1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

150 4. Constructions

we observe first that Adu: Iffi(1i) 3 x 1--+ U xu-^1 E Iffi(1i) is a well-defined
*-automorphism such that Adu IA= a. Moreover, for~' 'T/ E 1-i, we have
Adu(Bg,T))( = U(~(TJ, u-^1 (())) = U(~)a( (TJ, u-^1 (()))
= U(~)(U(TJ), () = eu(g),U(TJ)~
and hence Adu(Bg,T)) = eu(g),U(TJ)" It follows that a(IH) = Adu(IH) c IH
and that CJSoU = CJS o Adu on OC(1i). Therefore, if a E IH, we have
CJsou(a) = (CJs o Adu )(a)= CJs(a(a)) = a(a),
where the last equality follows from the covariance of (id, S). This proves
covariance of the representation (a, So U) of 1i into 0(1-i); the rest of the
proof is similar to the Toeplitz-Pimsner case. D

The following technical result will be handy in the future.

Proposition 4.6.23. Let A and B be unital C* -algebras with a common

unital subalgebra D and nondegenerate conditional expectations E-fj : A -+

D and Ejj: B -+ D. Let a be a *-automorphism on D. Define maps

<p = a o E-fj : A -+ D c A and 'lj; = a o Ejj : B -+ D c B. As in Example

4.6.11 1 we let T(1i~) be the universal C*-algebra generated by A and T 1

subject to the relation T*aT = c.p(a); let T(1i~) be the corresponding algebra

for B. Assume there is a u.c.p. map e: A-+ B such that Bin= idn and

Ejj o e = E-fj. Then 1 there is a u.c.p. map

8: T(1i~) -+ T(1i~)

such that

for every ao, ... , an EA.

Proof. Let JC = L^2 (B, Ejj) be the C*-correspondence over D with left
,,..._ --
action given by 7rK(d)b = a(d)b. Let :F(JC)^0 = ffin>l JC®n, where ® is the
interior tensor product over D. Since B naturally acts on JC (though this
action does not coincide with 7fK on D), we have B c Iffi(:F(JC)^0 ) (acting
on the first tensor component). Let TB E Iffi(:F(JC)^0 ) be the shift isometry:
TB((1 ® · · · ® (n) = l ® (1 ® · · · ® (n· Then, we have

T}3bTB = (7rK o Ejj)(b) = 'l/;(b)

for every b E B. Since the family of unitary operators Uz = ffin>l zn
implements the gauge action and the COmpreSSiOn to JC®l C :F(JC)O Sepa-
rates B from the "compact operators" span(BTBBTJ3B), we have T(1i~) ~
C*(B, TB) C Iffi(:F(JC)^0 ), by the gauge-invariant uniqueness theorem.