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Consider the following "condition (J)" on the norm of a (real or complex) Banach lattice $E$: whenever $x$ and $y$ are disjoint (i.e., $|x|\wedge |y|=0$) then

$\|x+y\|+\|x-y\|=2\|x\|+2\|y\|$.

Barry Johnson proved in 1964 that (J) holds in any complex Banach lattice $M(K)$, the bounded complex regular measures on a locally compact space $K$ (Proc. AMS 15, 1964, p. 186). It seems that the same should be true for any complex abstract $L$-space (a Banach lattice in which $\|x+y\|=\|x\|+\|y\|$ for any disjoint elements $\ge 0$) if you use the Kakutani representation.

(1) Is there a constructive proof of (J) for $L$-spaces, using only lattice-norm axioms and the additivity-of-norm condition and no representation theory?

(2) Must any Banach lattice satisfying (J) be an $L$-space?

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    $\begingroup$ Isn't (1) trivially true for $L$-spaces? In any Banach lattice $|x\pm y| = |x|+|y|$ when $x$ and $y$ are disjoint. $\endgroup$
    – Bill Johnson
    Commented Jan 2, 2015 at 14:16
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    $\begingroup$ Similarly, (J) immediately gives that $E$ is an $L$ space, since $\|x+y\| = \|x-y\|$ when $x$, $y$ are disjoint. Am I missing something? $\endgroup$
    – Bill Johnson
    Commented Jan 2, 2015 at 14:22
  • $\begingroup$ @Bill is correct. The comments answer the questions. If he had put them in an "Answer", I would have checked them. I was thrown off by Barry Johnson's proof of what turns out to be an essentially trivial fact about $L$-spaces. $\endgroup$
    – Fred Dashiell
    Commented Jan 2, 2015 at 17:41

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