2 00 9 Topological partial * - algebras : Basic properties and examples

Let A be a partial *-algebra endowed with a topology τ that makes it into a locally convex topological vector space A[τ ]. Then A is called a topological partial *-algebra if it satisfies a number of conditions, which all amount to require that the topology τ fits with the multiplier structure of A. Besides the obvious cases of topological quasi *-algebras and CQ*-algebras, we examine several classes of potential topological partial *-algebras, either function spaces (lattices of Lp spaces on [0, 1] or on R, amalgam spaces), or partial *-algebras of operators (operators on a partial inner product space, O*-algebras). E-mail: [email protected] [email protected] [email protected] UCL-IPT-97-16 November 1997 1 . Introduction and motivation A partial *-algebra is a vector space equipped with a multiplication that is only defined for certain pairs of elements. Many different species have cropped up in the recent mathematical literature, for instance, quasi *-algebras [29, 30], CQ*-algebras [15, 16] or various kinds of partial *-algebras of operators in Hilbert spaces, the so-called partial O*-algebras [6]-[12]. In all cases, there is an algebraic backbone, the abstract partial *-algebra, mentioned in [23] and developed in [6] and [9]. On top of that, a number of topological properties are introduced. For instance, partial O*-algebras were envisaged as generalizations of *-algebras of bounded operators (von Neumann algebras or C*-algebras) and of *-algebras of unbounded operators or O*-algebras [34]. Yet one element is missing in this picture, an abstract notion of topological partial *-algebra, that would encompass and unify all these examples. That such a concept is useful is illustrated by the following situation. Let (A,Ao) be a noncomplete topological quasi *-algebra, that is, Ao is a topological *-algebra, but the multiplication is only separately, not jointly continuous for the topology of Ao, and the latter is not complete. If A is the completion of Ao, then it is no longer an algebra in general, but only a partial algebra: a product AB is defined only (by continuity) if either A or B belongs to Ao. Let now πo be a *-representation of Ao by operators acting on a dense domain D(πo) in a Hilbert space H. This means, in particular, that πo maps Ao into L(D(πo)), i.e. the space of all closable operators A in H with domain D(A) = D(πo) and leaving it invariant. Now it is legitimate to ask how one could extend the representation πo from Ao to A, or at least to some larger subset of it. The obvious way would be by taking limits, using some notion of closability of the representation πo. But this implies in general extending πo beyond D(πo), since the extended operators need no longer map D(πo) into itself. In [14] we have perfromed such an extension, by operators in L(D(πo),D ), where D is the dual of D(πo) in a suitable topology. However, from the point of view of partial O*-algebras, a more natural framework for the extension is the space L(D(πo),H) of all closable operators A in H such that D(A) = D and D(A∗) ⊃ D, which is a partial *-algebra. However, in order to perform such an extension by closure, one clearly needs a more sophisticated topological structure on L(D(πo),H) than the one available in the current literature. As a matter of fact, a large number of interesting results have been obtained concerning partial O*-algebras, such as structure results, (GNS) representations, automorphisms and derivations (see [12] for a review and references to the original papers), but the interplay between the (partial) algebraic structure and the topological properties of partial O*-algebras has been largely ignored. It is the aim of the present paper to try and fill this gap. In other words, we want