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^*(SL₂()), such that there is only oneC^*-norm on the algebraic tensor productB^opB, 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^opB 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 NeumannII₁-factorN with separable predualN_* isQWEP if and only ifN is a von Neumann subfactor of the ultrapower of the hyperfiniteII₁-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