Weak-2-local symmetric maps on C*-algebras

2015/10/04 by Cabello, Juan Carlos, Peralta, Antonio M.
#FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1510.00915

Abstract

We introduce and study weak-2-local symmetric maps between C^*-algebras A and B as non necessarily linear nor continuous maps Δ: A→ B such that for each a,b∈ A and ϕ∈ B^*, there exists a symmetric linear map T_a,b,ϕ: A→ B, depending on a, b and ϕ, satisfying ϕΔ(a) = ϕT_a,b,ϕ(a) and ϕΔ(b) = ϕT_a,b,ϕ(b). We prove that every weak-2-local symmetric map between C^*-algebras is a linear map. Among the consequences we show that every weak-2-local ^*-derivation on a general C^*-algebra is a (linear) ^*-derivation. We also establish a 2-local version of the Kowalski-Słodkowski theorem for general C^*-algebras by proving that every 2-local ^*-homomorphism between C^*-algebras is a (linear) ^*-homomorphism.

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