Summary: For large classes of complex Banach spaces (mainly operator spaces) we consider orbits of finite rank elements under the group of linear isometries. These are (in general) real-analytic submanifolds of infinite dimension but of finite CR-codimension. We compute the polynomial convex hull of such orbits

$M$ explicitly and show as main result that every continuous CR-function on

$M$ has a unique extension to the polynomial convex hull which is holomorphic in a certain sense. This generalizes to infinite dimensions results from a recent joint paper of the author and

*D. Zaitsev* [Invent. Math. 153, No. 1, 45–104 (2003;

Zbl 1027.32032)].