Behaviour of holomorphic automorphisms on equicontinuous subsets of the space 𝒞(Ω,ℰ)

Isidro, J.

doi:10.1090/S0002-9939-99-04585-2

Behaviour of holomorphic automorphisms on equicontinuous subsets of the space $\mathcal C(\Omega ,E)$
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by J. M. Isidro PDF

Proc. Amer. Math. Soc. 127 (1999), 437-446 Request permission

Abstract:

Consider a compact Hausdorff topological space $\Omega$, a $\text {JB}^{\ast }$-triple $E$ and $F: = {\mathcal {C}}(\Omega , E)$, the $\text {JB}^{\ast }$-triple of all continuous $E$-valued functions $f\colon \Omega \to E$ with the pointwise operations and the norm of the supremum. Let ${\mathsf {G}}$ be the group of all holomorphic automorphisms of the unit ball $B_{F}$ of $F$ that map every equicontinuous subset lying strictly inside $B_{F}$ into another such a set. The real Banach-Lie group ${\mathsf {G}}$ and its Lie algebra are investigated. The identity connected component of ${\mathsf {G}}$ is identified when $E$ has the strong Banach-Stone property. This extends to the infinite dimensional setting a well known result concerning the case $E={\mathbb {C}}$.

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