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1.4. Double duals 5

x 0 e E IIB(X, M)*, where x EX, e E M* and x 0 e is defined by x 0 e(T) =

e(T(x)). Then IIB(X, M) is isometrically isomorphic to the dual of IIB(X, M)*,

whence it receives a weak-* topology. On bounded sets, this topology agrees

with the point-ultraweak (aka point-<r-weak) topology. That is, for a bounded

net convergence works as follows:

TA----+ T ~ e(TA(x))----+ e(T(x)), Vx EX, Ve EM*.

Thus the unit ball of IIB(X, M) is compact, by Alaoglu's Theorem, in the

point-ultraweak topology. Hence we obtain the following theorem (cf. The-

orem A.8).

Theorem 1.3.7. Let X be a Banach space, M be a von Neumann algebra

and TA: X ----+ M be a bounded net of linear maps. Then {TAhEA has a

cluster point in the point-ultraweak topology.

Representation theory. In contrast to the C* -case, representation theory

of von Neumann algebras is almost trivial: one can cut by a projection in the

commutant and that's about it. Of course, the precise statement is slightly

more complicated (see [183, Theorem IV.5.5]).

Theorem 1.3.8. Let M c IIB(1i) be a von Neumann algebra and 7r: M ----+
IIB(JC) be a normal representation. There exist a Hilbert space K and a pro-
jection Prr E IIB(1i 0 K) such that Prr commutes with M 0 1 C IIB(1i 0 K)
and n is unitarily equivalent to the representation M ----+ PrrIIB(1i 0 K)Prr,
m f---t Prr(m 01).

1.4. Double duals

The Banach space double dual of a C* -algebra A is a wild beast; it should

be approached with humility, even trepidity. Whatever it takes, though,

one must become acquainted with A** as it's an extremely useful universe

in which to work. See [183, Section III.2] for more.

The enveloping von Neumann algebra. Recall that the universal rep-

resentation of a C* -algebra A is

1fu= EB 1f<p: A--+IIB( EB L^2 (A,cp)) =lIB(1iu)·

<pES(A) <pES(A)

By definition, the enveloping von Neumann algebra of A is the double com-

mutant nu(A)". Thanks to the next result, we need not distinguish between

the double dual and the enveloping von Neumann algebra; we'll use A** to

denote either throughout this book.