DUAL ALGEBRA S

3

The strong operator topology (SOT) on ££(X) i s that locally convex, Hausdorf f

topology determined b y saying that a net { Tx} i n J?(X ) converge s strongly to T

0

if and onl y if, fo r al l x e X, ||(7 \ -

JT 0)JC||

- * 0 .

The followin g propositio n give s som e relationship s betwee n thes e topologies .

For mor e detail, see, for example, [18, p. 32].

PROPOSITION 1.6. The weak* topology and the ultraweak operator topology on

&(X) coincide. This topology, which we usually call the weak* topology, and the

strong operator topology are both stronger than the weak operator topology on

J?(X) and weaker than the norm topology on J?(X). The weak* topology and

strong operator topology are not comparable on S£(X). If 38 c Sf.{X) is any

(norm) bounded set, then the relative topologies induced on 38 by the weak* and

weak operator topologies coincide. Hence the unit ball in Sf(X) is compact and

metrizable in the weak operator topology. If X is finite dimensional, all of the

above-mentioned topologies coincide on J£(X).

It i s important t o know what form th e continuous linear functionals o n i f (X)

in thes e various topologie s take (cf. [18, p. 37] for more detail).

PROPOSITION

1.7. Suppose Jt is a linear manifold in ^f(X). Then the WOT-

closure and SOT-closure of Jt coincide. The WOT-continuous linear functionals on

Jt coincide with the SOT-continuous linear functionals on Jt', and these are exactly

the linear functionals of the form

f(T) = t (Tx

t

,yi) = t r r | £

Xl

9yM, T=J?,

where {x

i

}"^l and {jJJL x are any equipotent finite sequences from X. The

weak*-continuous linear functionals on J( are exactly those of the form

g(T) = £(Tx„y,) = tr l r| £ x, 0 j , ) J, l e J[,

where {x^JLi and {y

i

}°°s=l are any square-summable sequences from X. More-

over, the sequences {JC, } and {y t} maybechosen tosatisfy \\g\\ = Z)||jc

f-|[2

= EH^-H

2.

We come now to the subject of thes e lectures.

DEFINITION

1.8. Wit h X a s always, a dual algebra i s a subalgebr a o f if(Jf )

that contains 1 ^ an d is closed in the weak* topology on £?(X ).

EXAMPLES

1.9. The following are dual algebras;

(a) S£(X), wher e 1 di m Jf H 0;

(b) any finite-dimensiona l subalgebr a o f i f (X ) tha t contains \

x

, fo r example ,

the algebra of scala r operators;

(c) any von Neumann algebra on X whic h contains 1^;

(d) the commutant Sf' o f any subset & o f i f ( X ) ;