1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

2.5. C* -algebras associated to discrete groups 47

Theorem 2.5.11. Let <p: r ---t CC be a function with <p(e) = 1. The following

are equivalent:

(1) the function <p is positive definite;

(2) there exists a unitary representation Acp of r on a Hilbert space Hep

and a unit vector ~'P such that

<p(s) = (Acp(s)~cp' ~cp);

(3) the functional Wcp extends to a state on C*(I');

( 4) the multiplier mcp extends to a u. c. p. map on either C* (r) or Ct (r),

or extends to a normal u.c.p. map on L(I'). ·

Proof. (1):::::? (2): This follows from Definition 2.5.7.
(2):::::? (3): Trivial.
(3) :::::? (4): First we handle the von Neumann algebra case. We can
identify Wcp with a vector state in the universal representation C (r) c
JB(H) (i.e., the direct sum of a~l GNS representations). By Fell's absorption
principle, there is a unitary operator which conjugates Ct (r) ® 1 onto the
"diagonal" subalgebra of Ct (r) ® C (r); that is, the mapping a: Ct (r) ---t
Ct(r) ® C*(r) defined by

L CXt At f---7 L CXt (At ® t)

t t

is a -homomorphism and it extends to a normal -homomorphism (also
denoted a) from L(r) into L(r) ® JB(H) (since Fell's principle is spatially
implemented). Notice that mcp coincides with the continuous u.c.p. map

(idL(r) ® Wcp) 0 a: L(r) ---t L(r),

and this completes the von Neumann case (which evidently implies the re-
duced C*-algebra case as well).

For C(r) we consider the diagonal map C(r) ---t C(r) ® C(r), s f---7
s ® s, and repeat the argument above.

(4):::::? (1): If mcp is u.c.p., then for any finite sequence s1, ... , Sn Er,

[<p(si^1 sj)]ij = diag(s1, ... , sn)[mcp(si^1 sj)]ij diag(s1^1 , ... , s~^1 )

is positive since [s;^1 sj]ij E Mn(CC(r)) is positive. D

The proof of the following corollary amounts to checking whether or not
a particular function is positive definite, which we leave to you.

Corollary 2.5.12. If Ac r is an inclusion of groups and Ct(A) c Ct(r)
the resulting inclusion of C* -algebras, then there is a conditional expectation
EI: C~(r) ---t C~(A) obtained by "throwing away" all group elements outside