Centralizing traces and Lie triple isomorphisms on triangular algebras

Abstract

Let be a triangular algebra over a commutative ring and be the centre of . Suppose that is an -bilinear mapping and that is a trace of . We describe the form of satisfying the condition for all . The question of when has the standard form will be addressed. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism on to be almost standard. As applications, we characterize Lie triple isomorphisms of upper triangular matrix algebras and nest algebras. Some further research topics related to current work are proposed at the end of this article.

Keywords:

• centralizing trace
• Lie triple isomorphism
• commuting trace
• triangular algebra
• nest algebra

AMS Subject Classifications:

• 15A78
• 47L35

Acknowledgements

This research was done when the third author visited the School of Mathematics and Statistics at Beijing Institute of Technology in the summer of 2013. She takes this opportunity to express her sincere thanks to the School of Mathematics and Statistics and the Office of International Affairs at Beijing Institute of Technology for the hospitality extended to them during her visit. We are deeply grateful to a special Training Programme of International Exchange and Cooperation of the Beijing Institute of Technology. We are sincerely grateful to Professor Hwa-Long Gau for his kind considerations and warm help. We greatly appreciate the anonymous referee for a very thorough reading of the manuscript and for many valuable comments.

Additional information

Funding

The work of the first author is supported by the National Natural Science Foundation of China [grant number 11226068] and [grant number 11301195]. The work of the second author is partially supported by the National Natural Science Foundation of China [grant number 10871023].