1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

4.6. Cuntz-Pimsner algebras 147

B1. It follows that rn(μ)*xrn(v) E 7r(I(7r, 7 ))· Hence, for any "finite-rank

operator" e = l:i Ov.,μ,i on J{i!m, we have

O"n(e)XO"n(e) = L rn(vi)(rn(μi)*xrn(vj))rn(μj)* E O"n(lK(H®nI(7r,T)))

i,j

since rn(μi)*xrn(vj) E 7r(I(7r, 7 )) for every i,j. Now, let e>.. be an approximate

unit of lK(H®n) consisting of "finite-rank operators." Then, for any μ E

H®n, we have

limO"n(e>..)rn(μ) = limrn(e>..μ) = rn(μ),

>.. >..

since Tn is contractive and lime>..μ = μ (Remark 4.6.2). It follows that

lim>.. O"n(e>..)rm(μ) =rm(μ) for every m 2: n andμ E 7-l®m. Since x E Bn+l,

we have

X = li,fO"n(e>..)XO"n(e>..) E O"n(lK(H®nI(7r,T))).

This proves that B::;,n n Bn+l C O"n(OC(H®nI(7r,T))). D

Note that I(1r:F(?-t),T) = {O} for the Fock space representation (7r:F('Ji), T)
of 1-t'. on T(H). Indeed there is a conditional expectation onto 'Tr:F(?i)(A)
that annihilates B1 (see Theorem 4.6.6). It follows that I(ir,f) = {O} for a
universal representation (if, f-) since 1r:F(1i) is injective.

We say the representation ( 7r, r) admits a gauge action if there is an

action f3 of 'JI' = { z E CC : I z I = 1} on C* ( 7r, r) such that

f3z(7r(a)) = 7r(a) and f3z(r(~)) = zr(~)

for every z E 'JI', a E A and ~ E 7-t'.. Evidently the representation ( 7r :F(?i), T) of
1-t'. on 'T(H) admits a gauge action. A universal representation also admits
a gauge action because of universality. We now describe the fixed point
algebra C*(7r, r)/3 of a gauge action (3.

Lemma 4.6.17. Let (7r,r) be a representation that admits a gauge action
f3. Then, we have
C*(7r, r)/3 = LJ B::;,n·
n~O

Proof. Recall that C* ( 7r, r )f3 is the range of the conditional expectation E13
defined by

E13(x) = 1 f3z(x) dz.

For every μ E 7-t'.®m and v E H®n, we have

E13(rm(μ)rn(v)) = (1 Zm-n dz)rm(μ)rn(v) = Om,nTm(μ)rn(v)*.

Since C(7r, r) =span( AU { rm(μ)rn(v)} ), the assertion follows. D