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## 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 22 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.

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#### References

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