2-Local Derivations on Algebras of Matrix-Valued Functions on a Compactum

Ayupov S. A. , Arzikulov F. N.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 1.

Abstract:
The present paper is devoted to 2-local derivations. 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 \(H\). 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-lo\-cal 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 \mbox{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 \mbox{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.

Keywords: derivation, 2-local derivation, associative algebra, \(C^*\)-algebra, von Neumann algebra

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