Derivations on the algebra of operators in hilbert C*-modules

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.

Some Notes on Semidirect Products of Banach Algebras

“…Some results have also been obtained in the case of non-self-adjoint operator algebras. Christian [5] showed that every continuous derivation on a nest algebra on H to itself and to B(H) is inner and then this result generalized to some other forms among which we may refer to [18] and the references therein. Gilfeather and Smith have calculated the first cohomology group of some operator algebras called joins( [12], [13]).…”

The First Cohomology Group of Banach Inverse Semigroup Algebras With Coefficients in -Embedded Banach Bimodules

“…Some results have also been obtained in the case of non-self-adjoint operator algebras. Christian [4] showed that every continuous derivation on a nest algebra on H to itself and to B(H) is inner, and this result was generalised in [16]. However, the cohomology is nontrivial in general.…”

Derivations, Local and 2-Local Derivations on Some Algebras of Operators on Hilbert C*-Modules

“…There are few results in this topic. P. Li, D. Han and W. Tang [15] prove that each derivation on End * A (M) is inner, where M is a full Hilbert C*-module over a commutative unital C*-algebra A. M. Moghadam, M. Miri and A. Janfada [16] prove that each A-linear derivation on End A (M) is inner, where M is a full Hilbert C*-module over a commutative unital C*-algebra A with the property that there exist x 0 in M and f 0 in M ′ such that f 0 (x 0 ) = e.…”