Goursat's lemma in the context of Banach algebras
type: Item
instance of: scholarly article
title: Goursat's lemma in the context of Banach algebras
zbMATH Open document ID: 1336.46040
DOI: 10.1134/S000143461507007X
published in: Mathematical Notes
publication date: 2015-10-26T00:00:00Z
review text: The author introduces a way of integrating along a line segment in a Banach space \(X\). A function \(f\) from an open subset \(V\) of \(X\) to the space \(L(X)\) of bounded linear operators on \(X\) is said to be in the Goursat class if the integral of \(f\) around every triangle, whose convex hull is contained in \(V\), is equal to zero. When \(V\) is convex it is shown that the set of functions in the Goursat class coincides with those functions that are the anti-derivative of a function \(g\) from \(V\) into \(X\). In the case when \(f\) is differentiable, using \(f'_x\) to denote the Fréchet derivative of \(f\) at \(x\), it is shown that a necessary and sufficient condition for \(f\) to be in the Goursat class is that \(((f'_x)a)b=((f'_x)b)a\) for all \(x\) in \(V\) and all \(a,b\) in \(X\). When \(X\) is a real Banach algebra without unit and \(f: V\to X\) is differentiable, it is shown that the left multiplication operator, \(a\mapsto f(x).a\), is in the Goursat class if and only if \(f'_x.[a,b]=0\) where \([a,b]\) denotes the commutator of \(a\) and \(b\).
reviewed by: Christopher Boyd
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 6498241
zbMATH Keywords: Goursat's lemma
zbMATH Keywords: Banach algebra
zbMATH Keywords: Fréchet derivative
zbMATH Keywords: multiplication operator
author: E. A. Gorin
MaRDI profile type: Publication
Wikidata QID: Q124820342
cites work: Q3244654
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description: scientific article; zbMATH DE number 6498241