Peirce inner ideals in Jordan\(^*\)-triples
type: Item
instance of: scholarly article
title: Peirce inner ideals in Jordan\(^*\)-triples
zbMATH Open document ID: 0847.17028
published in: Journal of Algebra
publication date: 1996-06-05T00:00:00Z
review text: A complex Jordan*-triple \(A\) is said to be anisotropic if \(\{ aaa \}\neq 0\) whenever \(a\) is in \(A\setminus \{0\}\). Let \(A\) be an anisotropic complex Jordan*-triple, and \(J\) be an inner ideal of \(A\). The annihilator \(J^\perp\) and the kernel \(\text{Ker} (J)\) of \(J\) in \(A\) are defined by means of the equalities \(J^\perp:= \{a\in A: \{ JaA \}=0\}\) and \(\text{Ker} (J):= \{a\in A: \{JaJ \}= 0\}\), respectively, so that \(J^\perp\) is an inner ideal of \(A\) contained in \(\text{Ker} (J)\). \(J\) is said to be complemented in \(A\) if \(A= J\oplus \text{Ker} (J)\). The paper under review is devoted to studying complemented inner ideals \(J\) such that \(J^\perp\) is complemented too. In this case, if we put \(A_0 (J):= J^\perp\), \(A_1 (J):= \text{Ker} (J)\cap \text{Ker} (J^\perp)\), and \(A_2 (J): =J\), then we have \(A= A_0 (J) \oplus A_1 (J) \oplus A_2 (J)\). A classical and well-understood example of the situation above is the one when \(J= \{u \{uAu \}u\}\) for a given tripotent \(u\) in \(A\). In this particular setting, it is well-known that the so-called ``Peirce rules'' (i.e., \(\{A_1 (J) A_j (J) A_k (J)\} \subseteq A_{i+k-j} (J)\) when \(i+k-j\) is equal to 0, 1, or 2, and \(\{A_i (J) A_j (J) A_k (J)\}= \{0\}\) otherwise) hold. The authors prove that, in the general context considered here, (identifying pairs of symmetric relations) all except six of the ``generalized'' Peirce rules hold. More precisely, the only possible exceptions arise when \((i, j,k)\) is equal to \((0, 1,1)\), \((1,1, 0)\), \((1,0, 1)\), \((2,1,1)\), \((1,1,2)\), \((1,2,1)\), \((1,1,1)\), \((2,1,0)\), or \((0,1,2)\). Let \(A\) be a JBW*-triple, and \(J\) be an inner ideal of \(A\). Then \(A\) is anisotropic and, by previous results of the authors and \textit{K. McCrimmon} [J. Funct. Anal. 139, 196-224 (1996)], \(J\) is complemented in \(A\) if and only if it is weak*-closed in \(A\). Since \(J^\perp\) is automatically weak*-closed in \(A\) (for every \(J\)), if \(J\)…
Mathematics Subject Classification ID: 17C50
Mathematics Subject Classification ID: 17C10
Mathematics Subject Classification ID: 17C65
Mathematics Subject Classification ID: 46L89
zbMATH DE Number: 869799
zbMATH Keywords: Peirce decomposition
zbMATH Keywords: Peirce rules
zbMATH Keywords: complex Jordan*-triple
zbMATH Keywords: complemented inner ideals
zbMATH Keywords: JBW*-triple
zbMATH Keywords: generalized Peirce rules
author: C. M. Edwards
author: Gottfried T. Rüttimann
reviewed by: Ángel Rodríguez Palacios
MaRDI profile type: Publication
full work available at URL: jabr.1996.0051
OpenAlex ID: W2025336686
DOI: 10.1006/JABR.1996.0051
description: scientific article; zbMATH DE number 869799