Derivations on the algebra of operators in hilbert C*-modules |
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Authors: | Peng Tong Li De Guang Han Wai Shing Tang |
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Institution: | 1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P. R. China 2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA 3. Department of Mathematics, National University of Singapore, Singapore, 119076, Republic of Singapore
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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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Keywords: | Derivations inner derivations C*-algebras Hilbert C*-modules |
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