We investigate here the notion of a topological ring of quotients of a topological ring with respect to an arbitrary Gabriel (idempotent) filter of right ideals. We describe the topological ring of quotients first as a subring of the algebraic ring of quotients, and then show it is a topological bicommutator of a topological injective R-module. Unlike R. L. Johnson in [6] and F. Eckstein in [2] we do not always make the ring an open subring of its ring of quotients. This would exclude examples such as C(X), the ring of continuous real-valued functions on a compact space, and its ring of quotients as described in Fine, Gillman and Lambek [3].

Let Δ be the open unit disk in the complex plane and let be the group of automorphisms of Δ onto Δ, define by

The Banach spaces Hp = Hp(Δ), 1 ≦ p < ∞, are the Hardy spaces of functions analytic in Δ with their integral p means bounded,

The u-invariant of a field F, u = u(F), is defined to be the maximum of the dimensions of anisotropic quadratic forms over F. If F is a non-formally real field with a finite number q of square classes then it is known that u ≦ q. The purpose of this note is to give some necessary and sufficient conditions for equality in terms of the structure of the Witt ring of F.

The observations of this paper evolved from the concept of 'asymptotic nonexpansiveness' introduced by two of the writers in a previous paper [10]. Let X be a Banach space and K ⊆ X. A mapping T : K → K is called asymptotically nonexpansive if for each x, y ∊ K

where {ki} is a fixed sequence of real numbers such that ki→1 as i → ∞ . It is proved in [10] that if K is a bounded closed and convex subset of a uniformly convex space X then every asymptotically nonexpansive mapping T : K → K has a fixed point. This theorem generalizes the fixed point theorem of Browder-Göhde-Kirk [2 ; 12 ; 16] for nonexpansive mappings (mappings T for which ||T(x) — T(y)|| ≦ ||x — y||, x, y ∊ K) in a uniformly convex space. (A generalization along similar lines also has been obtained by Edelstein [4].)

A finite lattice L of order n is dismantlable [6] if there is a chain L1 ⊂ L2 ⊂ . . . ⊂ Ln = L of sublattices of L such that |Li| = i for every i = 1, 2, . . . , n. In [1] it was shown that every finite planar lattice is dismantlable. Furthermore, every lattice L with |L| ≦ 7 is dismantlable [6]; in fact, every large enough lattice contains a dismantlable sublattice with precisely n elements [4]. As well, such lattices are closed under the formation of sublattices and homomorphic images [6].

The principal result of this paper states that if X is a pre-Hilbert B-module over an arbitrary C*-algebra B, then the B-valued inner product on X can be lifted to a B-valued inner product on X″ (the B-dual of the B-dual X′ of X). Appropriate identifications allow us to regard X as a submodule of X″ and the latter in turn as a submodule of X′. In this sense, the inner product on X″ is an extension of that on X. As an example (and application) of this result, we consider the special case in which X is a right ideal of B and give a topological description of X″ when in addition B is commutative.