1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

448 A. Ultrafi.lters and Ultraproducts

Lemma A.9. The ultraproduct Mw is a von Neumann algebra and the

faithful tracial state rw is normal.

Proof. We prove that Mw coincides with the von Neumann algebra gen-

erated by the GNS representation associated with rw. It suffices to show

that the closed unit ball n of Mw is complete in the 2-norm - i.e., any

Cauchy sequence {xSk)}~ 1 in (n, 11112) is convergent. We may assume that

llxg:+l) - xg:) 112 < 2-k for every k. ·

Claim. Let k EN and suppose that (x~k))n E TIM is a lifting of xg:) with

ll(x~k))nll :::; 1. Then, we can find a lifting (x~k+l))n E TIM of xg:+l) such

that II (x~+l))nll :::; 1 and llx~+l) - x~) 112 < 2-k for all n.

Indeed, let (x~k+l))n E IT M be any lifting of xg:+l) with II (x~k+l))nll :::;

1. Then, we have { n E N : llx~k+l) - x~k) 112 < 2-k} E w. Hence, we can
   replace x~k+l) with x~k) for all n outside of the above set, without affecting
   Xw (k+l).
   Now, we inductively choose liftings (x~k))n E IT M of xg:) as in the claim.
   Then, for Xw = (x~n))n---+w E Mw, one immediately checks that
   00
   llxw - xSk) 112 = n---+w lim llx~n) - x~k) 112 :::; '"°' D 2-n = 2-(k-l).

Hence the sequence {xSk)}~ 1 converges to Xw.

Exercises

Exercise A.1. Prove Theorem A.8.

n=k

D

Exercise A.2. Using Theorem A.8, give a three line proof of Tychonoff's
theorem: The product of compact spaces is compact.

Exercise A.3. Check that £^00 (1) 3 f 1-+ limi---+U f(i) E CC is a character for
every ultrafilter U on I.

Exercise A.4. Prove that every projection e E Mw lifts to a projection
(en)n E IT M such that limn---+w r(en) = rw(e).

Exercise A.5. Prove that if Mis a factor (of type II 1 ), then so is Mw.
(Hint: Use the previous exercise.)