1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

100 3. Tensor Products

Define new c.p. maps by

and note that 11'¢~11 :S 1 + 8. Since 8 > 0 was arbitrary, it suffices to show

that '¢~ o !.pn -+ r.p in the point-strong operator topology.

Let x EA be any positive contraction. Then, since P;{:('lfnor.pn(x))P;{: :S

P?f'lfn (1), we have that ( '!fn o !.fJn ( x) )^112 P;{: -+ 0 in the strong operator topol-

ogy. Moreover, llPn('lfn o !.fJn(x))^112 ll^2 :S llPn'Zfn(l)Pnll :S 1+8. It follows

that Pn('!fn o !.fJn(x))P;{:-+ 0 in the strong operator topology. Therefore,

'¢~ o !.fJn(x) = Pn('!fn o !.fJn(x)) - Pn('!fn o !.fJn(x))P;f:--+ r.p(x)

in the strong operator topology. Since positive elements span A, we are

done. D

Proposition 3.8.3 (Radon-Nikodym). Assume f E S(A) is a state on a

C* -algebra A and let 7r f : A -+ JIB ( L^2 (A, f)) be the G NS representation. If~

is any positive linear functional on A such that e :Sf (i.e., e(a) :S f(a) for

all 0 ::; a E A), then there exists a unique operator Ye E 7r f (A)' such that

O::::: Ye::; 1 and

e(a) = (7rJ(a)yei, i),

for all a E A, where i is the canonical cyclic vector in L^2 (A,f) (though A

is not assumed unital).

Proof. Here is a sketch of the proof. First define a positive definite sesquilin-

ear form on L^2 (A, f) by

(a, b)e = e(b*a).

One must check that this is well-defined (use Cauchy-Schwarz), bounded on

a dense subspace and hence can be extended to all of L^2 (A, f). Then we

can find a unique operator Ye E IIB(L^2 (A, f)) such that

(a, b)e = (yea, b)

and one must check that this operator has the desired properties. D

We need one more lemma before coming to the first major theorem.
The statement is technical but amounts to saying that a certain property is
independent of the representation of the van Neumann algebra involved.

Lemma 3.8.4. Let A be a C* -algebra, M c JIB('h'.) be a von Neumann algebra
and r.p: A -+ M be a c.p. map. Assume that the product map

r.p x lM': A 8 M'-+ IIB('h'.), r.p ( ~ ai ® m~) = L r.p(ai)m~,

i i