dedicated to professor ivan vidav in honor of his eightieth birthday
Given a von Neumann algebra R on a Hilbert space H, the so-called R-topology is introduced into B(H), which is weaker than the norm and stronger than the ultrastrong operator topology. A right R-submodule X of B(H ) is closed in the R-topology if and only if for each b # B(H ) the right ideal, consisting of all a # R such that ba # X, is weak* closed in R. Equivalently, X is closed in the R-topology if and only if for each b # B(H ) and each orthogonal family of projections e i in R with the sum 1 the condition be i # X for all i implies that b # X.
1998 Academic Press
1. Introduction
Given a C*-algebra R on a Hilbert space H, a concrete operator right R-module is a subspace X of B(H) (the algebra of all bounded linear operators on H ) such that XR X. Such modules can be characterized abstractly as L -matricially normed spaces in the sense of Ruan [21],
[11] which are equipped with a completely contractive R-module multiplication (see [6] and[9]). If R is a von Neumann algebra, it is natural to study the R-submodules of B(H ) which are closed in the weak* topology. It turns out, however, that many properties of weak* closed modules are valid in fact for the larger class of so-called strong modules.
such that the sums i # I x i x i * and i # I a i * a i are convergent to bounded operators in the strong operator topology. All weak* closed modules are strong of course, and it turns out that all strong R-submodules of R (that is, right ideals) are necessarily weak* closed, but in general the class of all strong R-modules is much larger than the class of all weak* closed R-modules. For example, in the case R=C all norm closed R-submodules are strong.
Strong modules appeared first in a remark at the end of the paper [9] by Effros and Ruan under the name M -submodules. Later, in [15], a few article no.