Abstract and Applied Analysis
Volume 2011 (2011), Article ID 813723, 13 pages
http://dx.doi.org/10.1155/2011/813723
Research Article

Spatiality of Derivations of Operator Algebras in Banach Spaces

Quanyuan Chen1,2 and Xiaochun Fang1

1Department of Mathematics, Tongji University, Shanghai 200092, China
2Department of Information Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China

Received 5 June 2011; Revised 17 August 2011; Accepted 9 September 2011

Academic Editor: Wolfgang Ruess

Copyright © 2011 Quanyuan Chen and Xiaochun Fang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Suppose that is a transitive subalgebra of and its norm closure contains a nonzero minimal left ideal . It is shown that if is a bounded reflexive transitive derivation from into , then is spatial and implemented uniquely; that is, there exists such that for each , and the implementation of is unique only up to an additive constant. This extends a result of E. Kissin that “if contains the ideal of all compact operators in , then a bounded reflexive transitive derivation from into 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 into is spatial and implemented uniquely, if is a reflexive Banach space and contains a nonzero minimal right ideal .