The Embedding Theorem of an L0-Prebarreled Module into Its Random Biconjugate Space

We first prove Mazur’s lemma in a random locally convex module endowed with the locally L0-convex topology. Then, we establish the embedding theorem of an L0-prebarreled random locally convex module, which says that if (S,P) is an L0-prebarreled random locally convex module such that S has the countable concatenation property, then the canonical embedding mapping J of S onto J(S)⊂(Ss⁎)s⁎ is an L0-linear homeomorphism, where (Ss⁎)s⁎ is the strong random biconjugate space of S under the locally L0-convex topology.

Krein-Smulian-Type Theorems

Let (E, ℱ) be a weakly compactly generated Frechet space and let ℱ0 be another weaker Hausdorff locally convex topology on E. Let X be an ℱ-bounded compact subset of (E, ℱ0). The ℱ0-closed convex hull of X in E is then ℱ0-compact. We also give a new proof, without using Riemann–Lebesgue-integrable (Birkoff-integrable) functions, with the result that if (E, ∥ · ∥) is any Banach space and ℱ0 is fragmented by ∥ · ∥, then the same result holds. Furthermore, the closure of the convex hull of X in ℱ0-topology and in the original topology of E is the same.

THE RANDOM SUBREFLEXIVITY OF COMPLETE RANDOM NORMED MODULES

Combining respective advantages of the (ε, λ)-topology and the locally L0-convex topology we first prove that every complete random normed module is random subreflexive under the (ε, λ)-topology. Further, we prove that every complete random normed module with the countable concatenation property is also random subreflexive under the locally L0-convex topology, at the same time we also provide a counterexample which shows that it is necessary to require the random normed module to have the countable concatenation property.

THE STRICT TOPOLOGY ON THE DISCRETE LEBESGUE SPACES

Let Σ be a set and σ be a positive function on Σ. We introduce and study a locally convex topology β1(Σ,σ) on the space ℓ1(Σ,σ) such that the strong dual of (ℓ1(Σ,σ),β1(Σ,σ)) can be identified with the Banach space $(c_0(\Sigma ,1/\sigma ),\|\cdot \|_{\infty ,\sigma })$. We also show that, except for the case where Σ is finite, there are infinitely many such locally convex topologies on ℓ1(Σ,σ). Finally, we investigate some other properties of the locally convex space (ℓ1(Σ,σ),β1(Σ,σ)) , and as an application, we answer partially a question raised by A. I. Singh [‘L∞0(G)* as the second dual of the group algebra L1 (G) with a locally convex topology’, Michigan Math. J.46 (1999), 143–150].

Weakly compact operators and the strong* topology for a Banach space

The strong* topology s*(X) of a Banach space X is defined as the locally convex topology generated by the seminorms x ↦ ‖Sx‖ for bounded linear maps S from X into Hilbert spaces. The w-right topology for X, ρ(X), is a stronger locally convex topology, which may be analogously characterized by taking reflexive Banach spaces in place of Hilbert spaces. For any Banach space Y , a linear map T : X → Y is known to be weakly compact precisely when T is continuous from the w-right topology to the norm topology of Y. The main results deal with conditions for, and consequences of, the coincidence of these two topologies on norm bounded sets. A large class of Banach spaces, including all C*-algebras and, more generally, all JB*-triples, exhibit this behaviour.

UnboundedC*-seminorms, biweights, and*-representations of partial*-algebras: A review

The notion of (unbounded)C*-seminorms plays a relevant role in the representation theory of*-algebras and partial*-algebras. A rather complete analysis of the case of*-algebras has given rise to a series of interesting concepts like that of semifiniteC*-seminorm and spectralC*-seminorm that give information on the properties of*-representations of the given*-algebraAand also on the structure of the*-algebra itself, in particular whenAis endowed with a locally convex topology. Some of these results extend to partial*-algebras too. The state of the art on this topic is reviewed in this paper, where the possibility of constructing unboundedC*-seminorms from certain families of positive sesquilinear forms, called biweights, on a (partial)*-algebraAis also discussed.