Korovkin approximation in Waelbroeck algebras
Let A be a Waelbroeck algebra, i.e. a complex commutative locally multiplicatively convex Q-algebra. If B is a second Waelbroeck algebra, 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. For a subset T, we study its universal Korovkin closure defined by \(K_ u(T)=\{y\in A:\) For every Waelbroeck algebra B, every unital algebra homomorphism \(L: A\to B\) and every net \(L_{\alpha}: A\to B\) of spectral contractions the convergence \(\rho (L_{\alpha}x-Lx)\to 0\) (x\(\in T)\) implies the convergence \(\rho (L_{\alpha}y-Ly)\to 0\}\). A number of equivalent descriptions of the subspace \(K_ u(T)\) is given, including a characterization of \(K_ u(T)\) by a unique extension property of Radon measures concentrated on the structure space \(\Delta_ A\) or the Silov boundary \(\Gamma_ A\). The test set T is called a universal Korovkin system if \(K_ u(T)=A\). Various Choquet boundaries are introduced that prove useful for the characterization of universal Korovkin systems. Different representations of these boundaries as extreme points of convex sets are given; they allow, in particular, a determination of the extreme points ext \(\Omega\) \({}_ A\) of the set \(\Omega_ A\) of all spectral states on A and in this way solve a problem posed by Bonsall and Duncan in 1971. If the test set T is contained in a finite dimensional subspace, it turns out that all its Choquet boundaries coincide and are independent of the algebra A containing T. After a quick glance at the rôle of peak points and real peak points, we finally present a necessary condition for the existence of a finite universal Korovkin system in A that strengthens the only known condition and allows us, e.g., to conclude that the polydisc algebras \(A(D^ n)\) do not possess any finite universal Korovkin system.
scientific article; zbMATH DE number 3921187