On topologies of finite W*-algebras
Shôichirô Sakai · Illinois Journal of Mathematics · 1965
1. Let M be a W*-algebra (namely, a C*-algebra with a dual structure as a Banach space [4], [7]).We may consider the following five typical topologies on M" (1) the norm topology u as a Banach space; (2) the 5/Iackey topology r defined by uniform convergences on every relatively z(M,, M)-compact convex set of M,, where M, is the associated space of M (namely, M is the dual of M,); (3) the topology * defined by a family of semi-norms a, a all positive M,}, (x'x) / *(x) (xx*) / for x M; (4) the topology where a(x) and a defined by a family of semi-norms a all positive e M,}; (5) the weak*topology z (namely, z(M, M,) ).We can easily see that u r if M is infinite-dimensional.By intro- ducing the r-topology into W*-algebras, the author [7] simplified the proof that two topologies and have the same dual M, as a set--that is, we showed r * z, so that by the theorem of Mackey these four topologies have the same dual M, as a set.Considering this fact, the extremal property of the r-topology must be a powerful tool in the theory of W*-algebras.On the other hand, for the -, and z-topologies, we have nice concrete representations--in fact, the * (resp.and z) coincides with the strong*- operator topology--namely, the operator topology is defined by a family of semi-norms {11 x ]l, x* lille @} (resp.the strong operator topology and the weak operator topology) on bounded spheres, when M is faithfully repre- sented as a weakly closed *-algebra on a hilbert space @.Therefore, it is also important to have an analogous representation for the r-topology.In this note, we shall show a concrete representation of the r-topology of finite W*-algebras as follows" the r-topology of finite W*-algebras is equivalent to the -topology on bounded spheres.As a corollary of this result, we shall show that every z-continuous linear mapping of a finite W*-algebra into another W*-algebra is -continuous on bounded spheres.For non-finite W*-algebra, we have no solution; clearly is < on bounded spheres for non-finite ones, because s < on bounded spheres (cf.[5], [7]).Our conjecture is as follows" caa we conclude that the r-topology is equivalent to the *-topology on bounded spheres for all W*-algebras?2. Let M be a finite W*-algebra, M, the associated space of M. LEMMA 1.Let (fi) be a countable family of elements in M, then there is a