On 2-local isometries on continuous vector-valued function spaces

Abstract

A (not necessarily linear) mapping Φ from a Banach space X to a Banach space Y is said to be a 2-local isometry if for any pair x, y of elements of X, there is a surjective linear isometry T : X → Y such that T x = Φx and T y = Φ y. We show that under certain conditions on locally compact Hausdorff spaces Q , K and a Banach space E, every 2-local isometry on C 0 (Q , E) to C 0 (K , E) is linear and surjective. We also show that every 2-local isometry on p is linear and surjective for 1 p < ∞, p = 2, but this fails for the Hilbert space 2 .A (not necessarily linear) mapping Φ from a Banach space X to a Banach space Y is said to be a 2-local isometry if for any pair x, y of elements of X , there is a surjective linear isometry T : X → Y such that Φx = T x and Φ y = T y. The general question is whether Φ must itself be a surjective linear isometry. This type of problem is basic in that it asks whether a local assumption is enough to guarantee a more global conclusion. Early investigations along these lines involved derivations and automorphisms of operator algebras and were carried out by Kadison [8], Larson [9], and Larson and Sourour [10]. A set S of operators is called algebraically reflexive if S must contain every T which is local in this sense: given x in the domain, there is an S ∈ S such that T x = Sx. If the group G(X) of surjective linear isometries on X is algebraically reflexive, we will say that X is iso-reflexive. This language could also be applied to a pair (X, Y ) of Banach spaces if the isometries go from X to Y .Results concerning iso-reflexivity of certain operator algebras and function algebras have been obtained, about which [2,6,13], with their references, serve as a good introduction. In particular, Molnár and Zalar [12] showed that if Q is compact, Hausdorff, and first countable, then C (Q ) is iso-reflexive. Jarosz and Rao [6] extended this to the vector-valued case, proving that if Q is a first countable compact Hausdorff space and E is a uniformly convex and iso-reflexive Banach space, thenThe notion of 2-local is due to Šemrl [14] who was interested in dropping the linearity assumption for local automorphisms and derivations on L(H), the bounded linear operators on H , where H is a Hilbert space. To compensate for the loss of linearity, it was useful to require the local condition at two points. Molnár [11] showed that every 2-local isometry on L(H) is linear and so a surjective linear isometry. Gyory [5] showed that if Q is a first countable, σ -compact, (separable) locally compact Hausdorff space, then every 2-local isometry on C 0 (Q ) is a surjective linear isometry. That paper is the inspiration for the current note, in which we wish to consider the extension of Gyory's theorem to C 0 (Q , E) for an appropriate Banach space E. By C 0 (Q , E) we mean, of course, the continuous functions on Q to E which vanish at infinity and given the sup norm. In case E is the scalar field, we just write C 0 (Q ).Let us agree to say that a Banach space X is 2-iso-reflexive if every 2-local isometry...

2-Local standard isometries on vector-valued Lipschitz function spaces

“…Given a compact subset K ⊂ C, let A(K) denote the algebra of all complex-valued continuous functions on K which are holomorphic on the interior Int(K) of K. If K is a connected compact subset of C such that Int(K) has finitely many components and Int(K) = K, then every 2-local isometry (2-local automorphism) T on A(K) is a surjective linear isometry (respectively, automorphism). The same conclusion holds when K is the closure of a strictly pseudoconvex domain in C 2 with boundary of class C 2 (Hatori, Miura, Oka and Takagi [11]); ( ) If K is a σ-compact metric space and E is a smooth reflexive Banach space, then C 0 (K, E) is 2-iso-reflexive if and only if E is 2-iso-reflexive (Al-Halees and Fleming [1]); ( ) Every weak-2-local isometry between uniform algebras is linear (Li, Peralta, Wang and Wang [16]). 2-local derivations on C * -algebras have been studied in [21,2,3,19,20,8,9,14] and [15].…”

“…In fact, that condition is necessary to get the conclusion of these theorems. Actually, the same arguments given in the proof of [1,Theorem 3] can be applied to get the following result. Theorem 2.8.…”

2-local isometries on function spaces

“…We usually abbreviate G Π (E 1 , E 2 ) by G Π if E 1 and E 2 are clear from the context. Let Id [0,1] = π 0 : [0, 1] → [0, 1] be the identity function and π 1 = 1 − Id [0,1] . Put Π 0 = {π 0 , π 1 }.…”

“…Since U(1) − U(i) is a constant function we have √ 2 = |1 − i| = U(1) − U(i) Σ = U(1) − U(i) ∞ as 0 = L U (1)−U (i) = D 2 (U(1) − U(i)). Since U(1) and U(i) are constant functions we infer that U(i) = iU(1) or U(i) = −iU (1). Put U 0 = U(1)U.…”

Algebraic reflexivity of isometry groups and automorphism groups of some operator structures

“…One can find corresponding results and references in the book [21] (Sections 3.4, 3.5 and see also p. 24 in the Introduction). For more recent results we refer to the papers [1], [4], [9], [11]. In fact, although one of the main advantages of the concept of 2-locality is that the reflexivity of classes of transformations can be investigated in non-linear settings, in all the latter four papers the authors considered 2-local isometries, 2-local automorphisms, etc.…”