168 5. Exact Groups and Related Topicswidth Eis the subset Tube(E) in r x r given by
Tube(E) = {(s, t) EI' x I': sC^1 E E}.^2
By the generic term tube we mean a tube of width E for some finite subset
E c I'. The unif arm algebra (or the unif arm Roe algebra) C~ (I') of I' is
the C*-subalgebra of Illl(g^2 (r)) generated by C~(r) and g^00 (r). Thinking of
operators in JIB(g^2 (I')) as infinite matrices indexed by r, it is instructive to
convince yourself of the following fact: x = [xs,t]s,tEI' E JIB(g^2 (r)) belongs to
the *-algebra generated by ,\(C[r]) and g^00 (r) if and only if x is supported
in a tube (i.e., there exists a finite set E c r such that Xs,t = 0 whenever
(s, t) tj:; Tube(E)).^3
It turns out that the uniform Roe algebra is an old friend incognito.
Proposition 5.1.3. Let a: r ~ Aut(g^00 (r)) be the left translation action.
ThenProof. We may apply the construction of goo (r) ~ a,r r to any faithful rep-
resentation of goo (r), so we start with the canonical inclusion goo (r) c
JIB(g^2 (r)). (You may want to review Section 4.1 for the concrete construc-
tion of reduced crossed products.)
Define a unitary U: g^2 (r)0g^2 (r) ~ g^2 (r)0g^2 (r) by U(8x08y) = 8x08yx·
Now we compute
U7r(f)(8s 0 Ot) = U((at"^1 (!)8s) 0 Ot)
= U((f(ts)8s) 0 Ot)
= J(ts)8s 0 Ots
=Os 0 (f (ts)Ots)
= (10 J)(U(8s 0 8t)).
It follows that U 7r(f) U* = 1 0 f for all f E goo (r). A similar calculation
shows that U commutes with 1 0 ,\ 9 for all g E r and hence
U(g^00 (r) ~a,r r)U* =Cl 0 C*(g^00 (r), CHI'))~ C~(r).
D
Definition 5.1.4. A bounded function k: r x r ~ C is called a positive
definite kernel if the matrix [k(s, t)]s,tEi is positive for any finite subset
~er.(^2) 0ne should be careful about st- (^1) and s- (^1) t. We use here the right invariant tube so that
.A(s) is supported on a tube. However, when we deal with the Oayley graph later, we use the left
invariant metric to make the left multiplication action isometric.
(^3) !f you aren't familiar with this point of view, it is good to start with .e (^00) (I'); all of these
operators are supported in Tube({e}). Next consider an element from the group ring .A(IC[r]).
Such an operator is "constant down the diagonals," so which tube is it supported in?