Semisimple triangular AF algebras
type: Item
instance of: scholarly article
title: Semisimple triangular AF algebras
zbMATH Open document ID: 0808.47031
author: Allan Donsig
published in: Journal of Functional Analysis
publication date: 1993-04-28T00:00:00Z
review text: Necessary and sufficient conditions for a triangular AF algebra to be semisimple are given. In particular, a triangular AF algebra which can be written using the standard embedding infinitely often is semisimple; also a semisimple triangular AF algebra is given which does not have a presentation of this form. If two triangular AF algebras have the same Peters-Poon-Wagner diagonal invariant, then either both are semisimple or both are not. However, two algebras with the same diagonal invariant are given, where one has the Jacobson radical equal to the strong radical and the other does not. Semisimplicity can be characterized in terms of Power's fundamental relation. Generalizations of these results to triangular subalgebras of groupoid \(C^*\)-algebras are given. It is shown that there is a unique maximal bimodule over the diagonal which does not intersect the radical. Using this, it is shown that the Wedderburn principal theorem does not hold for all triangular AF algebras. A necessary condition for the theorem to hold is that the above mentioned unique bimodule is a subalgebra.
Mathematics Subject Classification ID: 47L30
zbMATH DE Number: 152045
zbMATH Keywords: semisimplicity
zbMATH Keywords: triangular AF algebra
zbMATH Keywords: Jacobson radical
zbMATH Keywords: groupoid \(C^*\)-algebras
zbMATH Keywords: radical
zbMATH Keywords: Wedderburn principal theorem
MaRDI profile type: Publication
full work available at URL: jfan.1993.1016
OpenAlex ID: W2020904909
DOI: 10.1006/JFAN.1993.1016
description: scientific article; zbMATH DE number 152045