RELATIVE YONEDA COHOMOLOGY FOR OPERATOR-SPACES

We provide two alternate presentations of the completely bounded Hochs child cohomology. One as a relative Yoneda cohomology, i.e., as equiva lence classes oi n-resolutions which are relatively split, and the sec ond as a derived Functor. The first presentation makes clear the impor tance or certain relative notions of injectivity, projectivity and ame nability which we introduce and study. We prove that every von Neumann algebra is relatively injective as a bimodule over itself rind conseq uently. H-cb(n)(M, M) = 0 for ally von Neumann algebra M. A result obt ained earlier by Christensen and Sinclair. We prove that the relativel y amenable C-algebras are precisely the nuclear C*-algebras, and henc e exactly those which are amenable as Banach algebras. In a similar ve in we prove that the only relatively projective C-algebras are finite dimensional. This result implies that the only C-algebras that are p rojective as Banach algebras are finite dimensional. a result first ob tained by Selivanov and Helemskii.In a slightly different direction we prove that B(H) viewed as a bimodule over the disk algebra with the l eft action given by multiplication by a coisometry and the right actio n given by multiplication by an isometry is an injective module. This result is in some sense a generalization of the Sz.-Nagy-Foias commuta nt lifting theorem or of the hypoprojectivity introduced by Douglas an d the author. (C) 1998 Academic Press.