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A *-algebra SI of linear operators with a common invariant dense domain & in a Hilbert space is studied relative to the order structure given by the cone Sl + of positive elements of SI (in the sense of positive sesquilinear form on &) and the p-topology defined as an inductive limit of the order norm p A (of the subspace SI X with A as its order unit) with A eSl + . In particular, for those SI with a countable cofinal sequence A, in ST + such that AT 1 eSI, the ,0-topology is proved to be order convex, any positive elements in the predual is shown to be a countable sum of vector states, and the bicommutant within the set B(&, &) of continuous sesquilinear forms on 3f is shown to be the ultraweak closure of SI. The structure of the commutant and the bicommutant are explicitly given in terms of their bounded operator elements which are von Neumann algebras and the commutant of each other. § 1. Introduction Our aim in this paper is to develop a theory of a certain class of *-algebras 91 of linear operators with a common invariant dense domain 2 in a Hilbert space 3? in parallel with theory of von Neumann algebras as a continuation of [7]. (Also see [14].) The set B(&, <&) of all continuous sesquilinear forms on @ (the continuity relative to the collection of norms @ 3 x»-> ||^4x|[, A e 91) plays the role of the set L(j^) of all bounded linear operators on 3? in theory of von Neumann algebras. For those 91 satisfying Condition I described below, we can give the decomposition of continuous linear forms into positive components, i.e. the strong normality of the positive cone 9l + , the description of positive elements in the predual of 91 and the notion of commutant, for which the bicommutant coincides with the ultraweak closure.Under weaker Condition I 0 or IQ, also described below, structure of the

Abstract. Let C(X; R) the algebra of continuous real valued functions defined on a locally compact space X. We consider linear subspaces A ⊂ C(X; R) having the following property: there is a sequence (Φj) j∈N of positive functions in A with limx→∞ Φj(x) = +∞ for every j ∈ N, such that A consists of functions f ∈ C(X; R) bounded above for the absolute value by an homothetic of some Φn (n depends on each f ). Dominated convergence of a sequence (gn) n≥1 in A is an estimation of the form |gn(x) − g(x)| ≤ εn|h(x)| for allx ∈ X and all n ∈ N where gn, g, h ∈ A and εn → 0 as n → ∞. We extend the Stone-Weierstrass theorem to subalgebras or lattices B ⊂ A and we show that the dominated convergence for sequences is exactly the convergence of sequences when A is endowed with a locally convex (DF)-space topology.

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