Abstract
We study local uniform convexity and Kadec‐Klee type properties in K‐interpolation spaces of Lorentz couples. We show that a wide class of Banach couples of (commutative and) non‐commutative Lorentz spaces possess the (so‐alled) (DGL)‐property originally introduced by Davis, Ghoussoub and Lindenstrauss in the context of renorming order continuous Banach latties. This property is used as a key tool to show that local uniform convexity and certain Kadec‐Klee type properties in non‐commutative symmetric spaces of measurable operators may be inferred from corresponding properties of the parameter space of the K‐interpolation method. Further applications are given to renorming properties of separable symmetric Banach function spaces and their non‐commutative counterparts.
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