Composition operators between algebras of uniformly continuous functions

Abstract

A characterization of uniform continuity for maps between unit balls of real Banach spaces is given in terms of universal properties. Also, it is shown that the classes of compact and weakly compact composition operators induced by such maps agree.

Realcompactness and spaces of vector-valued functions

“…But, on the other hand, we can prove as in [7,Theorem 2.3] (see also the Remark after it) that if f • h ∈ C * u (Y ) whenever f ∈ C * u (X), then h is uniformly continuous. Since the same argument works also for h −1 , the corollary is proved.…”

Linear isometries on subalgebras of uniformly continuous functions

“…Also with the same proof as in [5,Theorem 2.3], but using Lemma 4.1 instead, we have the following result.…”

“…In [5], Lacruz and Llavona characterized uniform continuity for maps between the unit balls of two Banach spaces, Kx and Ky, in terms of composition operators of C u (Kx) into C u (Ky). In this paper, we obtain similar results for weakly normal subalgebras of C u {Kx) and C U (KY), and use them to describe the linear isometries T between such subalgebras.…”

“…Let (x n ), (x^) be sequences in K x with the property that lim\f(x n )-f(x' n )\ = 0, for every f 6 A u (Kx)-Then lim \\x n -x' n \\ = 0.Proof. The idea of the proof of[5, Lemma 2.2] remains valid for this theorem. Just the first step of that proof needs some changes.…”

Kaplansky Theorem for completely regular spaces

“…Define a linear map ψ(f ) = φ(φ −1 (1)f ) = f • τ from U C b (X) into U C(Y ). Using the arguments in [16,Theorem 2.3], we can show that τ is uniformly continuous. Similarly, τ −1 is also uniformly continuous.…”