Numerical Ranges of Elements of Topological Algebras

Abstract

<p>Giles and Husain have extended the concept of numerical range of an element of a normed algebra to locally multiplicatively convex algebra and have studied. We make a further study of it. We shown that for a normal element, a, of a complete unital lmc*-algebra the numerical range of a, V(A,{p};a) is equal to the convex hull of Sp(A,a) and we apply it to compute the numerical ranges of certain elements of certain complete lmc*-algebras. The relation between the numerical range V(A, {p}; x) and various growth conditions on the resolvent (x-λ)‾¹ is discussed.</p> <p>We also extend the concept of numerical range of elements from Banach algebras to pseudo-banach algebras introduced by Allan, Dales and Mcclure. We show that the numerical range of an element of a pseudo-banach algebra is a compact convex subset of the complex plane containing the spectrum of that element and that the spectral radius is equal to the numerical radius.</p> <p>We also study certain extreme positive maps of B*-algebras, Bp*-algebras and show them to be equal to the nonzero multiplicative linear functionals of the respective algebras. It is shown that the boundary of the numerical range of an element is the spectrum of that element of these topological algebras.</p>