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Let $$\mathfrak {M}$$ M be a von Neumann algebra, and let $$\mathfrak {T}:\mathfrak {M} \rightarrow \mathfrak {M}$$ T : M → M be a bounded linear map satisfying $$\mathfrak {T}(P^{2}) = \mathfrak {T}(P)P + \Psi (P,P)$$ T ( P 2 ) = T ( P ) P + Ψ ( P , P ) for each projection P of $$\mathfrak {M}$$ M , where $$\Psi :\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}$$ Ψ : M × M → M is a bi-linear map. If $$\Psi $$ Ψ is a bounded l-semi Hochschild 2-cocycle, then $$\mathfrak {T}$$ T is a left centralizer associated with $$\Psi $$ Ψ . By applying this conclusion, we offer a characterization of left $$\sigma $$ σ -centralizers, generalized derivations and generalized $$\sigma $$ σ -derivations on von Neumann algebras. Moreover, it is proved that if $$\mathfrak {M}$$ M is a commutative von Neumann algebra and $$\sigma :\mathfrak {M} \rightarrow \mathfrak {M}$$ σ : M → M is an endomorphism, then every bi- $$\sigma $$ σ -derivation $$D:\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}$$ D : M × M → M is identically zero.
ANNALI DELL'UNIVERSITA' DI FERRARA – Springer Journals
Published: Jun 21, 2017
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