A. UltraE.lters and Ultraproducts 447by
rp(f) = ~im f(i).
i->U
It is not too hard to check that rp is a character (i.e., a -homomorphism) on
goo (I). Conversely, if rp is a nonzero character on goo (I), then the family of
subsets J such that rp(XJ) = 1 is an ultrafilter on I, which gives the character
rp, whence there is a natural bijection between the set of ultrafilters on I
and the set of characters on g^00 (I) (which, in turn, is isomorphic to the
Stone-Cech compactification f3I of I).
Now we assume that I is a directed set and U is a cofinal ultrafilter.
Let X, Y be Banach spaces and (n)EJ be a bounded net in IIB(X, Y). Since
bounded subsets of Y are pre-compact in the weak ~topology, we cari define
a map Tu : X -+ Y by
Tu(x) =weak-Fm n(x).
i->U
This turns out to be a bounded linear map in the set of point-weak* cluster
points of the net (Ti)iEI.
Ultraproducts. Let U be an ultrafilter on a set I. Let (Xi)iEJ be a net of
Banach spaces. We denote by IlXi the g^00 -direct sum of the Xi's and let
Nu be the Banach subspace of U-null nets:
Nu= {(xi)iEI E rrxi: i->U ~im !!xiii= 0}.
iEJThe ultraproduct Banach space of (Xi)iEJ is defined as Xu = (ITXi)/Nu.
We write xu, or (xi)i_.u, for the element represented by (xi)iEJ· It is a nice
exercise to check that llxull = limu llxill· If Ai= Xi are all C-algebras, then
the ultraproduct Au is again a C -algebra. If Hi = Xi are all Hilbert spaces,
then the ultraproduct Hu is again a Hilbert space such that (rJU, fo) =
limu\rJi, ei)· If Ac IIB(Hi), then Au c IIB(Hu) with aueu = (aiei)i->U·
Ultraproducts of von Neumann algebras are not quite as straightforward
as the C*-case. To keep the notation consistent1, let w be a free ultrafilter
on N. Let (M, T) be a von Neumann algebra with a faithful tracial state T
(or, more generally, a sequence of von Neumann algebras Mn with faithful
tracial states Tn)· Let NS:) be the norm-closed ideal of IT M, given by
NS:)= {(xn)nEN E IT M: n->w lim llxnlb = O},
nENwhere llxn[[2 = T(x~xn)^112. The (tracial) ultraproduct of (M, T) is defined
to be MW= err M)/NS:); it has a faithful tracial state Tw given by Tw(xw) =
limw T(xn)· We note that [[xw[[2 := Tw(x~xw)^112 = limn->w llxn[[2.
1 It is a lamentable habit, but operator algebraists tend to use the symbol w for an ultrafilter
on N.