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

Abstract

Abstract. 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 behave differently from surjective real-linear isometries and 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.

Real-Multilinear Isometries on Function Algebras

“…According to [7], there exist a nonempty subset Y j of Y , a subset K j of Y j , a continuous surjective…”

“…The proof is a modification of the proof of [7,Lemma 4.1]. Since for each (f 1 , ..., f k ) ∈ α 1 V x1 ×...×α k V x k , the maximum modulus set of T (f 1 , ..., f k ), M T (f1,...,f k ) , is a compact subset of the one point compactification Y ∞ of Y , it is enough to check that the family {M T (f1,...,f k ) : (f 1 , ..., f k ) ∈ α 1 V x1 ×...×α k V x k } has the finite intersection property.…”

Transformations preserving the norm of means between positive cones of general and commutative $C^*$-algebras

“…Indeed, Tonev and Yates [32] consider such maps between uniform algebras and give sufficient conditions to ensure them being composition operators. See also, among others, [11,14,15]. However, we have the following example of a norm-additive map between positive cones of continuous functions which is not a generalized composition operator.…”

“…The celebrated Mazur-Ulam theorem [19] states that if T is a surjective map between normed spaces E and F preserving the norm of differences, i.e., T x − T y = x − y , x, y ∈ E, then T is automatically affine. Being aware of this classical result, one may ask what happens if T preserves the norm of sums instead of differences, i.e., if T x + T y = x + y , x, y ∈ E. Such transformations are called norm-additive maps and were studied, e.g., in [4,10,14,32]. In the above setting, this turns out to be an easy problem.…”

On some isometries and other preservers

“…The study of norm-additive maps on an additive semigroup of a normed space is a very active and lively area. As for earlier results on investigations in this directions, especially on function algebras, we mention the series of publications [37,38,43,44,87].…”