Continuous maps preserving Jordan triple products from 𝔾𝕃₁ to 𝔾𝕃₂ and 𝔾𝕃₃

Abstract

Let ${\mathbb{G}\mathbb{L}}_n$ be a complex general linear group. In this paper we determine the structure of all Jordan triple product maps from ${\mathbb{G}\mathbb{L}}_1$ to ${\mathbb{G}\mathbb{L}}_m$ for $m=2,3$. The results are applied to determine the structure of all continuous Jordan triple product maps between ${\mathbb{G}\mathbb{L}}_1$ and ${\mathbb{G}\mathbb{L}}_m$ for $m=2,3$. These are the continuous maps $\varphi :{\mathbb{G}\mathbb{L}}_1\to {\mathbb{G}\mathbb{L}}_m$ which satisfy $\varphi \left(zwz\right)=\varphi \left(z\right)\varphi \left(w\right)\varphi \left(z\right),z,w\in {\mathbb{G}\mathbb{L}}_1.$

COMMUNICATED BY:

• L. Molnar

Keywords:

• General linear group
• Jordan triple product map

2010 Mathematics Subject Classifications:

• 15A60
• 15A86

Acknowledgments

The authors are thankful to the referee for carefully reading the paper and for his valuable suggestions and remarks.

Disclosure statement

No potential conflict of interest was reported by the authors.