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 ) ;
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