Banach spaces with the 𝐿¹-Banach-Stone property

Greim, Peter

doi:10.1090/S0002-9947-1985-0768743-4

Banach spaces with the $L^ 1$-Banach-Stone property
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Trans. Amer. Math. Soc. 287 (1985), 819-828 Request permission

Abstract:

It has previously been shown that separable Banach spaces $V$ with trivial $L$-structure have the ${L^1}$-Banach-Stone property, i.e. every surjective isometry between two Bochner spaces ${L^1}({\mu _i},V)$ induces an isomorphism of the two measure algebras. We remove the separability restriction, employing the topology of the measure algebra’s Stonean space. The result is achieved via a complete description of the $L$-structure of ${L^1}(\mu ,V)$.

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2

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