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DOI: 10.23671/VNC.2018.1.11396 2Local Derivations on Algebras of MatrixValued Functions on a Compactum
Abstract:
The present paper is devoted to 2local derivations. In 1997, P. Semrl introduced the notion of 2local derivations and described 2local derivations on the algebra \(B(H)\) of all bounded linear operators on the infinitedimensional separable Hilbert space \(H\). After this, a number of paper were devoted to 2local maps on different types of rings, algebras, Banach algebras and Banach spaces. A similar description for the finitedimensional case appeared later in the paper of S. O. Kim and J. S. Kim. Y. Lin and T. Wong described 2local derivations on matrix algebras over a finitedimensional 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 2local derivations on the algebra \(B(H)\) of all linear bounded operators on an arbitrary Hilbert space \(H\) and proved that every 2local derivation on \(B(H)\) is a derivation. Then there appeared several papers dealing with 2local derivations on associative algebras. In the present paper 2lo\cal derivations on various algebras of infinite dimensional matrixvalued functions on a compactum are described. We develop an algebraic approach to investigation of derivations and \mbox{2local} derivations on algebras of infinite dimensional matrixvalued functions on a compactum and prove that every such 2local derivation is a derivation. As the main result of the paper it is established that every \mbox{2local} 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 matrixvalued functions on a compactum.
Keywords: derivation, 2local derivation, associative algebra, \(C^*\)algebra, von Neumann algebra
Language: Russian
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For citation: Ayupov S. A., Arzikulov F. N. 2Local derivations on algebras of matrixvalued functions on a compactum. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no. 1, pp.3849. DOI 10.23671/VNC.2018.1.11396 ← Contents of issue 
 

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