Imaginaries in Hilbert spaces
type: Item
instance of: scholarly article
title: Imaginaries in Hilbert spaces
zbMATH Open document ID: 1059.03021
author: Alexander Berenstein
author: Itaï Ben Yaacov
published in: Archive for Mathematical Logic
publication date: 2004-12-16T00:00:00Z
review text: The paper is a contribution to the model theory of Hilbert spaces. The authors work in a ``big'' Hilbert space \(\mathcal H\) and consider it as a multi-sorted structure whose sorts are the balls \(\{v: \| v\| \leq n\}\), for \(n<\omega\). Atomic predicates are \(r\leq\| \lambda_1x_1+\dots+\lambda_kx_k\| \leq s\), where \(r,s\in\mathbb R^{>0}\), all \(\lambda_i\) are in \(\mathbb R\) or \(\mathbb C\) (depending of the ground field of \(\mathcal H\)), and all \(x_i\) are of the same sort. Positive formulas are defined to be positive Boolean combinations of atomic formulas. Then with respect to positive formulas \(\mathcal H\) is a universal domain of a compact abstract theory, in the sense of \textit{I.~Ben-Yaacov} [J. Math. Log. 3, No.~1, 85--118 (2003; Zbl 1028.03034)]. A partial type is a set of positive formulas, possibly with parameters in \(\mathcal H\). A type-definable set is a set of realizations in \(\mathcal H\) of a partial type. An imaginary is the equivalence class \(a_E\) of a (possibly infinite!) tuple \(a\) modulo a type-definable equivalence relation \(E\). The goal of the paper is to characterize imaginaries in \(\mathcal H\). A type-definable set is called bounded if its cardinality is smaller than \(| \mathcal H| \). For imaginaries \(a\) and \(b\), we say that \(b\) is definable (bounded) over \(a\) if the set of realizations of \(\text{tp}(b/a)\) is a singleton (bounded set). The set of all \(b\) definable (bounded) over \(a\) is called the definable (bounded) closure of \(a\) (in symbols, \(\text{dcl}(a)\) and \(\text{bdd}(a)\), respectively). It is shown that \(\text{bdd}(a)\) is inter-definable with a unique Hilbert space \(H\subseteq\mathcal H\). Moreover, given \(H\), there is a Galois correspondence between the imaginaries \(a\) such that \(\text{bdd}(a)\) is inter-definable with \(H\) and the compact subgroups of \(U(H)\), the unitary group of \(H\) with the strong topology. The correspondence is given by \(a\mapsto\text{Gal}(a)\) and \(…
reviewed by: Oleg Belegradek
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C95
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 2123161
zbMATH Keywords: elimination of imaginaries
zbMATH Keywords: model theory of Hilbert spaces
MaRDI profile type: Publication
full work available at URL: s00153-003-0200-4
OpenAlex ID: W2023176173
DOI: 10.1007/S00153-003-0200-4
description: scientific article; zbMATH DE number 2123161