# Characterization of some derivations on von Neumann algebras via left centralizers

Characterization of some derivations on von Neumann algebras via left centralizers 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ANNALI DELL'UNIVERSITA' DI FERRARA Springer Journals

# Characterization of some derivations on von Neumann algebras via left centralizers

, Volume 64 (1) – Jun 21, 2017
12 pages      /lp/springer-journals/characterization-of-some-derivations-on-von-neumann-algebras-via-left-0WmIZxix5j
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general; Analysis; Geometry; History of Mathematical Sciences; Numerical Analysis; Algebraic Geometry
ISSN
0430-3202
eISSN
1827-1510
DOI
10.1007/s11565-017-0290-2
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

ANNALI DELL'UNIVERSITA' DI FERRARASpringer Journals

Published: Jun 21, 2017

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