2-local Isometries Research Articles

2-LOCAL ISOMETRIES OF SOME NEST ALGEBRAS

AbstractLet H be a complex separable Hilbert space with $\dim H \geq 2$ . Let $\mathcal {N}$ be a nest on H such that $E_+ \neq E$ for any $E \neq H, E \in \mathcal {N}$ . We prove that every 2-local isometry of $\operatorname {Alg}\mathcal {N}$ is a surjective linear isometry.

2-Local isometries on vector-valued differentiable functions

2-Local isometries on vector-valued differentiable functions

Approximate local isometries of derivative Hardy spaces

For any 1 ≤ p ≤ ∞, let Sp () be the space of holomorphic functions f on such that f′ belongs to the Hardy space Hp (), with the norm ∥f∥∑ = ||f||∞ +||f′|| p . We prove that every approximate local isometry of Sp () is a surjective isometry and that every approximate 2-local isometry of Sp () is a surjective linear isometry. As a consequence, we deduce that the sets of isometric reflections and generalized bi-circular projections on Sp () are also topologically reflexive and 2-topologically reflexive.

Local and 2-local isometries between absolutely continuous function spaces

Local and 2-local isometries between absolutely continuous function spaces

A SPHERICAL VERSION OF THE KOWALSKI–SŁODKOWSKI THEOREM AND ITS APPLICATIONS

AbstractLi et al. [‘Weak 2-local isometries on uniform algebras and Lipschitz algebras’, Publ. Mat.63 (2019), 241–264] generalized the Kowalski–Słodkowski theorem by establishing the following spherical variant: let A be a unital complex Banach algebra and let $\Delta : A \to \mathbb {C}$ be a mapping satisfying the following properties: (a) $\Delta $ is 1-homogeneous (that is, $\Delta (\lambda x)=\lambda \Delta (x)$ for all $x \in A$ , $\lambda \in \mathbb C$ ); (b) $\Delta (x)-\Delta (y) \in \mathbb {T}\sigma (x-y), \quad x,y \in A$ . Then $\Delta $ is linear and there exists $\lambda _{0} \in \mathbb {T}$ such that $\lambda _{0}\Delta $ is multiplicative. In this note we prove that if (a) is relaxed to $\Delta (0)=0$ , then $\Delta $ is complex-linear or conjugate-linear and $\overline {\Delta (\mathbf {1})}\Delta $ is multiplicative. We extend the Kowalski–Słodkowski theorem as a conclusion. As a corollary, we prove that every 2-local map in the set of all surjective isometries (without assuming linearity) on a certain function space is in fact a surjective isometry. This gives an affirmative answer to a problem on 2-local isometries posed by Molnár [‘On 2-local *-automorphisms and 2-local isometries of B(H)', J. Math. Anal. Appl.479(1) (2019), 569–580] and also in a private communication between Molnár and O. Hatori, 2018.

Two-local isometries on vector-valued differentiable functions

In this paper, we investigate the 2-local isometries between the vector-valued differentiable function spaces and , where X, Y are open subsets of and E is a smooth, strictly convex, reflexive and 2-iso-reflexive Banach space (in particular, when and , satisfies such conditions). We show that such maps can be written as weighted composition operators.

2-Iso-reflexivity of pointed Lipschitz spaces

We show that in the case in which X and Y are uniformly concave complete pointed metric spaces, every 2-local isometry Δ from Lip0(X) to Lip0(Y) admits a representation as the sum of a weighted composition operator and a homogeneous Lipschitz functional on, at least, a subspace Y0 of Y which is isometric to Y. Moreover, Δ is both linear and surjective when X is also separable.

2-local isometries of non-commutative Lorentz spaces

Let mathcal M be a von Neumann algebra equipped with a faithful normal finite trace tau, and let Sleft( mathcalM, tauright) be an ast -algebra of all tau -measurable operators affiliated with mathcal M . For x in Sleft( mathcalM, tauright) the generalized singular value function mu(x):trightarrow mu(tx), t0, is defined by the equality mu(tx)infxp_mathcalM:, p2pp in mathcalM, , tau(mathbf1-p)leq t. Let psi be an increasing concave continuous function on 0, infty) with psi(0) 0, psi(infty)infty, and let Lambda_psi(mathcal M,tau) left x in Sleft( mathcalM, tauright): x _psi int_0inftymu(tx)dpsi(t) infty right be the non-commutative Lorentz space. A surjective (not necessarily linear) mapping V:, Lambda_psi(mathcal M,tau) to Lambda_psi(mathcal M,tau) is called a surjective 2-local isometry, if for any x, y in Lambda_psi(mathcal M,tau) there exists a surjective linear isometry V_x, y:, Lambda_psi(mathcal M,tau) to Lambda_psi(mathcal M,tau) such that V(x) V_x, y(x) and V(y) V_x, y(y). It is proved that in the case when mathcalM is a factor, every surjective 2-local isometry V:Lambda_psi(mathcal M,tau) to Lambda_psi(mathcal M,tau) is a linear isometry.

