Banach spaces with biholomorphically equivalent unit balls are isomorphic

Kaup, Wilhelm; Upmeier, Harald

doi:10.1090/S0002-9939-1976-0422704-3

Banach spaces with biholomorphically equivalent unit balls are isomorphic
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by Wilhelm Kaup and Harald Upmeier PDF

Proc. Amer. Math. Soc. 58 (1976), 129-133 Request permission

Abstract:

It is shown that in every complex Banach space $E$ with open unit ball $D \subset E$ there is a closed $C$-linear subspace $V \subset E$ such that $V \cap D$ is the orbit of the origin $0 \in E$ under the group ${\operatorname {Aut}}(D)$ of all biholomorphic automorphisms of $D$. In particular two complex Banach spaces are isometrically equivalent if and only if their open unit balls are biholomorphically equivalent.

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