Holomorphic functions on symmetric Banach manifolds of compact type are constant
By a deep theorem of Kaup, every bounded symmetric domain can be identified with the open unit ball \(B_Z\) of a Banach space \(Z\) which has the structure of a \(JB^*\)-triple \((Z,\{.,.,.\})\). Then \((Z,-\{.,.,.\})\) is a \(J^*\)-triple and -- again by Kaup -- gives rise to a simply connected symmetric Banach manifold \(\widetilde{M}_Z\) called the dual of \(B_Z\). When \(Z\) is finite-dimensional, \(\widetilde{M}_Z\) is compact. For infinite-dimensional \(Z\) this is no longer the case: \(\widetilde{M}_Z\) is modeled locally on \(Z\). However, \(\widetilde{M}_Z\) has come to be known as the symmetric manifold of compact type associated to \(Z\) because it shares some properties with true compact manifolds. In this paper the authors prove a key conjecture reinforcing the compact-like behaviour of \(\widetilde{M}_Z\): every holomorphic function \(f:\widetilde{M}_Z\longrightarrow C\) is constant. The result is proved first for the quasi-invertible manifold associated to a \(JB^*\)-triple, and then extended to \(\widetilde{M}_Z\) through a description of its local structure.
scientific article; zbMATH DE number 1336830