Inventiones mathematicae

, Volume 112, Issue 1, pp 449–489

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

  • Eberhard Kirchberg

DOI: 10.1007/BF01232444

Cite this article as:
Kirchberg, E. Invent Math (1993) 112: 449. doi:10.1007/BF01232444


We show the existence of a block diagonal extensionB of the suspensionS(A) of the reduced groupC*-algebraA = Cr*(SL2(ℤ)), such that there is only oneC*-norm on the algebraic tensor productBopB, 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 productBopB 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 constantΛG=1 then the universal groupC*-algebraC*(G) is not exact but the reduced groupC*-albegraCr*(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.

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Eberhard Kirchberg
    • 1
  1. 1.Mathematisches InstitutHeidelbergGermany