On the spectral theory for Rickart ordered *-algebras
type: Item
instance of: scholarly article
zbMATH Open document ID: 1261.46050
author: Dmitry Sh. Goldstein
author: Alexander A. Katz
author: Roman B. Shklyar
publication date: 2012-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1012.5262
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46K05
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 06B99
zbMATH DE Number: 6077741
zbMATH Keywords: *-algebra
zbMATH Keywords: self-adjoint element
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: spectral theorem
zbMATH Keywords: \(\sigma\)-complete lattice
MaRDI profile type: Publication
arXiv ID: 1012.5262
arXiv classification: math.OA
arXiv classification: math.FA
title: On the spectral theory for Rickart ordered *-algebras
published in: Review Bulletin of the Calcutta Mathematical Society
review text: The authors introduce and study the concept of a Rickart ordered *-algebra (\(RO^*\)-algebra). A Rickart *-algebra is an algebra \(A\) with an involution such that for any element \(x\) in \(A\) its right annihilator is a right principal ideal generated by a projection. If, moreover, additional conditions are added (every positive element has a square root contained in its bicommutant, and some suprema of finite sums of positive elements are self-adjoint), then it is said to be an \(RO^*\)-algebra. The authors define bounded elements in \(A\) and show that the set of all these elements is an \(RO^*\)-algebra which, in a certain norm, is a \(C^*\)-algebra. They show that if \(A\) is commutative, then the set of its self-adjoint elements is a conditionally \(\sigma\)-complete lattice. It is also shown that a maximal commutative subalgebra of \(A\) is an \(RO^*\)-algebra and this result is used for obtaining the spectral decomposition of a self-adjoint element of \(A\).
reviewed by: Wiesław Tadeusz Żelazko
description: scientific article; zbMATH DE number 6077741