Isometries on spaces of absolutely continuous functions in a noncompact framework

In this paper we deal with surjective linear isometries between spaces of scalar-valued absolutely continuous functions on arbitrary (not necessarily closed or bounded) subsets of the real line (with at least two points). As a corollary, it is shown that when the underlying spaces are connected, each surjective linear isometry of these function spaces is a weighted composition operator, a result which generalizes all the previous known results concerning such isometries

Norm-additive in modulus maps between function algebras

The main purpose of this paper is to characterize norm-additive in modulus, not necessarily linear, maps defined between function algebras (not necessarily unital or uniformly closed). In fact, for function algebras A and B on locally compact Hausdorff spaces X and Y , respectively, we study surjections T,S : A −→ B satisfying ∥|Tf| + |Sg|∥Y = ∥|f| + |g|∥X for all f, g ∈ A

Multi-linear isometries on spaces of vector-valued continuous functions

In this paper we study multilinear isometries defined on certain subspaces of vector-valued continuous functions. We provide conditions under which such maps can be properly represented. Our results contain all known results concerning linear and bilinear isometries defined between spaces of continuous functions. The key result is a vector-valued version of the additive Bishop’s Lemma, which we think has interest in itself

Diameter preserving mappings between function algebras

In this paper we study the behaviour of linear diameter preserving mappings when defined between subalgebras of continuous functions. Namely, we obtain a representation of such mappings as the sum of a weighted composition operator and a linear functional on, at least, the Choquet boundaries of the algebras under consideration. In particular, we give a complete description when we consider several classical function algebras

Real-Linear Isometries and Jointly Norm-Additive Maps on Function Algebras

In this paper, we describe into real-linear isometries defined between (not necessarily unital) function algebras and show, based on an example, that this type of isometries behaves differently from surjective real-linear isometries and from classical linear isometries. Next we introduce jointly norm-additive mappings and apply our results on real-linear isometries to provide a complete description of these mappings when defined between function algebras which are not necessarily unital or uniformly closed.Research of J. J. Font was partially supported by Universitat Jaume I (Projecte P1-1B2014-35)

Diameter preserving maps on function spaces

In this paper we describe, under certain assumptions, surjective diameter preserving mappings when defined between function spaces, not necessarily algebras, thus extending most of the previous results for these operators. We provide an example which shows that our assumptions are not redundant.Universitat Jaume I (Projecte P1·1B2014-35) and Generalitat Valenciana (Projecte AICO/16/030

Korovkin-Type Results on Convergence of Sequences of Positive Linear Maps on Function Spaces

In this paper, we deal with the convergence of sequences of positive linear maps to a (not assumed to be linear) isometry on spaces of continuous functions. We obtain generalizations of known Korovkin-type results and provide several illustrative examples

Linear and Multilinear Isometries in a Noncompact Framework

Both classical linear and multilinear isometries defined between subalgebras of bounded continuous functions on (complete) metric spaces are studied. Particularly, we prove that certain such subalgebras, including the subalgebras of uniformly continuous, Lipschitz or locally Lipschitz functions, determine the topology of (complete) metric spaces. As consequence, it is proved that the subalgebra of Lipschitz functions determines the Lipschitz in the small structure of a complete metric space. Furthermore, we provide a weighted composition representation for multilinear isometries from similar subalgebras on (not necessarily complete) metric spaces. We apply this general representa- tion to obtain more specific ones for subalgebras of uniformly continuous and Lipschitz functions