Strongly asymptotic morphisms on separable metrisable algebras
type: Item
instance of: scholarly article
title: Strongly asymptotic morphisms on separable metrisable algebras
zbMATH Open document ID: 0983.46041
author: Edwin Beggs
published in: Journal of Functional Analysis
publication date: 2002-04-07T00:00:00Z
review text: \textit{A. Connes} and \textit{N. Higson} [C. R. Acad Sci., Paris, Ser. I 311, No. 2, 101-106 (1990; Zbl 0717.46062)], dealing with the category of separable \(C^{*}\)-algebras, defined an asymptotic morphism to be a family \((f_{t})\) of mappings from \(A\) to \(B\), say \(t \in [0,\infty)\), such that, as \(t \to \infty\), \(f_{t}\) tends to act like a homomorphism, without the assumption that \(f_{t}\) converges as \(t \to \infty\). The methods are closely related to those of continuous fields of \(C^{*}\)-algebras. They defined a composition \( (g \circ_{\varphi} f)_{t}(a) = g_{\varphi(t)}(f_{t}(a))\) of asymptotic morphisms \(f_{t}:A \to B\) and \(g_{t}:A \to \)C where \(\varphi\) is an increasing positive valued function which acts as a change of parameter for \(g\) from \(t\) to \(\varphi(t)\). Two asymptotic morphisms \(f\) and \(g\) with values in \(B\) are called homotopy equivalent if there exists an asymptotic morphism \(F_{t} : A \to C([0,1]) \otimes B\) such that \(F_{t}(a)(0) = f_{t}(a)\) and \(F_{t}(a)(1) = g_{t}(a)\) for all \(t \geq 0\) and \(a \in A\). The author adapts the above to separable metrisable algebras; the difficulty is to ensure that the composition of asymptotic morphisms remains an asymptotic morphism. He gives sufficiency conditions to take account of the metric and the operations of algebraic and scalar multiplication (and involution, if necessary) and defines strong asymptotic continuity of asymptotic morphisms. To construct a reparametrisation \(\varphi\) he uses the fact that for separable metric spaces every cover has a countable subcover, and he shows that strong asymptotic continuity of \(f_{t}\) and \( g_{t}\) implies that of \((g \circ_{\varphi} f)_{t}\) . This leads to a definition of strongly asymptotic morphisms of algebras for which multiplication is continuous with respect to a translation-invariant metric. For the composition to be well-defined at the level of homotopy class a change of parameter should not affect the…
reviewed by: Aubrey Wulfsohn
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46A03
Mathematics Subject Classification ID: 46A16
zbMATH DE Number: 1542413
zbMATH Keywords: metrisable algebra
zbMATH Keywords: morphism
zbMATH Keywords: asymptotics
zbMATH Keywords: homotopy
zbMATH Keywords: locally convex
zbMATH Keywords: Lindelöf principle
zbMATH Keywords: involution
zbMATH Keywords: continuous fields of \(C^{*}\)-algebras
zbMATH Keywords: category of separable \(C^{*}\)-algebras
zbMATH Keywords: asymptotic morphisms
zbMATH Keywords: separable metrisable algebras
zbMATH Keywords: strong asymptotic continuity
zbMATH Keywords: pointwise bounded asymptotic morphisms
zbMATH Keywords: compatability of compositions
zbMATH Keywords: asymptotic integral
zbMATH Keywords: exact sequence
zbMATH Keywords: asymptotic equivalence
zbMATH Keywords: suspension
MaRDI profile type: Publication
full work available at URL: jfan.2000.3651
OpenAlex ID: W2087522764
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DOI: 10.1006/JFAN.2000.3651
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description: scientific article; zbMATH DE number 1542413