Partial * - Algebras of Closable Operators : a Review

This paper reviews the theory of partial *-algebras of closable operators in Hilbert space (partial O*-algebras), with some emphasis on partial GW*-algebras. First we discuss the general properties and the various types of partial *-algebras and partial O*-algebras. Then we summarize the representation theory of partial *-algebras, including a generalized Gel'fand-Naimark-Segal construction; the main tool here is the notion of positive sesquilinear form, that we study in some detail (extendability, normality, order structure,. . .). Finally we turn to automorphisms and derivations of partial O*-algebras, and their mutual relationship. The central theme here is to nd conditions that guarantee spatiality. Ever since the pioneering days of Heisenberg's matrix mechanics, operator algebras have played a prominent role in quantum theories. Besides the traditional Hilbert space formulation of Schrr odinger, Dirac and von Neumann, the algebraic formulation has also become standard, for instance in quantum statistical mechanics (see e.g. the monograph of Bratteli and Robinson 25]). However the algebras used in this context consist invariably of bounded operators (typically, representations of abstract C*-algebras and their bicommutants, that is, von Neumann algebras). In particular, the latter play a crucial role in the Tomita-Takesaki theory 67, 70]. Yet it is often more natural in physical applications to consider unbounded operators, e.g. generators of symmetry groups, such as position, momentum, energy, angular momentum , etc. In that case it is usually assumed that all the relevant operators have a common dense invariant domain. Take for instance a single nonrelativistic spinless particle. In the Schrr odinger representation, with Hilbert space L 2 (IR 3), the canonical variables are represented by the operators ~ Q and ~ P, both unbounded and obeying the canonical commutation relations Q j ; P k ] = ii jk : The natural domain associated to this system is of course Schwartz space S(IR 3), and the corresponding *-algebra L y (S) consists of all operators A with domain S, such that AS S and A*S S. In the same way, one obtains the CCR algebra over S(IR 3n) for a system of n spinless particles. If spin and statistics are taken into account, the algebra is L y (W), where W is obtained from S by simple modiications. This is indeed the natural arena for nonrelativistic quantum mechanics, as demonstrated forcefully by Dubin and Hennings 30]. Elaboration of this model led eventually the theory of algebras of unbounded operators, also known …