Pullback and pushout constructions in \(C^*\)-algebra theory
type: Item
instance of: scholarly article
title: Pullback and pushout constructions in \(C^*\)-algebra theory
zbMATH Open document ID: 0944.46063
published in: Journal of Functional Analysis
publication date: 2000-09-20T00:00:00Z
review text: The paper is dedicated to the category of \(C^*\)-algebras with *-homomorphisms as morphisms. Many of the results require proper morphisms. ``Pullback'' and ``pushout'' are certain commutative diagrams of \(C^*\)-algebras, often used in the theory of extensions. The author undertakes a thorough study of such diagrams in relation to restricted direct sums and amalgamated free products of \(C^*\)-algebras. It is shown that pullback and pushout diagrams are stable under tensoring with a fixed algebra and under crossed products with a fixed group. From the abstract: ``The crowning result is that if three short exact sequences of \(C^*\)-algebras are given, with appropriate morphisms between the sequences allowing for pullback or pushout constructions at the levels of ideals, algebras and quotients, then the three new \(C^*\)-algebras will again form a short exact sequence under some mild extra conditions. As a generalization of a theorem of T. A. Loring, it is shown that each morphism between a pair of \(C^*\)-algebras, combined with its extension to the stabilized algebras, gives rise to a pushout diagram''.
Mathematics Subject Classification ID: 46M15
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L85
Mathematics Subject Classification ID: 46L09
zbMATH DE Number: 1367849
zbMATH Keywords: category of \(C^*\)-algebras
zbMATH Keywords: *-homomorphisms as morphisms
zbMATH Keywords: proper morphisms
zbMATH Keywords: restricted direct sums
zbMATH Keywords: amalgamated free products
zbMATH Keywords: pullback and pushout diagrams are stable under tensoring with a fixed algebra and under crossed products with a fixed group
zbMATH Keywords: short exact sequence
author: Gert Kjaergård Pedersen
reviewed by: Khristo N. Boyadzhiev
MaRDI profile type: Publication
full work available at URL: jfan.1999.3456
OpenAlex ID: W2001928959
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DOI: 10.1006/JFAN.1999.3456
description: scientific article; zbMATH DE number 1367849