Noncommutative end theory
type: Item
instance of: scholarly article
title: Noncommutative end theory
zbMATH Open document ID: 0921.46079
author: Charles A. Akemann
author: Søren Eilers
published in: Pacific Journal of Mathematics
publication date: 1997-09-16T00:00:00Z
review text: The corona algebra \(M({\mathfrak A})/{\mathfrak A}\) contains essential information on the global structure of \({\mathfrak A}\), as demonstrated for instance by Busby theory. It is an interesting and surprisingly difficult task to determine the ideal structure of \(M({\mathfrak A})/{\mathfrak A}\) by means of the internal structure of \({\mathfrak A}\). Toward this end, we generalize Freudenthal's classical theory of ends of topological spaces to a large class of \(C^*\)-algebras. However, mirroring requirements necessary already in the commutative case, we must restrict attention to \(C^*\)-algebras \({\mathfrak A}\) which are \(\sigma\)-unital and have connected and locally connected spectra. Furthermore, we must study separately a certain pathological behavior which occurs in neither commutative nor stable \(C^*\)-algebras. We introduce a notion of sequences determining ends in such a \(C^*\)-algebra \({\mathfrak A}\) and pass to a set of equivalence classes of such sequences, the ends of \({\mathfrak A}\). We show that ends are in a natural 1-1 correspondence with the set of components of \(M({\mathfrak A})/{\mathfrak A}\), hence giving a complete description of the complemented ideals of such corona algebras. As an application we show that corona algebras of primitive \(\sigma\)-unital \(C^*\)-algebras are prime. Furthermore, we employ the methods developed to show that, for a large class of \(C^*\)-algebras, the end theory of a tensor product of two nonunital \(C^*\)-algebras is always trivial.
Mathematics Subject Classification ID: 46L85
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 54D40
Mathematics Subject Classification ID: 46M05
zbMATH DE Number: 1060823
zbMATH Keywords: noncommutative topology
zbMATH Keywords: \(C^*\)-algebras
zbMATH Keywords: corona algebras
zbMATH Keywords: end theory
zbMATH Keywords: Freudenthal compactification
zbMATH Keywords: Busby theory
zbMATH Keywords: tensor product
MaRDI profile type: Publication
full work available at URL: pjm.1998.185.47
OpenAlex ID: W2073351041
DOI: 10.2140/PJM.1998.185.47
description: scientific article; zbMATH DE number 1060823