Derivations and derivable maps on von Neumann algebras. Issue 15 (18th November 2021)

• Record Type: Journal Article
• Authors: An, Runling; Cai, Yaru
• Is Part Of: Linear & multilinear algebra. Volume 69:Issue 15(2021)
• Page Start: 2806
• Page End: 2812
• Publication Date: 2021-11-18
• Subjects: von Neumann algebras -- derivations -- derivable maps -- full-derivable points; 46L57 -- 47B47; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2021.1913087
• Languages: English
• ISSNs: 0308-1087

Abstract

Abstract : Let A be a von Neumann algebra, a linear map δ : A → A is said to be derivable at Ω ∈ A, if δ ( A B ) = δ ( A ) B + A δ ( B ) for every A, B ∈ A with A B = Ω . In this paper, we show that a norm-continuous linear map δ : A → A is derivable at an arbitrary but fixed operator Ω ∈ A if and only if there exists a linear derivation τ : A → A such that δ ( A ) = τ ( A ) + δ ( I ) A for all A ∈ A, where δ ( I ) is in the centre of A and δ ( I ) Ω = 0 . When A is a von Neumann algebra with no summands of type I 1 or a properly infinite von Neumann algebra, we obtain the same result by weakening the linearity and norm-continuity assumption of δ into additivity.