2-Local isometries between Banach algebras of continuous functions with involution

We give a description of 2-local real isometries between $$C(X,\tau )$$ and $$C(Y,\eta )$$ where X and Y are compact Hausdorff spaces, X is also first countable and $$\tau $$ and $$\eta $$ are topological involutions on X and Y, respectively. In particular, we show that every 2-local real isometry T from $$C(X,\tau )$$ to $$ C(Y,\eta )$$ is a surjective real linear isometry whenever X is also separable.

2-Local isometries between spaces of functions of bounded variation

Given two subsets X and Y of the real line with at least two points, we apply results on surjective linear isometries between Banach spaces of all functions of bounded variation BV(X) and BV(Y) to show that every 2-local isometry $$T:BV(X)\longrightarrow BV(Y)$$ is a constant multiple of an isometric linear algebra isomorphism. Moreover, similar results are given for the closed subspaces of BV(X) and BV(Y) consisting of all continuous (resp. absolutely continuous) functions when X and Y are compact.

On 2-local *-automorphisms and 2-local isometries of [formula omitted

It is an important result of Šemrl which states that every 2-local automorphism of the full operator algebra over a separable Hilbert space is necessarily an automorphism. In this paper we strengthen that result quite substantially for *-automorphisms. Indeed, we show that one can compress the defining two equations of 2-local *-automorphisms into one single equation, hence weakening the requirement significantly, but still keeping essentially the conclusion that such maps are necessarily *-automorphisms.

Isometries of perfect norm ideals of compact operators

It is proved that for every surjective linear isometry $V$ on a perfect Banach symmetric ideal $\mathcal C_E\neq \mathcal C_2$ of compact operators, acting in a complex separable infnite-dimensional Hilbert space $\mathcal H$ there exist unitary operators $u$ and $v$ on $\mathcal H$ such that $V(x)=uxv$ or $V(x) = ux^tv$ for all $x\in \mathcal C_E$, where $x^t$ is the transpose of an operator $x$ with respect to a fixed orthonormal basis in $\mathcal H$. In addition, it is shown that any surjective 2-local isometry on a perfect Banach symmetric ideal $\mathcal C_E \neq \mathcal C_2$ is a linear isometry on $\mathcal C_E$.

Generalized 2-local isometries of spaces of continuously differentiable functions

Let C(n) (I) denote the Banach space of n-times continuously differentiable functions on I = [0, 1], equipped with the norm ||f||n = max { |f(0)|, |fʹ(0)|, … , |f(n–1)(0)|, ||f(n)||∞} (f ) ∈ C(n) (I)), where ||·||∞ is the supremum norm. We call a map T : C(n) (I) → C(n)(I) a 2-local real-linear isometry if for each pair f, g in C(n)(I), there exists a surjective real-linear isometry Tf,g : C(n)(I) → C(n)(I) such that T(f) = Tf,g(f) and T(g) = Tf,g(g). In this paper we show that every 2-local real-linear isometry of C(n)(I) is a surjective real-linear isometry. Moreover, a complete description of such maps is presented.

2-Local Isometries on Spaces of Lipschitz Functions

AbstractLet (X, d) be a metric space, and let Lip(X) denote the Banach space of all scalar-valued bounded Lipschitz functions ƒ on X endowed with one of the natural normswhere L(ƒ) is the Lipschitz constant of ƒ. It is said that the isometry group of Lip(X) is canonical if every surjective linear isometry of Lip(X) is induced by a surjective isometry of X. In this paper we prove that if X is bounded separable and the isometry group of Lip(X) is canonical, then every 2-local isometry of Lip(X) is a surjective linear isometry. Furthermore, we give a complete description of all 2-local isometries of Lip(X) when X is bounded.

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

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 = Φ x 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 ℓ p is linear and surjective for 1 ⩽ p < ∞ , p ≠ 2 , but this fails for the Hilbert space ℓ 2 .

2-Local isometries and 2-local automorphisms on uniform algebras

2-Local isometries and 2-local automorphisms on uniform algebras

2-LOCAL ISOMETRIES OF SOME OPERATOR ALGEBRAS

AbstractAs a consequence of the main result of the paper we obtain that every 2-local isometry of the $C^*$-algebra $B(H)$ of all bounded linear operators on a separable infinite-dimensional Hilbert space $H$ is an isometry. We have a similar statement concerning the isometries of any extension of the algebra of all compact operators by a separable commutative $C^*$-algebra. Therefore, on those $C^*$-algebras the isometries are completely determined by their local actions on the two-point subsets of the underlying algebras.AMS 2000 Mathematics subject classification: Primary 47B49