Renormings and fixed point property in non-commutative \(L_1\)-spaces. II: Affine mappings
A Banach space has the fixed point property for affine mappings (A-FPP) if every affine nonexpansive mapping from a closed bounded convex subset of \(X\) into itself has a fixed point. The authors note that, since Banach spaces containing asymptotically isometric copies of \(\ell_1\) fail to have the A-FPP, every nonreflexive subspace of \(L_1[0,1]\) or, more generally, every nonreflexive subspace of a noncommutative \(L_1\)-space associated with a finite von Neumann algebra with its usual norm fails to have the A-FPP. The main result in the article is that, if \(\mathcal{M}\) is a finite von Neumann algebra, then the noncommutative \(L_1\)-space \(L_1(\mathcal{M})\) can be renormed to have the A-FPP. In particular, \(L_1[0,1]\) can be endowed with a norm \(|\!|\!|\cdot |\!|\!|\) equivalent to \(\|\cdot\|_1\) so that \((L_1[0,1], |\!|\!|\cdot |\!|\!|)\) has the A-FPP. Whether \(L_1[0,1]\) can be renormed to have the fixed point property for nonexpansive mappings is still an open question.NEWLINENEWLINE For Part I see [ibid. 74, No. 10, 3091--3098 (2011; Zbl 1230.46010)].
scientific article; zbMATH DE number 6057313
scientific article