Continuous Jordan triple endomorphisms of $$\pmb {\mathbb{G}\mathbb{L}_{2}(\mathbb {C})}$$ G L 2 ( C )

S. E. Ghasempouri, A. KhaliliAsboei and S. S. SalehiAmiri ()
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S. E. Ghasempouri: University of New Brunswick
A. KhaliliAsboei: Farhangian University
S. S. SalehiAmiri: Isalamic Azad University

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 2, 691-708

Abstract: Abstract Let $$\mathbb{G}\mathbb{L}_{n}(\mathbb {C})$$ G L n ( C ) be the complex general linear group of degree n. In this paper, we present the general form of all continuous endomorphisms of $$\mathbb{G}\mathbb{L}_{2}(\mathbb {C})$$ G L 2 ( C ) with respect to the Jordan triple product. These are the continuous maps $$ \varphi :\mathbb{G}\mathbb{L}_{2}(\mathbb {C}) \rightarrow \mathbb{G}\mathbb{L}_{2}(\mathbb {C})$$ φ : G L 2 ( C ) → G L 2 ( C ) which satisfy $$\begin{aligned} \varphi (ABA) = \varphi (A) \varphi (B) \varphi (A) ,~~~~ A, B \in \mathbb{G}\mathbb{L}_{2}(\mathbb {C}). \end{aligned}$$ φ ( A B A ) = φ ( A ) φ ( B ) φ ( A ) , A , B ∈ G L 2 ( C ) . As a result, we present the general form of all continuous homomorphisms and automorphisms of $$\mathbb{G}\mathbb{L}_{2}(\mathbb {C})$$ G L 2 ( C ) .

Keywords: General linear group; Isometry; Jordan triple product map; 15A60; 15A86 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-023-00395-1

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