Spatiality of Derivations of Operator Algebras in Banach Spaces

Quanyuan Chen and Xiaochun Fang

Abstract and Applied Analysis, 2011, vol. 2011, issue 1

Abstract: Suppose that 𝒜 is a transitive subalgebra of B(X) and its norm closure 𝒜¯ contains a nonzero minimal left ideal ℐ. It is shown that if δ is a bounded reflexive transitive derivation from 𝒜 into B(X), then δ is spatial and implemented uniquely; that is, there exists T ∈ B(X) such that δ(A) = TA − AT for each A ∈ 𝒜, and the implementation T of δ is unique only up to an additive constant. This extends a result of E. Kissin that “if 𝒜¯ contains the ideal C(H) of all compact operators in B(H), then a bounded reflexive transitive derivation from 𝒜 into B(H) is spatial and implemented uniquely.” in an algebraic direction and provides an alternative proof of it. It is also shown that a bounded reflexive transitive derivation from 𝒜 into B(X) is spatial and implemented uniquely, if X is a reflexive Banach space and 𝒜¯ contains a nonzero minimal right ideal ℐ.

Date: 2011
References: Add references at CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1155/2011/813723

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2011:y:2011:i:1:n:813723

Access Statistics for this article

More articles in Abstract and Applied Analysis from John Wiley & Sons
Bibliographic data for series maintained by Wiley Content Delivery ().