1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

46 2. Nuclear and Exact C* -Algebras

Proposition 2.5.8. Let A c r be a subgroup. There is a canonical inclusion

C*(A) c C*(r).

Proof. By universality, there is a canonical *-homomorphism 1f : C* (A) -+

C*(r). Our task is to show it's injective.

We may assume A is countable. Then C*(A) has a faithful state <p which

we think of as a positive definite function on A. Now extend c.p to all of r

by defining cp(s) = 0, for alls tJ. A. Our proof will be complete once we see

why this extension is positive definite on r (since the GNS representation

of C*(A) with respect to <p will be a subrepresentation of 1f composed with

the GNS representation of C*(r) with respect to the extension of c.p tor).

The key observation is that [cp(s-^1 t)]s,tE.;y is block diagonal with respect

to the left coset decomposition-i.e., ifs, t Er and belong to different cosets

({:} s-^1 t tJ. A), then cp(s-^1 t) = 0 - for every finite set .;y-c r. Since a block

diagonal matrix is positive if and only if every block is positive, we may

assume that F c gA for some fixed g Er. Checking positive definiteness is

now trivial. D

An analogous result holds in the reduced case too. The key point is that

right cosets give a direct sum decomposition

and hence the left regular representation of r, when restricted to A, is a mul-

tiple of the left regular representation of A (multiplicity equals the number

of cosets). This implies

Proposition 2.5.9. If A c r is a subgroup, then C~(A) c C~(r) canoni-

cally.

We will need one more important fact about positive definite functions:

they naturally give rise to completely positive maps at the C*-level. First a

bit more notation.

Definition 2.5.10. Let c.p: r -+ CC be a function. We define a corresponding
linear functional wcp : qr] -+ CC by

wcp(L">:ttt) = L cp(t)at

tEr tEr

and multiplier mcp : qr] -+ qr] by

mcp(L att) = L cp(t)att.

tEr tEr