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.
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.DOI Code:
10.1285/i15900932v11p331
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