Finite universal Korovkin systems in tensor products of commutative Banach algebras
type: Item
instance of: scholarly article
title: Finite universal Korovkin systems in tensor products of commutative Banach algebras
zbMATH Open document ID: 0764.46052
DOI: 10.1007/BF02844688
author: Michael Pannenberg
publication date: 1992-09-27T00:00:00Z
review text: Let \(A\), \(B\) be commutative unital Banach algebras. A linear map \(L: A\to B\) is called a spectral contraction, if \(\rho(Lx)\leq\rho(x)\) (\(x\in A\)), where \(\rho\) denotes the spectral radius. A subset \(T\) of \(A\) is said to be a universal Korovkin system for \(A\) if the convergence of \(L_ \alpha x\) to \(Lx\) for all \(x\in T\) already implies \(L_ \alpha y\to Ly\) for all \(y\in A\), whenever \(L_ \alpha: A\to B\) is a net of spectral contractions taking values in an arbitrary uniform algebra \(B\) and \(L: A\to B\) is a unital algebra homomorphism. The main interest in Korovkin approximation theory is focussed on the case of finite universal Korovkin systems. The author answers the question whether the existence of a finite universal Korovkin system is a hereditary property with respect to formation of tensor products in the category of commutative unital Banach algebras. Theorem 2.5. Let \(A\), \(B\) be commutative unital Banach algebras. (i) \(A\widehat {\otimes}_ \pi B\) possesses a finite universal Korovkin system \(\Leftrightarrow A\) as well as \(B\) possess a finite universal Korovkin system. (ii) If \(A\) is a uniform algebra, we have: \(A\widehat {\otimes}_ \varepsilon B\) possesses a finite universal Korovkin system \(\Leftrightarrow A\) as well as \(B\) posess a finite universal Korovkin system. Further, the author discusses some natural applications of that theorem to Banach algebras of algebra-valued continuous functions. The results remain valid in the context of Waalbroeck algebras (i.e. complete unital locally multiplicatively convex \(Q\)-algebras), for Korovkin approximation in Waalbroeck algebras see \textit{M. Pannenberg} [Math. Ann. 274, 423-437 (1986; Zbl 0576.41014)]. Finally, the author states permanence results for direct sums, ideals and quotient algebras.
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 46J05
zbMATH DE Number: 60569
zbMATH Keywords: spectral contraction
zbMATH Keywords: spectral radius
zbMATH Keywords: universal Korovkin system
zbMATH Keywords: uniform algebra
zbMATH Keywords: unital algebra homomorphism
zbMATH Keywords: Korovkin approximation
zbMATH Keywords: universal Korovkin systems
zbMATH Keywords: hereditary property
zbMATH Keywords: tensor products
zbMATH Keywords: category of commutative unital Banach algebras
zbMATH Keywords: Banach algebras of algebra-valued continuous functions
zbMATH Keywords: Waalbroeck algebras
zbMATH Keywords: complete unital locally multiplicatively convex \(Q\)-algebras
reviewed by: M. Fritzsche
MaRDI profile type: Publication
cites work: Q3660348
cites work: Q3667430
cites work: On the universal convergence sets
cites work: Q3719215
cites work: Q3948089
cites work: Approximation and Abstract Boundaries
cites work: Q5637432
cites work: Q3215824
cites work: Q5598133
cites work: The Slice Product of Function Algebras
cites work: Q3941808
cites work: Constructive Techniques in Rational Approximation
cites work: Uniform cross norms and tensor products of Banach algebras
cites work: Q4092470
cites work: A Choquet Boundary for the Product of Two Compact Spaces
cites work: Produits tensoriels topologiques et espaces nucléaires
cites work: Q5680616
cites work: Q3730448
cites work: A characterization of homomorphisms in certain Banach involution algebras
cites work: Q3721143
cites work: Korovkin approximation in Waelbroeck algebras
cites work: A discrete integral representation for polynomials of fixed maximal degree and their universal Korovkin closure.
cites work: A characterization of C(X) among algebras on planar sets by the existence of a finite universal Korovkin system
cites work: On integral representation and the Choquet boundary for convolution algebras of measures
cites work: Q4066086
cites work: Q4774586
cites work: Tensor products of commutative Banach algebras
cites work: Q5557339
cites work: Topological vector spaces and algebras
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 60569