Finite universal Korovkin systems in tensor products of commutative Banach algebras
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.
category of commutative unital Banach algebras
Banach algebras of algebra-valued continuous functions
complete unital locally multiplicatively convex \(Q\)-algebras
scientific article; zbMATH DE number 60569
scientific article