On Certain Properties of Numerical Radius Preservers in Nonunital Banach Algebra

Abstract

Numerical radius preserver problem is one of the linear preserver problems which have been studied over decades. However, this problem still remains interesting as it has not been solved in totality. Only partial solutions have been obtained in special cases like unital Banach algebras and C*-algebras among others. No solutions have been obtained in the nonunital cases. In this paper, we characterize certain properties of numerical radius preservers in nonunital Banach algebras. In particular, we show that every numerical radius preserver is surjective, continuous and real linear. Moreover, they preserve multiplicativity, commutativity, involution and is a *-isomorphism.