Weak-local derivations and homomorphisms on C*-algebras

Let Ω be a compact Hausdorff space and let A be a C * -algebra. We prove that if every weak-2-local derivation on A is a linear derivation and every derivation on C(Ω, A) is inner, then every weak-2-local derivation ∆ : C(Ω, A) → C(Ω, A) is a (linear) derivation. As a consequence we derive that, for every complex Hilbert space H, every weak-2-local derivation ∆ : C(Ω, B(H)) → C(Ω, B(H)) is a (linear) derivation. We actually show that the same conclusion remains true when B(H) is replaced with an atomic von Neumann algebra. With a modified technique we prove that, if B denotes a compact C * -algebra (in particular, when B = K(H)), then every weak-2-local derivation on C(Ω, B) is a (linear) derivation. Among the consequences, we show that for each von Neumann algebra M and every compact Hausdorff space Ω, every 2-local derivation on C(Ω, M ) is a (linear) derivation.

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“…Proof. By [8,Theorem 3.4] it is enough to prove that ∆ is linear. This linearity certainly holds if and only if δ t ∆ is linear for all t ∈ Ω.…”

Section: Weak-2-local Derivations On C(ω) ⊗ Amentioning

confidence: 99%

Stability of derivations under weak-2-local continuous perturbations

Jordá

Peralta

2016

Aequat. Math.

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show abstract

“…Theorem 4.3 in [1] implies that T is a (continuous) Jordan * -homomorphism on B(H). A remarkable result of E. Størmer, assures that T is a * -homomorphism or a * -anti-homomorphism (cf.…”

Section: Proof Of Theorem 22mentioning

confidence: 99%

“…According to the terminology in [1], every mapping T in the hypothesis of this Theorem is a extreme-strong-local * -automorphism on B(H). Theorem 4.3 in [1] implies that T is a (continuous) Jordan * -homomorphism on B(H).…”

Section: Proof Of Theorem 22mentioning

confidence: 99%

“…It is shown in [1] that bilocal * -automorphisms and extreme-strong-local * -automorphisms on B(H) define the same applications. Theorem 4.3 in [1] established that every extreme-strong-local * -automorphism on an atomic von Neumann algebra is a Jordan * -homomorphism.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 4.3 in [1] established that every extreme-strong-local * -automorphism on an atomic von Neumann algebra is a Jordan * -homomorphism. The conclusion of this result is no longer true for general von Neumann algebras, even nor for von Neumann algebras which are bidual Banach spaces (compare [1,comments preceding Theorem 4.3]). …”

Section: Introductionmentioning

confidence: 99%

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