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

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We show the existence of a block diagonal extensionB of the suspensionS(A) of the reduced groupC *-algebraA = C * r (SL 2(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 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 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.

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  1. Mathematisches Institut, Im Neuenheimer Feld 288, W-6900, Heidelberg, Germany

    Eberhard Kirchberg

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Oblatum 13-IV-92

Work partially supported by DFG

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Kirchberg, E. On non-semisplit extensions, tensor products and exactness of groupC *-algebras. Invent Math 112, 449–489 (1993). https://doi.org/10.1007/BF01232444

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