Central Extension of Mappings on von Neumann Algebras
نویسندگان
چکیده
Let M be a von Neumann algebra and ρ : M → M be a ∗-homomorphism. Then ρ is called a centrally extendable ∗-homomorphism (CEH) if there is a maximal abelian subalgebra (masa) M of the commutant M of M and a surjective ∗-homomorphism φ : M → M such that φ(Z) = ρ(Z) for all Z in the center of M. A ∗-ρderivation δ : M → M is called a centrally extendable ∗-ρ-derivation (CED) if there is a masa M of M such that δ has a norm preserving extension δ̃ : C(M,M) → C(M,M) which is a ∗-ρ̃-derivation for some ∗-homomorphism ρ̃ : C(M,M) → C(M,M) as an extension of ρ, where C(M,M) is the C-algebra generated by M ∪ M. In this paper we give some sufficient conditions for a ∗-homomorphism to be a CEH and prove that δ is a CED if and only if ρ is a CEH. Thus the study of ρ-derivations on arbitrary von Neumann algebras is reduced to the case of type I von Neumann algebras. 2000 Mathematics Subject Classification: Primary 46L57; Secondary 46L05, 47B47
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