Group C*-algebras without the completely bounded approximation property

TL;DR

This paper demonstrates that certain algebraic structures associated with simple Lie groups of real rank at least 2 lack the completely bounded approximation property, highlighting limitations of previous generalizations.

Contribution

It proves that the Fourier algebra and reduced C*-algebras of these groups do not possess the approximation properties previously conjectured to hold.

Findings

Fourier algebra A(G) lacks a multiplier bounded approximate unit.

Reduced C*-algebra of lattices in these groups lacks the completely bounded approximation property.

Results do not extend from rank 1 to higher rank simple Lie groups.

Abstract

It is proved that: (1) The Fourier algebra A(G) of a simple Lie group G of real rank at least 2 with finite center does not have a multiplier bounded approximate unit. (2) The reduced C*-algebra of any lattice in a non-compact simple Lie group of real rank at least 2 with finite center does not have the completely bounded approximation property. Hence, the results obtained by J. de Canniere and the author for SO(n,1), n at least 2, and by M. Cowling for SU(n,1) do not generalize to simple Lie groups of real rank at least 2.