On Banach algebras with a Jordan involution

Abstract

Let A be a Banach algebra. By a Jordan involution $x→ x^#$ on A we mean a conjugate-linear mapping of A onto A where $x^##= x$ for all x in A and $$(xy + yx)^# = x^#y^# +y^#x^#$$ for all $x, y$ 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 $x ∈ A$ is $#$ -normal if x permutes with $x^#$ and $#$ -self-adjoint if $x = x^#$. Let y be $#$-normal. Then $$2 (y^#y)^# = (y^#y + yy^#)^# =2y^#y$$ so that $y^#y$ is $#$-self-adjoint. By [5, pp. 481-2]we know that $$(xⁿ)^# = (x^#)ⁿ$$ for all $x∈ A$ and all positive integers n. Also $e^# = e$ if A has an identity e