On non-semisplit extensions,tensor products and exactness of groupC *-algebras

Summary We show the existence of a block diagonal extensionB of the suspensionS(A) of the reduced groupC ^*-algebraA = C _r ^* (SL ₂()), such that there is only oneC ^*-norm on the algebraic tensor productB ^op B, butB is not nuclear (even not exact). Thus the class of exactC ^*-algebras is not closed under extensions.The existence comes from a new established tensorial duality between the weak expectation property (WEP) of Lance and the local variant (LLP) of the lifting property.We characterize the local lifting property of separable unitalC ^*-algebrasA as follows:A has the local lifting property if and only if Ext (S(A)) is a group, whereS(A) is the suspension ofA.If moreoverA is the quotient algebra of aC ^*-algebra withWEP (for brevity:A isQWEP) but does not satisfyLLP then there exists a quasidiagonal extensionB of the suspensionS(A) by the compact operators such that on the algebraic tensor productB ^op B there is only oneC ^*-norm.The question if everyC ^*-algebra isQWEP remains open, but we obtain the following results onQWEP: AC ^*-algebraC isQWEP if and only ifC ^** isQWEP. A von NeumannII ₁-factorN with separable predualN _* isQWEP if and only ifN is a von Neumann subfactor of the ultrapower of the hyperfiniteII ₁-factor. IfG is a maximally almost periodic discrete non-amenable group with Haagerup's Herz-Schur multiplier constant _G =1 then the universal groupC ^*-algebraC ^*(G) is not exact but the reduced groupC ^*-albegraC _r ^* (G) is exact and isQWEP but does not satisfyWEP andLLP.We study functiorial properties of the classes ofC ^*-algebras satisfyingWEP, LLP resp. beingQWEP.As applications we obtain some unexpected relations between some open questions onC ^*-algebras.Oblatum 13-IV-92Work partially supported by DFG