Inventiones mathematicae

, Volume 112, Issue 1, pp 449–489 | Cite as

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

  • Eberhard Kirchberg


We show the existence of a block diagonal extensionB of the suspensionS(A) of the reduced groupC*-algebraA = C r * (SL2(ℤ)), 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 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 constantΛ 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.


Tensor Product Local Variant Compact Operator Quotient Algebra Lift Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Eberhard Kirchberg
    • 1
  1. 1.Mathematisches InstitutHeidelbergGermany

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