This paper investigates the completion of the maximal Op*-algebra L + (D) of (possibly) unbounded operators on a dense domain D in a Hilbert space. It is assumed that D is a Frechet space with respect to the graph topology. Let D + denote the strong dual of D, equipped with the complex conjugate linear structure. It is shown that the completion of L + (D} (endowed with the uniform topology) is the space of continuous linear operators X (D, D + ) . This space is studied as an ordered locally convex space with an involution and a partially defined multiplication. A characterization of bounded subsets of D in terms of self-adjoint operators is given. The existence of special factorizations for several kinds of operators is proved. It is shown that the bounded operators are uniformly dense in § 1. Introduction Non-normable topological *-algebras satisfying various completeness conditions have been studied in several papers (see, e.g., [7,8,10,12,22,23,31,34]). However, these conditions are not fulfilled for the maximal *-algebra L + (D] of (possibly unbounded) operators on a dense linear subspace D of some Hilbert space H (for precise definitions, see Section 2). On the other hand, L + (D) is one of the most important unbounded operator algebras because it contains all *-algebras of operators on a fixed domain D.It is the aim of this paper to study the completion of L + (D) with respect to the uniform topology. We assume throughout that D is a Frechet space in the topology defined by the graph norms of operators belonging to L + (D}. However, some of the results can be obtained for more general domains D by the same proofs (see Remark 3 after Proposition 3.8 and the remarks after Proposition 5.1 and Corollary 5.6).Among [24,25]. However, it is closely related to the product of operators on partial inner product spaces which was defined in [3]. Linear spaces with a partially defined multiplication were previously considered also in [4,5,6,11].The pattern of the paper is as follows. In Section 2, we recall some definitions, notations, and some known or easy results. The study of the space JC(D, D + ] will be continued in [21]. In particular, we show there that X(D 9 D + ] is the second strong dual of its subspace of completely continuous operators. In [20, 33], the methods of the present paper are applied to the investigation of closed ideals in L + (D\ Acknowledgements

An operator-theoretic approach to invariant integrals on non-compact quantum spaces is introduced on the examples of quantum ball algebras. In order to describe an invariant integral, operator algebras are associated to the quantum space which allow an interpretation as "rapidly decreasing" functions and as functions with compact support. If an operator representation of a first order differential calculus over the quantum space is known, then it can be extended to the operator algebras of integrable functions. The important feature of the approach is that these operator algebras are topological spaces in a natural way. For suitable representations and with respect to the bounded and weak operator topologies, it is shown that the algebra of functions with compact support is dense in the algebra of closeable operators used to define these algebras of functions and that the infinitesimal action of the quantum symmetry group is continuous. §1. IntroductionThe development of quantum mechanics at the beginning of the past century resulted in the discovery that nuclear physics is governed by noncommutative quantities. Recently, there have been made various suggestions that spacetime may be described by non-commutative structures at Planck scale. Within this approach, quantum groups might play a fundamental role.

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This is an expository paper on the importance and applications of GB *-algebras in the theory of unbounded operators, which is closely related to quantum field theory and quantum mechanics. After recalling the definition and the main examples of GB *-algebras we exhibit their most important properties. Then, through concrete examples we are led to a question concerning the structure of the completion of a given C *-algebra A0[ • 0], under a locally convex *-algebra topology τ , making the multiplication of A0 jointly continuous. We conclude that such a completion is a GB *-algebra over the τ-closure of the unit ball of A0[ • 0]. Further, we discuss some consequences of this result; we briefly comment the case when τ makes the multiplication of A0 separately continuous and illustrate the results by examples.