Actually I want an involution on Banach algebra B such that
- ||x*|| $\le$ c.||x|| $\forall$ x in B and some c > 1
- $\exists$ y in B such that ||y*|| $\ne$ ||y||
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But, $y^{*}=y$, so $|y|\le c|y^|$. – Berci Oct 30 '19 at 08:33
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But I want such kind of involution which is not isometry. (i) $\frac{1}{c}$.||x|| $\le$ ||x|| $\le$ c.||x|| $\forall$ x in B and some c > 1( for continuity of involution ). (ii) $\exists$ y in B such that ||y|| $\neq$ ||y|| ( for non-isometry ) – Manabendra Giri Jan 04 '20 at 05:15