I'm trying to define an isometric isomorphism $T:M([0,1])\to M([0,1])$ that is not weak-star continuous (by $M([0,1])$ I mean the Banach space of regular Borel measures). How I can build one? One observation is that $T$ is not weak-star continuous if the adjoint mapping $T^*$ doesn't map $C([0,1])$ onto itself.
2026-04-24 02:49:08.1776998948
Isometries on the Banch Space M([0,1]) of regular Borel Measures
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Perhaps one could try the following: Take $\varphi:[0,1]\to [0,1]$ which permutes two points and leaves the others fixed. Given $\mu\in \mathcal{M}[0,1]$ and $A$ Borel, define $T\mu(A)=\mu(\varphi(A))$. One can check $w^*$ discontinuity by recalling the strong relationship between the topology of $[0,1]$ and the $w^*$ topology on the Dirac point evaluation functionals.