Tingley's problems on uniform algebras

The Mazur–Ulam property in ℓ∞-sum and c0-sum of strictly convex Banach spaces

Given a C ∗ -algebra A , let S ( A + ) denote the set of positive elements in the unit sphere of A . Let H 1 , H 2 , H 3 , and H 4 be complex Hilbert spaces, where H 3 and H 4 are infinite-dimensional and separable. In this article, we prove a variant of Tingley’s problem by showing that every surjective isometry Δ : S ( B ( H 1 ) + ) → S ( B ( H 2 ) + ) (resp., Δ : S ( K ( H 3 ) + ) → S ( K ( H 4 ) + ) ) admits a unique extension to a surjective complex linear isometry from B ( H 1 ) onto B ( H 2 ) (resp., from K ( H 3 ) onto K ( H 4 ) ). This provides a positive answer to a conjecture recently posed by Nagy.

We introduce a vector-valued version of a uniform algebra, called the vector-valued function space over a uniform algebra. The diameter two properties of the vector-valued function space over a uniform algebra on an infinite compact Hausdorff space are investigated. Every nonempty relatively weakly open subset of the unit ball of a vector-valued function space $$A(K, (X, \tau ))$$ over an infinite dimensional uniform algebra has diameter two, where $$\tau $$ is a locally convex Hausdorff topology on a Banach space X compatible to a dual pair. Under the assumption of X equipped with the norm topology being uniformly convex and the additional condition that $$A\otimes X\subset A(K, X)$$ , it is shown that Daugavet points and $$\Delta $$ -points on A(K, X) over a uniform algebra A are the same, and they are characterized by the norm-attainment at a limit point of the Shilov boundary of A. In addition, a sufficient condition for the convex diametral local diameter two property of A(K, X) is also provided. Similar results also hold for an infinite dimensional uniform algebra.

Let l < p < ∞, p ≠ 2 and α > 0. In what follows sp(α) will denote the space of all real or complex sequences for which(1.1)In this paper we show that the spaces sp(α) are Banach spaces under the natural norm and in fact share many properties that the usual lp spaces have. Our main results give characterizations of the surjective isometries of sp(α). These turn out to be quite different than the results for lp. For example, we show that for α ≠ 1, an operator T is a surjective isometry if and only if T is a modulus one multiple of the identity. The methods used are valid for both real and complex scalars. They involve the use of a disjoint support condition together with a property of semi inner products. In the complex case the information on isometries allows us to give complete descriptions of the Hermitian operators as well as the adjoint abelian operators.

A characterization of isometries of complex Musielak-Orlicz spaces $L_Φ$ is given. If $L_Φ$ is not a Hilbert space and $U : L_Φ → L_Φ$ is a surjective isometry, then there exist a regular set isomorphism τ from (T,Σ,μ) onto itself and a measurable functio

Abstract. The necessary and sufficient conditions for stability of abstract difference schemes in Hilbert and Banach spaces are formulated. Contrary to known stability results we give stability conditions for schemes with non-self-adjoint operator coefficients in a Hilbert space and with strongly positive operator coefficients in a Banach space. It is shown that the parameters of the sectorial spectral domain play the crucial role. As an application we consider the Richardson iteration scheme for an operator equation in a Banach space, in particulary the Richardson iteration with precondition for a finite element scheme for a non-selfadjoint operator. The theoretical results are also the basis when using the regularization principle to construct stable difference schemes. For this aim we start from some simple scheme (even unstable) and derive stable schemes by perturbing the initial operator coefficients and by taking into account the stability conditions. Our approach is also valid for schemes with unbounded operator coefficients.

We show that any super-reflexive Banach space is a $\Lambda$-space (i.e., the weak-polynomial convergence for sequences implies the norm convergence). We introduce the notion of $\kappa$-space (i.e., a Banach space where the weak-polynomial convergence for sequences is different from the weak convergence) and we prove that if a dual Banach space $Z$ is a $\kappa$-space with the approximation property, then the uniform algebra $A(B)$ on the unit ball of $Z$ generated by the weak-star continuous polynomials is not tight.

In this survey we report on very recent results about some non-linear geometrical properties of many classes of real and complex Banach spaces and uniform algebras, including the ball algebra \(\fancyscript{A}_u(B_X)\) of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex Banach space \(X\). These geometrical properties are: Polynomial numerical index, Polynomial Daugavet property and Bishop-Phelp-Bollobas property for multilinear mappings.

Many of the fundamental research problems in the geometry of normed linear spaces can be loosely phrased as: Given a Banach space X and a class of Banach spaces Y does X contain a subspace Y ∈ Y? As a Banach space X is determined by its unit ball B x ≡ { x ∈ X :‖ x ‖ ≤ 1 } the problem can be rephrased in terms of the geometry of convex sets: Can a given unit ball B x be sliced with a subspace to obtain a set in some given class of unit balls? A result of this type is the famous theorem of Dvoretzky (see also [L], [M6], [M4], [MS], [FLM]).

A projection P0 on a complex Banach space is generalized 3- circular if its linear combination with two projections P1 and P2 having coefficients λ1 and λ2, respectively, is a surjective isometry, where λ1 and λ2 are distinct unit modulus complex numbers different from 1 and P0⊕P1⊕P2=I. Such projections are always contractive. In this paper, we prove structure theorems for generalized 3-circular projections acting on the spaces of all n×n symmetric and skew-symmetric matrices over C when these spaces are equipped with unitary congruence invariant norms.

Isometric shifts on [formula omitted

It has previously been shown that separable Banach spaces $V$ with trivial $L$-structure have the ${L^1}$-Banach-Stone property, i.e. every surjective isometry between two Bochner spaces ${L^1}({\mu _i},V)$ induces an isomorphism of the two measure algebras. We remove the separability restriction, employing the topology of the measure algebra’s Stonean space. The result is achieved via a complete description of the $L$-structure of ${L^1}(\mu ,V)$.

Reflexivity of the Automorphism and Isometry Groups of the Suspension of [formula omitted]([formula omitted

A universal theorem for stability of ε-isometries of Banach spaces

In this paper, we study the extension of isometries between the unit spheres of normed space E and l p (p > 1). We arrive at a conclusion that any surjective isometry between the unit sphere of normed space l p (p > 1) and E can be extended to be a linear isometry on the whole space l p (p > 1) under some condition.