Are there any natural Banach algebras with some natural operator $A$ where $\Vert A \Vert \ne \Vert A^* \Vert$?
2026-03-25 20:19:10.1774469950
A natural Banach algebra where adjoint doesn't preserve the norm?
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Sure. Take a $C^\ast$-algebra $A$ and a faithful $\ast$-representation $\pi:A \to B(H)$. NOw, consider $V$ an invertible operator with $\| V \| \, \| V^{1} \| \leq K$ and let $\varphi: A \to B(H)$ be $\mathrm{Ad}_V \circ \pi = V \pi(\cdot) V^{-1}$. Define the norm $\| a \|_V = \| \varphi(a) \|_{B(H)}$. $A$ is still a Banach algebra under that norm but the norm is not $\ast$-invariant.