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*