Holomorphic functions on symmetric Banach manifolds of compact type are constant
type: Item
instance of: scholarly article
title: Holomorphic functions on symmetric Banach manifolds of compact type are constant
zbMATH Open document ID: 0937.46042
DOI: 10.1007/PL00004681
author: Seán Dineen
author: Pauline Mellon
published in: Mathematische Zeitschrift
publication date: 1999-09-29T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46H70
zbMATH DE Number: 1336830
zbMATH Keywords: symmetric manifold
zbMATH Keywords: compact type
zbMATH Keywords: JB*-triple
reviewed by: Ignacio Zalduendo
MaRDI profile type: Publication
full work available at URL: pl00004681
OpenAlex ID: W2092405391
description: scientific article; zbMATH DE number 1336830