Abstract :
In the Banach space setting s-complete Boolean algebras of projections
briefly, B.a..or, equivalently, their representation as spectral measures,
were intensively studied by W. Badew1, 2x. Such objects are a natural
extension of the fundamental notion of the resolution of the identity of a
normal operator in Hilbert space. Since the concepts involved are of a
topological and order theoretic nature there is no difficulty in extending
the study of such objects to the setting of locally convex Hausdorff spaces
briefly, lcHs.w10, 12x. If the underlying lcHs X is a separable Fr´echet
space, then it is known that such a B.a. M is necessarily a closed subset of
L X.with respect to the strong operator topology ts; seew8, Proposition 3;
9, Theorem 5 i.x. Here L X. denotes the space of all continuous linear
operators of X into itself. There are many sufficient conditions known on
both X and M which guarantee that if M is s-complete, then it is
necessarily a ts-closed subset of L X.; seew7]9x, for example.
Despite the quite general criteria referred to above it is easy to exhibit
examples where M fails to be ts-closed in L X.. For instance, let X
denote the non-separable Hilbert space l 2 w0, 1x.. Let S denote the family
of Borel subsets ofw0, 1x. Define, for each EgS, the selfadjoint projec-
tion P E.gL X. by P E.xsxEx for each xgX. It is straightforward
to verify that Ms P E.; EgS4 is a s-complete B.a. To see that M is
not ts-closed, let F:w0, 1xbe any non-Borel set and let F F. denote the
family of all finite subsets of F directed by inclusion. Then P E.; Eg
F F.4 is a net in M which is ts-convergent to the selfadjoint projection
QgL X.of multiplication by xF . However, QfM. Despite the fact that
M is not ts-closed it is routine to verify that M is sequentially ts-closed in
L X.. That is, if Pn4`ns1 :M is any sequence which is ts-convergent in
L X., say to the element TgL X., then actually TgM.
364
0022-247Xr97 $25.00
Copyright Q 1997 by Academic Press
All rights of reproduction in any form reserved.
BOOLEAN ALGEBRAS OF PROJECTIONS 365
Dr. Susumu Okada posed the question of whether it is always the case
that a Bade s-complete B.a. M:L X. is sequentially ts-closed. The aim
of this note is to show, under quite mild completeness restrictions on
L X., that the answer is yes in the case that the s-complete B.a. M is
purely atomic. An example is given showing that the answer is no in
general.
