On Banach algebras with a Jordan involution

Abstract

Let A be a Banach algebra. By a Jordan involution on A we mean a conjugate-linear mapping of A onto A where for all x in A and (Error rendering LaTeX formula) for all in A. Of course any involution is automatically a Jordan involution. An easy example of a Jordan involution which is not an involution is given, for the algebra of all complex two-by-two matrices, by $$≤ft(\begin{array}{cc} a & b \\ c & d \\ \end{array}\right) # = ≤ft( \begin{array}{cc} \bar{a} & \bar{b }\\ \bar{c} & \bar{d} \\ \end{array}\right)$$ In this note we provide one instance where a Jordan involution is compelled to be an involution. Say is -normal if x permutes with and -self-adjoint if . Let y be -normal. Then (Error rendering LaTeX formula) so that is -self-adjoint. By [5, pp. 481-2]we know that (Error rendering LaTeX formula) for all and all positive integers n . Also if A has an identity e .