Some remarks on spectral states ofLMC-algebras

We generalize the notion of a spectral state (as introduced for Banach algebras by Moore, Bonsall and Duncan) to the context of locally multiplicatively-convex (LMC) algebras by proceeding in a way analogous to the generalization of numerical range theory from Banach algebras toLMC-algebras carried out by Giles and Koehler. Among the results obtained in this note are integral representations of spectral states by probability measures on the structure space ofA and the determination of the extreme points of the convex set\(\Omega _A \) of all spectral states on a commutativeLMC-algebraA (which is related to different Choquet boundaries) as well as a characterization of symmetric involutions by the coincidence of the notions of positive state and spectral state and a characterization of theQ-property by the weak-^*-boundedness of\(\Omega _A \). The paper finishes with two elementary commutativity criteria involving spectral states and two Korovkin-type theorems for the approximation of unital algebra homomorphisms by σ-equicontractive nets of linear operators mapping anLMC-algebraA into theLMC-algebra of all continuous complex-valued functions on a completely regular spaceX.