Rapid Decay and the Metric Approximation Property

The central point of our proof is an observation that the proof of the same property for free groups due to Haagerup [2] transfers directly to this more general situation. A discrete group Γ satisfies property (RD) (Rapid Decay) with respect to a length function l on Γ if the operator norm of any element of the group ring can be uniformly majorised by a Sobolev norm determined by l. In detail, this means the following. The left action of a group Γ on itself extends to the convolution action of the group ring CΓ on the Hilbert space l(Γ). This is the left regular representation λ of Γ which embeds the group ring in the C-algebra B(l(Γ)) of all bounded linear operators on l(Γ). The reduced C-algebra C r (Γ) is the C -subalgebra of B(l(Γ)) generated by λ(CΓ). For any positive real number s we define a Sobolev norm associated with the length function l by