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.
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![x, y](http://siba-ese.unile.it/plugins/generic/latexRender/cache/2317793a8de61ab32c0f17adff9ea8d4.png)
![x ∈ A](http://siba-ese.unile.it/plugins/generic/latexRender/cache/c252938788bf411b91f463ce8e9cc462.png)
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![#](http://siba-ese.unile.it/plugins/generic/latexRender/cache/01abfc750a0c942167651c40d088531d.png)
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![#](http://siba-ese.unile.it/plugins/generic/latexRender/cache/01abfc750a0c942167651c40d088531d.png)
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![e<sup>#</sup> = e](http://siba-ese.unile.it/plugins/generic/latexRender/cache/dc9d3fd1de02cb87b03d3888a4efd49c.png)
DOI Code:
10.1285/i15900932v11p331
Full Text: PDF