DUAL ALGEBRAS
5
# = AlgLat(^) . A n operato r T i n if(Jf ) i s reflexive i f 7T
r
, th e smalles t
WOT-closed algebr a containing T and 1^, is reflexive.
Whether a n algebr a f% o r a n operato r T i s reflexive is of considerabl e interest ,
because reflexiv e algebra s an d operators , a s on e ca n easil y se e fro m th e defini -
tions, hav e a goo d suppl y o f nontrivia l invarian t subspaces . This topi c is studie d
in Chapter IX .
The following elementary facts about preannihilators are extracted from [26a] .
PROPOSITION
1.16 . / / Ji is a weak*-closed subspace of &(Jf\ then Jt is
WOT-closed if and only if
x
Jt is the | | \\ x-closure of the finite-rank operators in
xJt,
PROPOSITION
1.17 . If s/c if(Jf ) is a dual algebra, then s4 is reflexive if and
only if
x
s& is the smallest subspace of V
x
containing all the rank-one operators in
PROPOSITION
1.18. / / Sf is a subset of J?(J(?) then 9" is a dual algebra whose
preannihilator ± (£f') is the subspace of ^
x
generated by the set {KS SK:
KeVuSeS? } . In particular, if T e .£? ( Jf ), then x (sf±) is the || ^-closure of
the linear manifold {TK KT: KG ^l), and hence T has a n.h.s. if and only if
there exists a rank-one operator x ® y and a sequence { Kn}°^m,l of operators in £l
such that |K * Gy)~ (TK
n
- K
H
T)\\X - 0 .
We tur n no w t o th e proposition whic h show s why dual algebra s ar e named a s
they are. For a proof, see [16, Proposition 2.1 and Corollary 2.2].
PROPOSITION
1.19 . Let X be a complex Banach space and let Jt be a weak*
closed subspace of X* with preannihilator x Jt. Then X/ x Jt is a Banach space
whose dual (X/
x
Jt)* can be identified with Jt. In particular, with Jf as always,
if Ji is a weak* closed linear manifold in if(Jf) , then $ x(X)/
x
Ji = Q^ is a
Banach space whose dual space can be identified with Jt'. Under this identification
the pairing between Jf and Q^ is given by the bilinear functional (T, [L]) =
tr(7X), T e Jf
9
[L] G Q^r, where, as usual, we write [L] for the coset in Q^ of
an element L e ^(JT) .
Henceforth i n thes e lecture s w e wil l routinel y mak e th e identificatio n o f Q%
and Jt withou t further discussion .
The followin g theore m is frequently use d in wha t follows . For a proof, see [16,
§2].
THEOREM
1.20. / / Jt is a linear manifold in &(Jl?), then Jt is weak*-closed if
and only if Jt intersects the closed unit ball in Sf(Jff) in a weak*-closed set. If Ji
is weak *-closed, then the weak * topology that accrues to Jt as the dual space of
Qjf coincides with the relative weak* topology that accrues to Jt as a subspace of
£?(Jf?). Furthermore, if Jt is weak*-closed, X is a separable complex Banach
space, and $ : X* -^ Jt is a linear mapping, then $ is continuous when X* and Jt
are given their weak* topologies if and only if whenever {xn}^1 is a sequence in X*
Previous Page Next Page