Über Banachverbandsalgebren mit multiplikativen Zerlegungseigenschaft. (Banach lattice algebras with multiplicative decomposition property)
type: Item
instance of: scholarly article
title: Über Banachverbandsalgebren mit multiplikativen Zerlegungseigenschaft. (Banach lattice algebras with multiplicative decomposition property)
zbMATH Open document ID: 0691.46032
DOI: 10.1007/BF01951573
published in: Acta Mathematica Hungarica
publication date: 1988-01-01T00:00:00Z
review text: Let A be a real Banach lattice algebra and denote its canonical complexification by \(A_{{\mathbb{C}}}\). Let \(L^ r(A,A')\) denote the order- complete Banach lattice of regular linear mappings (i.e. expressible as difference of positive linear mappings) of A into its topological dual and let M: A\({}'\to L^ r(A,A')\) be the linear operator defined by \[ (M\mu)(x)(z)=\mu (xz)\quad for\quad all\quad x,z\in A,\quad \mu \in A'. \] The same symbol M is used for its canonical complexification. A multiplicative decomposition property for \(A_{{\mathbb{C}}}\) was defined by the author in J. Funct. Anal. 37, 382-400 (1988; Zbl 0478.46017), and is proved to be equivalent to the property used by \textit{J. L. Taylor} [Trans. Am. Math. Soc. 119, 150-166 (1965; Zbl 0128.355)] in his definition of abstract convolution measure algebras (whose Banach structures are L-spaces) and also equivalent to M being a lattice homomorphism. The latter result is evident from the tensor-product formulation of his property by \textit{J. L. Taylor} [Trans. Am. Math. Soc. 135, 105-113 (1969; Zbl 0189.140)]. A Banach lattice algebra is called an L-algebra of its underlying set is an abstract L-space (which the author usually, but not always, calls an AL-space). The author does not bother to define AL-algebras, a species of L-algebras, which he used in the earlier article (loc. cit.). The absolute kernel N(\(\mu)\) of a positive linear form \(\mu\) on A is defined to be \(\{\) \(x\in A: \mu (| x|)=0\}\). For a positive non-trivial submultiplicative form A/N(\(\mu)\) is an L-algebra, denoted \(A^{\mu}\), and if \(\mu\) is multiplicative it is an AL-algebra. If A has the multiplicative decomposition property then also \(A^{\mu}\) does. The author is thus able to use the identification of the dual of an L-space of continuous functions on an extremely disconnected set K. The results of this paper generalize those of the author's previous article (loc. cit.) where he used AL-algebras, and are an attempt …
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 46H10
zbMATH DE Number: 4131036
zbMATH Keywords: submultiplicative positive linear forms
zbMATH Keywords: Banach lattice algebra
zbMATH Keywords: order- complete Banach lattice of regular linear mappings
zbMATH Keywords: AL-algebras
zbMATH Keywords: multiplicative decomposition property
zbMATH Keywords: Gelfand spectral theory
zbMATH Keywords: multiplicative linear forms
zbMATH Keywords: central homomorphisms
zbMATH Keywords: maximal structural ideals
zbMATH Keywords: absolute kernels
author: Egon Scheffold
MaRDI profile type: Publication
cites work: Finite dimensional convolution algebras
cites work: Representations of L -Algebras
cites work: Extensions and Liftings of Positive Linear Mappings on Banach Lattices
cites work: Q4049667
cites work: Fixräume regulärer Operatoren und ein Korovkin-Satz
cites work: Über komplexe Banachverbandsalgebren
cites work: Banachverbandsalgebren mit einer natürlichen Wedderburn-Zerlegung
cites work: The Structure of Convolution Measure Algebras
cites work: Noncommutative convolution measure algebras
full work available at URL: https://doi.org/10.1007/bf01951573
OpenAlex ID: W2081851489
description: scientific article; zbMATH DE number 4131036