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2-Local derivations on algebras of matrix-valued functions on a compactum
Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 1, pp. 38-49 Cet article a éte moissonné depuis la source Math-Net.Ru

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The present paper is devoted to 2-local derivations. In 1997, P. Ŝemrl 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 $H$. After this, a number of paper were devoted to 2-local maps on different types of rings, algebras, Banach algebras and Banach spaces. A similar description for the finite-dimensional case appeared later in the paper of S. O. Kim and J. S. Kim. Y. Lin and T. Wong described 2-local derivations on matrix algebras over a finite-dimensional division ring. Sh. A. Ayupov and K. K. Kudaybergenov suggested a new technique and have generalized the above mentioned results for arbitrary Hilbert spaces. Namely they considered 2-local derivations on the algebra $B(H)$ of all linear bounded operators on an arbitrary Hilbert space $H$ and proved that every 2-local derivation on $B(H)$ is a derivation. Then there appeared several papers dealing with 2-local derivations on associative algebras. In the present paper 2-local derivations on various algebras of infinite dimensional matrix-valued functions on a compactum are described. We develop an algebraic approach to investigation of derivations and 2-local derivations on algebras of infinite dimensional matrix-valued functions on a compactum and prove that every such 2-local derivation is a derivation. As the main result of the paper it is established that every 2-local derivation on a $*$-algebra $C(Q, M_n(F))$ or $C(Q,\mathcal{N}_n(F))$, where $Q$ is a compactum, $M_n(F)$ is the $*$-algebra of infinite dimensional matrices over complex numbers (real numbers or quaternoins) defined in section 1, $\mathcal{N}_n(F)$ is the $*$-subalgebra of $M_n(F)$ defined in section 2, is a derivation. Also we explain that the method developed in the paper can be applied to Jordan and Lie algebras of infinite dimensional matrix-valued functions on a compactum. 
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Sh. A. Ayupov; F. N. Arzikulov. 2-Local derivations on algebras of matrix-valued functions on a compactum. Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 1, pp. 38-49. http://geodesic.mathdoc.fr/item/VMJ_2018_20_1_a4/

[1] Akhiezer N. I., Glazman I. M., Theory of Linear Operators in Hilbert Space, Transl. from the Russian, Dover Publ., Inc., N. Y., 1993 | MR | Zbl

[2] Arzikulov F. N., “Infinite order and norm decompositions of $C^*$-algebras”, Int. J. Math. Anal., 2:5 (2008), 255–262 | MR

[3] Arzikulov F. N., “Infinite norm decompositions of $C^*$-algebras”, Oper. Theory Adv. Appl., 220, Springer AG, Basel, 2012, 11–21 | DOI | MR

[4] Arzikulov F. N., “Infinite order decompositions of $C^*$-algebras”, SpringerPlus., 5:1 (2016), 1–13 | DOI

[5] Ayupov Sh. A., Kudaybergenov K. K., “2-local derivations and automorphisms on $B(H)$”, J. Math. Anal. Appl., 395 (2012), 15–18 | DOI | MR | Zbl

[6] Bresar M., “Jordan derivations on semiprime rings”, Proc. Amer. Math. Soc., 104 (1988), 1003–1006 | DOI | MR | Zbl

[7] Kim S. O., Kim J. S., “Local automorphisms and derivations on $M_n$”, Proc. Amer. Math. Soc., 132 (2004), 1389–1392 | DOI | MR | Zbl

[8] Lin Y., Wong T., “A note on 2-local maps”, Proc. Edinb. Math. Soc., 49 (2006), 701–708 | DOI | MR | Zbl

[9] Sakai S., $C^*$-Algebras and $W^*$-Algebras, Springer-Verlag, Berlin etc., 1971 | MR | Zbl

[10] Šemrl P., “Local automorphisms and derivations on $B(H)$”, Proc. Amer. Math. Soc., 125 (1997), 2677–2680 | DOI | MR