Local uniform convexity and Kadec--Klee type properties in \(K\)-interpolation spaces. II
A Banach space \((X, \| \cdot \| )\) is said to be locally uniformly convex if, for any sequence \((x_n) \subset X\), \(\| x_n\| \rightarrow \| x\| \) and \(\| x_n + x\| \rightarrow 2\| x\| \) imply that \(x_n \rightarrow x\) in norm. M.~Kadec proved that every separable Banach space can be renormed to be locally uniformly convex. This result was extended to weakly compactly generated spaces by S.~Troyanski. \textit{W. J. Davis}, \textit{N.~Ghoussoub}, and \textit{J.~Lindenstrauss} [Trans. Am. Math. Soc. 263, 531--540 (1981; Zbl 0479.46010)] showed that a Banach lattice has an equivalent uniformly convex lattice norm if and only if order continuous. In the present paper, the authors continue their investigation of renormings of (commutative and) non-commutative function spaces from [J. Funct. Spaces Appl. 2, No. 2, 125--173 (2004; Zbl 1060.46014)]. For instance, they prove: if \((M, \tau)\) is a semifinite von Neumann algebra and \(E\) is a separable symmetric Banach function space on \([0,\tau(1))\), then the space \(E(M,\tau)\) has an equivalent fully symmetric locally uniformly convex norm. Here, we say that a non-commutative function space \(F\) is symmetric (fully symmetric) if, for \(x, y \in F\), we have \(\| x\| \leq \| y\| \) if \(\mu_x \leq \mu_y\) (respectively, \(\mu_x \prec \mu_y\)), where \(\mu_x\) and \(\mu_y\) are the generalized singular value functions, and \(\prec\) is the Hardy--Littlewood--Polya submajorization. The main tool in proving this result (as well as some other theorems involving uniform convexity and related Kadec--Klee properties) is the DGL property (so named after the paper of Davis, Ghoussoub, and Lindenstrauss quoted above) of compatible couples of Banach lattices. This property describes a connection between properties of \(K\)-functionals and convergence in a certain topology. It is shown that many pairs of Lorentz spaces \(\Lambda_\phi\) (equipped with the norm \(\| f\| = \int f^* d \phi\)) have the DGL property.
scientific article; zbMATH DE number 2147913