8. Ayupov Sh.A., Arzikulov F.N. (2018) Description of 2-local and local derivations on some Lie rings of skew-adjoint matrices. Linear and Multilinear Algebra. 9(26). Pp. З1-17

9. Ayupov Sh.A., Arzikulov F.N. (2017) 2-Local Derivations on AW*-Algebras of Type I. Lobachevskii Journal of Mathematics. 1(38). Pp. 148-161.

10. Ayupov Sh.A., Arzikulov F.N. (2018) 2-Local derivations on algebras of matrix-valued functions on a compact. Vladikavkazskiy matematicheskiy jurnal. Vol. 20. Issue 1. Pp. 38-49.

11. Ayupov Sh.A., Arzikulov F.N., Umrzaqov N.M. (2019) Local derivations on associative and Jordan matrix algebras. O`zbek matematika jurnali. 4. Pp. 13-33.

#### Recommended Citation

Arzikulov, Farhodjon N. and Solijanova, Kamola A.
(2019)
"Description of 2-local two sided multiplication on an algebra of matrices,"
*Scientific Bulletin. Physical and Mathematical Research*: Vol. 1
:
Iss.
2
, Article 9.

Available at:
https://uzjournals.edu.uz/adu/vol1/iss2/9