Abstract
Let M be a full Hilbert C*-module over a C*-algebra A, and let End* A (M) be the algebra of adjointable operators on M. We show that if A is unital and commutative, then every derivation of End* A (M) is an inner derivation, and that if A is σ-unital and commutative, then innerness of derivations on “compact” operators completely decides innerness of derivations on End* A (M). If A is unital (no commutativity is assumed) such that every derivation of A is inner, then it is proved that every derivation of End*A(L n (A)) is also inner, where L n (A) denotes the direct sum of n copies of A. In addition, in case A is unital, commutative and there exist x 0, y 0 ∈ M such that <x 0, y 0〉 = 1, we characterize the linear A-module homomorphisms on End* A (M) which behave like derivations when acting on zero products.
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The first author is supported by National Natural Science Foundation of China (Grant No. 11171151) and Natural Science Foundation of Jiangsu Province of China (Grant No. BK2011720); the third author is supported by Singapore Ministry of Education Academic Research Fund Tier 1 (Grant No. R-146-000-136-112)
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Li, P.T., Han, D.G. & Tang, W.S. Derivations on the algebra of operators in hilbert C*-modules. Acta. Math. Sin.-English Ser. 28, 1615–1622 (2012). https://doi.org/10.1007/s10114-012-0172-6
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DOI: https://doi.org/10.1007/s10114-012-0172-6