Relative Yoneda Cohomology for Operator Spaces

We provide two alternate presentations of the completely bounded Hochschild cohomology. One as a relative Yoneda cohomology, i.e., as equivalence classes ofn-resolutions which are relatively split, and the second as a derived functor. The first presentation makes clear the importance of certain relative notions of injectivity, projectivity and amenability which we introduce and study. We prove that every von Neumann algebra is relatively injective as a bimodule over itself and consequently,Hncb(M, M)=0 for any von Neumann algebraM. A result obtained earlier by Christensen and Sinclair. We prove that the relatively amenableC*-algebras are precisely the nuclearC*-algebras, and hence exactly those which are amenable as Banach algebras. In a similar vein we prove that the only relatively projectiveC*-algebras are finite dimensional. This result implies that the onlyC*-algebras that are projective as Banach algebras are finite dimensional, a result first obtained by Selivanov and Helemskii. In a slightly different direction we prove thatB(H) viewed as a bimodule over the disk algebra with the left action given by multiplication by a coisometry and the right action given by multiplication by an isometry is an injective module. This result is in some sense a generalization of the Sz.-Nagy-Foias commutant lifting theorem or of the hypoprojectivity introduced by Douglas and the author.