Geometric properties of some Calderon-Lozanovskii spaces and Orlicz-Lorentz spaces

Geometry of Calderon-Lozanovskii spaces E(phi) in the case of a sigma-finite measure and a Banach function space with the Fatou property is studied. It is proved that if the space of order continuous elements E(alpha) not equal {0} and phi is not an element of Delta(2)(E), then E(phi) contains an order isometric copy of l(infinity). If E(alpha) not equal E then it is proved that E(phi) contains an order almost isometric copy of l infinity for every Orlicz function phi. Under the assumption that E is uniformly monotone it is proved that epsilon(0)(E(phi)) less than or equal to epsilon(0)(L(phi)), where epsilon(0)(E(phi)) and epsilon(0)(L(phi)) stand respectively for the characteristic of convexity of E(phi) and L(phi) (the Orlicz space). As a consequence of this inequality, the characteristic of convexity of Orlicz-Lorentz space Lambda(phi,omega) is computed in the case when the Orlicz function phi is strictly convex. This generalizes the criterion for uniform rotundity of Lambda(phi,omega) given in [Ka3]. Criteria for strict monotonicity, local uniform monotonicity and uniform monotonicity of Orlicz-Lorentz spaces Lambda(phi,omega) are also given. Finally, uniform non-squareness, B-convexity and superreflexivity of Lambda(phi,omega) are studied.