Let $S$ and $T$ be continuous linear operators on an ultraprime Banach algebra $A$. We show that if $S$, $T$, and $ST$ are close to satisfy the derivation identity on $A$, then either $S$ or $T$ approaches to zero. If $T$ is close to satisfy the derivation identity and $[T(a),a]$ is near the centre of $A$ for each $a \in A$, then either $T$ approaches to zero or $A$ is nearly commutative. Further, we give quantitative estimates of these phenomena.
References
J. Alaminos, J. Extremera and A. R. Villena, Hyperreflexivity of the derivation space of some group algebras, II, Bull. London Math. Soc., 44 (2012), 323-325. MR2914610 1239.47026 10.1112/blms/bdr096 J. Alaminos, J. Extremera and A. R. Villena, Hyperreflexivity of the derivation space of some group algebras, II, Bull. London Math. Soc., 44 (2012), 323-325. MR2914610 1239.47026 10.1112/blms/bdr096
M. Mathieu, Rings of quotients of ultraprime Banach algebras, with applications to elementary operators, Conference on Automatic Continuity and Banach Algebras (Canberra), 1989, pp. 297-317, Proc. Centre Math. Anal. Austral. Nat. Univ., Vol. 21, Austral. Nat. Univ., Canberra, 1989. M. Mathieu, Rings of quotients of ultraprime Banach algebras, with applications to elementary operators, Conference on Automatic Continuity and Banach Algebras (Canberra), 1989, pp. 297-317, Proc. Centre Math. Anal. Austral. Nat. Univ., Vol. 21, Austral. Nat. Univ., Canberra, 1989.
M. Mathieu, Elementary operators on prime $C^*$-algebras, I. Math. Ann., 284(2) (1989), 223-244. 0648.46052 10.1007/BF01442873 M. Mathieu, Elementary operators on prime $C^*$-algebras, I. Math. Ann., 284(2) (1989), 223-244. 0648.46052 10.1007/BF01442873
M. Mathieu, Posner's second theorem deduced from the first, Proc. Amer. Math. Soc., 114(3) (1992), 601-602. MR1075948 0746.16030 M. Mathieu, Posner's second theorem deduced from the first, Proc. Amer. Math. Soc., 114(3) (1992), 601-602. MR1075948 0746.16030
E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100. MR95863 0082.03003 10.1090/S0002-9939-1957-0095863-0 E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100. MR95863 0082.03003 10.1090/S0002-9939-1957-0095863-0
