A CHARACTERISATION OF CENTRAL ELEMENTS IN -ALGEBRAS

Abstract

Wu [‘An order characterization of commutativity for$C^{\ast }$-algebras’,Proc. Amer. Math. Soc.129(2001), 983–987] proved that if the exponential function on the set of all positive elements of a$C^{\ast }$-algebra is monotone in the usual partial order, then the algebra in question is necessarily commutative. In this note, we present a local version of that result and obtain a characterisation of central elements in$C^{\ast }$-algebras in terms of the order.