Lattices of uniformly continuous functions

Rings of Uniformly Continuous Functions

There is a natural bijective correspondence between the compactifications of a Tychonoff space X, the totally bounded uniformities on X, and the unital C∗-subalgebras of C∗(X) (the algebra of bounded continuous complex valued functions on X) with what we call the completely regular separation property. The correspondence of compactifications with totally bounded uniformities is well know and ca...

Lattices of Continuous Functions

Between continuous and uniformly continuous functions on R n ∗

We study classes of continuous functions on R that can be approximated in various degree by uniformly continuous ones (uniformly approachable functions). It was proved in [BDP1] that no polynomial function can distinguish between them. We construct examples that distinguish these classes (answering a question from [BDP1]) and we offer appropriate forms of uniform approachability that enable us ...

Completion of Lattices of Semi-continuous Functions

If U and V are topologies on an abstract set X, then the triple (X, U, V) is a bitopological space. Using the theorem of Priestley on the representation of distributive lattices, results of Dilworth concerning the normal completion of the lattice of bounded, continuous, realvalued functions on a topological space are extended to include the lattice of bounded, semi-continuous, real-valued funct...

Uniformly summing sets of operators on spaces of continuous functions

Let X and Y be Banach spaces. A set ᏹ of 1-summing operators from X into Y is said to be uniformly summing if the following holds: given a weakly 1-summing sequence (x n) in X, the series n T x n is uniformly convergent in T ∈ ᏹ. We study some general properties and obtain a characterization of these sets when ᏹ is a set of operators defined on spaces of continuous functions.