1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

120 4. Constructions

Lemma 4.1.8. Let 'ljJ be a faithful state on B. Then idA ® 'ljJ: A® B -+ A

is faithful.

Proof. Observe that {f ® g : f EA, g E B} c (A® B)* separates the
points of A® B. Indeed, A® BC IIB(?-i ® K) and vector states arising from
elementary tensors h ® k E 'H ® K separate all of IIB(?-i ® K).

So, if x E (A® B)+ is nonzero, we can find a state r.p on A such that

( r.p ® idB) ( x) E B is nonzero (and positive). Since. 1/J is faithful, we have

0<1/J((r.p ® idB)(x)) = r.p((idA ® 1/J)(x)),

which implies (idA ® 'lj;)(x) is nonzero. D

Proposition 4.1.9. The map E: Cc(r,A)-+ A, E("2:, 8 a 8 A 8 ) = ae, extends

to a faithful conditional expectation from A Xla,r r onto A. In particular,

max sEr !las llA ::; II "as L_,; As llA><l°' ' rr·
sEr
Proof. Let (u, idA, 1-i) be a covariant representation. By Fell's absorption
principle, we may view A XI a,r r as the C* -algebra generated by A ® 1 and
(u ® >..)(r) - in particular, it is a subalgebra of IIB(?-i) ® c;(r). The key
observation is that in this representation our map E is nothing but the
restriction of idllll(1-l) ® r, where r is the canonical faithful tracial state on
c;(r) (which is clear since r(>..s) = 0, whenever s f. e). Thus the previous
lemma implies that E is faithful..
Finally, note that a 8 = E(z>..~) for z = "2:, 8 a 8 A 8 • This implies the as-
serted inequality, so the proof is complete. · p
Remark 4.1.10. Note that E :, AX1a,rr-+ A is r-equivariant: E(>..sz°>..;^1 ) =
a. 8 (E(z)) for every s Er and z EA Xla,r r.
More generally, if a. and /3 are actions of r on sets X and, respectively,
Y, we will say a map qi: X-+ Y is r-equivariant if qi o a.g = /3g o qi for al~
g Er.

Exercises
Exercise 4.1.1. Let r: I'-+ Aut(C) be the trivial action. Since CC= JIB(C),
use Proposition 4.1.5 to show that <C Xlr,r r ~ c;(r). While you are at it,
observe that C X1 7 r:::::: C*(I').

Exercise 4.1.2. Prove that if T: r -+ Aut(A) is the trivial action (i.e.,
Tg = id A for all g E r), then

A Xlr,r r ~A® c;(r).

What is the corresponding result for universal crossed products? (Hint:
Think of universal properties.)