390 13. WEP and LLP
generated by two elements and ISi can be 3.) Since EBm#n 7rm @1rn does not
have a nonzero invariant vector, we have
sup{ II L 7rm(s) 0 7rn(s)ll : m # n} < ISi,
sES
by Lemma 12.1.9. D
Remark 13.5.5. Here is another proof. A deep theorem of Haagerup and
Thorbj0rnsen [81] states that there is an embedding
00 00
n=l n=l
Let g1, ... , g1~ (k ~ 2) be the free generators of IF'k and take for each i a
lifting (ui(n))~=l E Il~=l Mn(CC) of 7r(gi) such that Ui(n) are unitary for all
n. We set U-i(n) = ui(n)*: Then, by Fell's absorption principle, we have
±k ±k
limsup II L ui(m) 0 ui(n)ll =II L >-.(gi) 0 ui(n)ll
m-+oo i=±l i=±l
±k
= II L >-.(gi) II
i=±l
< 2k,since lFk is nonamenable (Theorem 2.6.8). Therefore, passing to an appro-
priate subsequence, we obtain a coding family of unitary matrices.
Remark 13.5.6. The reduced group C-algebra O~(IFk) is a non-QD C-
subalgebra of rr~=l Mn(<C)/ EB~=l Mn(<C). It turns out that any coding
family of unitary k-tuples gives rise to such a non-QD subalgebra. This
observation of S. Wassermann [195] will be revisited in Chapter 17, but
let's sketch a proof now ..
Let {(ui(n))i=l, ... ,k: n EN} be a coding family, M = IT~=lMN(n)(<C)
and Ui = ( Ui ( n)) ~=l E M. Denote by 7r : M ---* M / K the quotient map,
where K = EEl~=lMN(n)(CC) and set Vi= 7r(ui)· Then we have
k k
II L Vi 0 VillM/K@M/K:::; II L Ui 0 VillM@M/K
~1 ~1
k
= mEN sup II L. ui(m) 0 villM N(m) (C)®M/K
i=l
k
= mEN sup limsup n-+oo II L. ui(m) 0 ui(n)llM N(m) (C)®M N(n) (C)
i=l
< k.