Math. Z. 156, 19-47 (1977) Mathematische Zeitschrift 9 by Springer-Verlag 1977 Injective Banach Lattices Richard Haydon Brasenose College,Oxford OX13LB, England 1. Introduction A Banach lattice E (over the field of reals) is said to be injective if, for every Banach lattice G, every closed linear sublattice F of G and every positive linear operator u: F--*E, there is a positive linear extension v: G-.E with IIv[I = Ilull. This definition of injectivity is the standard one if we agree to work with the category Balatl of Banach lattices and positive linear contractions, and to take as embeddings all isometric linear lattice homomorphisms. The concept was first examined by Lotz [7], who showed that lattices of the following classes are injective: (I) ~(S), where S is stonian (that is, compact and extremally disconnected); (II) (AL)-spaces. It is the second of these which shows that there really are significant differences between this theory and the corresponding ideas for Banach spaces. We recall that the Nl-spaces, which are the injective objects in the category Ban~ of Banach spaces and linear contractions, can be characterized in a number of ways (see for instance p. 160 of [6]): (a) E is a Na-space if and only if E has the so-called (0% 2)-intersection property; (b) E is a ~a-space of and only if E is linearly isometric to a space cg(S), with S stonian; (c) E" is a Na-space if and only if E has the (4, 2)-intersection property. (A Banach space E has the (n, 2)-intersection property if, in E, every collection of n balls, which are pairwise non-disjoint, has a nonempty intersection, and that E has the (o% 2)-I.P. if it has the (n, 2)-I.P. for every, possibly infinite, cardinal n.) A significant advance in the study of injective Banach lattices was made by Cartwright [3] when he was able to formulate an "order intersection property", and prove, among other things, a result which closely parallels (c) above. Cartwright's property, which I shall call simply "property (C)" can be most conveniently stated as a "splitting property".