1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

44 2. Nuclear and Exact C*-Algebras

right regular representation p: r , IBS(£^2 (I')) - i.e., L(I') = p(C[r])' and
L(I')' = p(C[I'])" (see Theorem 6.1.4).
Another way of describing L(I') is as the set of TE IBS(£^2 (I')) such that
T is constant down the diagonals - meaning that for every s, t, x, y E r
such that ts-^1 = yx-^1 , we have (Tbs, bt) = (T8x, by).^12 A simple calculation
shows that every unitary As E IBS(£^2 (I')) is constant down all diagonals; hence
any finite linear combination has this property; thus anything in the weak
closure C~ (I')'^1 = L(I') does too. The converse, that every such operator is
a weak limit of something in C~ (I'), uses the bicommutant theorem. Indeed,
assume TE IBS(£^2 (I')) and assume there exist scalars {as}sEr c tC such that
(T8g, 8h) = ahg-1 for all g, h Er. A simple calculation shows (Tpsbg, 8h) =
(psT8g, 8h),·for alls EI', and hence TE p(C[I'])' = L(I').
Here is a useful consequence of these remarks.
Proposition 2.5.4. Assume I' is countably infinite and 7r: C~ (I') -t IBS(1-i)
is a representation on a separable Hilbert space 1-l. Then l EB 7r is approxi-
mately unitarily equivalent to t, where l: C~ (I') , IllS ( £^2 (I')) is the canonical
inclusion. In particular, every state on C~ (I') can be approximated by vector
states coming from £^2 (I').

Proof. To invoke Voiculescu's Theorem (version 1.7.3), we only need to

know that CHI') n JK(£^2 (I')) = {O}. In fact, this is even true for L(I') since

no nonzero compact operator can be constant down the diagonals. (The

details are left to the reader.) D

Another important representation-theoretic fact is Fell's absorbtion prin-

ciple. Roughly, it states that the left regular representation absorbs all other

representations tensorially.^13

Theorem 2.5.5 (Fell's absorption principle). Let 7r be a unitary represen-

tation of I' on 1-i. Then, .\ ® 7r is unitarily equivalent to.\® 11-i·

The proof amounts to writing down the proper unitary and doing some

calculations. We'll do the first part: Define a unitary U on £^2 (I') ® 1i by

L Ot ® ~t f---7 L Ot ® 7r(t)fr

t t

In certain situations it is important to know about positive definite func-

tions on r.

(^12) "What? Why is that 'constant down the diagonals'?" you may wonder. Well, if r = Z
and you write down the matrix of such an operator (with respect to the canonical basis), you'll
see that it really is constant down the diagonals.
(^13) Fell's principle and the proof of Theorem 2.5.11 depend on the basics of tensor products
(found in the first three sections of Chapter 3).