Derivations and 2-local derivations on matrix algebras over commutative algebras

2016/11/03 by Huang, Wenbo, Li, Jiankui, Qian, Wenhua
#47B47 #47L35 #FOS: Mathematics #Operator Algebras (math.OA) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1611.00871

Abstract

We characterize derivations and 2-local derivations from M_n(A) into M_n(M), n ≥ 2, where A is a unital algebra over ℂ and M is a unital A-bimodule. We show that every derivation D: M_n(A) → M_n(M), n ≥ 2, is the sum of an inner derivation and a derivation induced by a derivation from A to M. We say that A commutes with M if am=ma for every a\inA and m\inM. If A commutes with M we prove that every inner 2-local derivation D: M_n(A) → M_n(M), n ≥ 2, is an inner derivation. In addition, if A is commutative and commutes with M, then every 2-local derivation D: M_n(A) → M_n(M), n ≥ 2, is a derivation.

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