On rings of operators

Abstract

Let 5C be a complex Hubert Space, J3(3C) the ring of bounded operators on 3C, E an abelian symmetric subring of B(3Z) containing the identity which is closed in the weak operator topology, E\ the commutant of E, and suppose E\ has a cyclic vector £o which we normalize so that |£o| =1. Diximier [l] has shown that E (respect. Ei), as a Banach space, is the dual of the Banach space R (respect. Ri) of all linear forms on E (respect. E\) that are continuous in the ultra-strong topology of E (respect. Ei). In this note we show that every TÇzR is also continuous in the weak operator topology of E, from which it follows that a linear functional T on E is continuous in either the weak, ultraweak, strong, or ultrastrong topologies if and only if it is continuous in all four simultaneously. In the process, we obtain an integral representation for such T, which we later use in a theorem on centrally reducible positive functionals on E%.We denote the maximal ideal space of E by M, and for A, B, • • • £E, we denote the corresponding Gel'fand transforms by a, 6, • • • . Then A->a is an isometric isomorphism from £ onto C(M). Consequently, every bounded linear functional on E has the formwhere v is a complex Borel measure on M uniquely determined by T.Of special interest are functionals of the form A -»(i4 £, £), where | is a vector of 3C. We denote by v$ the measure corresponding to the vector £, and by \x the measure v^. for every AÇzE.

Kinetic Expression with Diffusion Correction for Ammonia Synthesis on Industrial Catalyst

“…In Equations 6, 7, and 8 the temperature, T, is in degrees Kelvin and the pressure, P, is in atmospheres. [The fugacity of a component may be calculated from an equation of state (Porcelli, 1967), but, unfortunately, no applicable volumeexplicit equations of state have been found in the literature, so the procedure is relatively time-consuming. We have studied this method using both the Beattie-Bridgman and the Redlich-Kwong equations of state and have concluded that the simple equations given above are precise.…”

An equivalent formulation of the invariant subspace conjecture

“…The proofs of our main results require a great deal of the theory of direct integral decompositions for von Neumann algebras. The necessary literature for our use of the theory can be found in [3], [5], [6] and [8]. In what follows, E will denote a weakly closed symmetric subring of B(H) such that its commutant E has a cyclic vector for H. M denotes the maximal ideal space of E and for BeE the mapping B -• È(m) denotes the Gelfand transform of B.…”