On Banach algebras with a Jordan involution

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