{"title":"Horofunctions and metric compactification of noncompact Hermitian symmetric spaces","authors":"Cho-Ho Chu, María Cueto-Avellaneda, Bas Lemmens","doi":"10.1007/s10231-023-01419-7","DOIUrl":null,"url":null,"abstract":"<div><p>Given a Hermitian symmetric space <i>M</i> of noncompact type, we show, among other things, that the metric compactification of <i>M</i> with respect to its Carathéodory distance is homeomorphic to a closed ball in its tangent space. We first give a complete description of the horofunctions in the compactification of <i>M</i> via the realisation of <i>M</i> as the open unit ball <i>D</i> of a Banach space <span>\\((V,\\Vert \\cdot \\Vert )\\)</span> equipped with a particular Jordan structure, called a <span>\\(\\textrm{JB}^*\\)</span>-triple. We identify the horofunctions in the metric compactification of <span>\\((V,\\Vert \\cdot \\Vert )\\)</span> and relate its geometry and global topology, via a homeomorphism, to the closed unit ball of the dual space <span>\\(V^*\\)</span>. Finally, we show that the exponential map <span>\\(\\exp _0 :V \\longrightarrow D\\)</span> at <span>\\(0\\in D\\)</span> extends to a homeomorphism between the metric compactifications of <span>\\((V,\\Vert \\cdot \\Vert )\\)</span> and <span>\\((D,\\rho )\\)</span>, preserving the geometric structure, where <span>\\(\\rho \\)</span> is the Carathéodory distance on <i>D</i>. Consequently, the metric compactification of <i>M</i> admits a concrete realisation as the closed dual unit ball of <span>\\((V,\\Vert \\cdot \\Vert )\\)</span>.</p></div>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annali di Matematica Pura ed Applicata","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10231-023-01419-7","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Given a Hermitian symmetric space M of noncompact type, we show, among other things, that the metric compactification of M with respect to its Carathéodory distance is homeomorphic to a closed ball in its tangent space. We first give a complete description of the horofunctions in the compactification of M via the realisation of M as the open unit ball D of a Banach space \((V,\Vert \cdot \Vert )\) equipped with a particular Jordan structure, called a \(\textrm{JB}^*\)-triple. We identify the horofunctions in the metric compactification of \((V,\Vert \cdot \Vert )\) and relate its geometry and global topology, via a homeomorphism, to the closed unit ball of the dual space \(V^*\). Finally, we show that the exponential map \(\exp _0 :V \longrightarrow D\) at \(0\in D\) extends to a homeomorphism between the metric compactifications of \((V,\Vert \cdot \Vert )\) and \((D,\rho )\), preserving the geometric structure, where \(\rho \) is the Carathéodory distance on D. Consequently, the metric compactification of M admits a concrete realisation as the closed dual unit ball of \((V,\Vert \cdot \Vert )\).
期刊介绍:
This journal, the oldest scientific periodical in Italy, was originally edited by Barnaba Tortolini and Francesco Brioschi and has appeared since 1850. Nowadays it is managed by a nonprofit organization, the Fondazione Annali di Matematica Pura ed Applicata, c.o. Dipartimento di Matematica "U. Dini", viale Morgagni 67A, 50134 Firenze, Italy, e-mail annali@math.unifi.it).
A board of Italian university professors governs the Fondazione and appoints the editors of the journal, whose responsibility it is to supervise the refereeing process. The names of governors and editors appear on the front page of each issue. Their addresses appear in the title pages of each issue.