We investigate some problems of the following type: For which sets H is it true that if f is in a given class F of periodic functions and the difference functions ∆hf(x) = f(x + h) − f(x) are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by H(F ,G), that is, H(F ,G) = {H ⊂ R/Z : (∃f ∈ F \ G) (∀h ∈ H) ∆hf ∈ G}, we try to charact...

We introduce unitary representations of continuous groupoid on continuous fields of Hilbert spaces. We investigate some properties of these objects and discuss some of the standard constructions from representation theory in this particular context. An important rôle is played by the regular representation. We conclude by discussing some operator algebra associated to continuous representations...

The purpose of this paper is to consider the class of continuous collections of mutually exclusive compact continuous curves in the plane. Throughout this paper we shall denote by G or G with a subscript or superscript a collection of this class with the property that G with respect to its elements as points is a nondegenerate compact closed point set. G has then the significance of being both ...

We prove that the semigroup operation of a topological semigroup S extends to a continuous semigroup operation on its the Stone-Čech compactification βS provided S is a pseudocompact openly factorizable space, which means that each map f : S → Y to a second countable space Y can be written as the composition f = g ◦ p of an open map p : X → Z onto a second countable space Z and a map g : Z → Y ...

and Applied Analysis 3 (1) either f or g is continuous; (2) there exists F satisfying the condition φ, for all x, y ∈ X, such that d (f (x) , g (y)) ≤ F (d (x, y) , d (x, f (x)) , d (y, g (y))) , (4) then f, g have unique common fixed points. Proof. Let g be continuous, x 0 arbitrary in X, x n , y n the sequence ofX, and x n = (fg) n (x 0 ) = fg (x n − 1) , y n = g(fg) n−1 (x 0 ) ,

Let C(X,G) denote the group of continuous functions from a topological space X into a topological group G with the pointwise multiplication and the compact-open topology. We show that there is a natural topology on the collection of normal subgroups A(X) of C(X,G) of the M {f C(X,G): f(p) e} which is P analogous to the hull-kernel topology on the commutative Banach algegra C(X) of all continuou...

We define, for a locally compact quantum group G in the sense of Kustermans– Vaes, the space of LUC (G) of left uniformly continuous elements in L∞(G). This definition covers both the usual left uniformly continuous functions on a locally compact group and Granirer’s uniformly continuous functionals on the Fourier algebra. We show that LUC (G) is an operator system containing the C∗-algebra C0(...