On numerical ranges of elements of locally<i>m</i>-convex algebras

Abstract

The concept of numerical range is extended from normed algebras to locally ra-convex algebras. It is shown that the approximating relations between the numerical range and the spectrum of an element are preserved in the generalization. The set of elements with bounded numerical range is characterized and the relation between boundedness of the spectrum and of the numerical range is discussed. The Vidav-Palmer theory is generalized to give a characterization of *-algebras by numerical range.