In the present paper a notion of additive local multiplication on the algebra of matrices over an arbitrary field is introduced and investigated. It is proved that every additive local left multiplication on this matrix algebra is a left multiplication. Also, in the present paper, a notion of additive local Jordon multiplication on a Jordon algebra of symmetric matrices is also introduced and investigated. It is proved that every additive local Jordon multiplication on the Jordon algebra of symmetric matrices over the field of rational numbers is a Jordon multiplication. The fact that every continuous additive local Jordon multiplication and every linear additive local Jordon multiplication on the Jordon algebra of symmetric matrices over the field of complex or real numbers is a Jordon multiplication is also proved. Recall that a local derivation as defined us follows: given an algebra , a linear map is called a local derivation if for every there exists a derivation such that . In 1990 R.Kadison introduced the concept of local derivation and proved that each continuous local derivation from a von Neumann algebra into its dual Banach bemodule is a derivation. B.Jonson extends the above result by proving that every local derivation from a -algebra into its Banach bimodule is a derivation. In particular, Johnson gives an automatic continuity result by proving that local derivation of a -algebra into a Banach -bimodule are continuous even if not assumed a priori to be so. Based on these results, many authors have studied local derivations on operator algebras. The present paper develops a pure algebraic approach to the investigation of multiplication operators and local multiplications on associative and Jordon algebras. For this propose we introduce a notion of additive local left multiplication on an algebra as follows: given an algebra , an additive map is called additive local left multiplication, if for every there exists an element in such that . We prove that fo an arbitrary field and of matrices over every additive local left multiplication on the algebra is a left multiplication, ie. there exists such that . Similarly, every additive local right multiplication on the algebra is a right multiplication, ie. there exists such that . Also we introduce a notion of additive local Jordon multiplication on a Jordon algebra, and, prove that for an arbitrary field and the Jordon algebra of symmetric matrices over , every additive local Jordon multiplication on the Jordon algebra is a Jordon multiplication, ie. there exists such that . The results above allow us to assert that for finite ring genereted by its identity element and the addition (in particular the ring ). or the ring if integre numbers and the ring of matrices over every local left multiplication on the ring is a left multiplication. Similarly, every local right multiplication on the ring is a right multiplication. Similarly, for a finite ring generated by its identity element and the addition or the ring of integral numbers and the Jordon ring of symmetric matrices over every local Jordon multiplication on the Jordon ring is a Jordon multiplication.
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Arzikulov, F. N. and Umrzakov, N. M.
"LOCAL MULTIPLICATION OPERATORS ON ALGEBRAS OF MATRICES,"
Scientific Bulletin. Physical and Mathematical Research: Vol. 1
, Article 12.
Available at: https://uzjournals.edu.uz/adu/vol1/iss1/12