The present paper deals with a question of M. Mbekhta concerning partial isometries on Banach spaces.

This survey features some recent developments concerning the bounded approximation property in Banach spaces. As a central theme, we discuss the weak bounded approximation property and the approximation property which is bounded for a Banach operator ideal. We also include an overview around the related long-standing open problem: Is the approximation property of a dual Banach space always metric?

We prove a conjecture of Wojtaszczyk that for 1 ≤ p < ∞, p ≠ 2, $H_p\left(\right)$ does not admit any norm one projections with dimension of the range finite and greater than 1. This implies in particular that for 1 ≤ p < ∞, p ≠ 2, $H_p$ does not admit a Schauder basis with constant one.

In this note, we prove that any “bounded” isometries of separable metric spaces can be represented as restrictions of linear isometries of function spaces $C\left(Q\right)$ and $C\left(\Delta \right)$, where $Q$ and $\Delta$ denote the Hilbert cube ${[0,1]}^{\infty }$ and a Cantor set, respectively.

This paper deals with a few, not widely known, aspects of Kottman's constant of a Banach space and its symmetric and finite variations. We will consider their behaviour under ultrapowers, relations with other parameters such as Whitley's or James' constant, and connection with the extension of c₀-valued Lipschitz maps.

We show that a Banach space X has an infinite dimensional reflexive subspace (quotient) if and only if there exist a Banach space Z and a non-isomorphic one-to-one (dense range) Tauberian (co-Tauberian) operator form X to Z (Z to X). We also give necessary and sufficient condition for the existence of a Tauberian operator from a separable Banach space to c0 which in turn generalizes a result of Johnson and Rosenthal. Another application of our result shows that if X** is separable, then there exists...

Let X and Y be locally compact Hausdorff spaces, let E and F be Banach spaces, and let T be a linear isometry from C₀(X,E) into C₀(Y,F). We provide three new answers to the Banach-Stone problem: (1) T can always be written as a generalized weighted composition operator if and only if F is strictly convex; (2) if T is onto then T can be written as a weighted composition operator in a weak sense; and (3) if T is onto and F does not contain a copy of $\ell {₂}^{\infty }$ then T can be written as a weighted composition...

We study the Bishop-Phelps-Bollobás property for numerical radius (for short, BPBp-nu) of operators on ℓ₁-sums and ${\ell }_{\infty }$-sums of Banach spaces. More precisely, we introduce a property of Banach spaces, which we call strongly lush. We find that if X is strongly lush and X ⊕₁ Y has the weak BPBp-nu, then (X,Y) has the Bishop-Phelps-Bollobás property (BPBp). On the other hand, if Y is strongly lush and $X{\oplus }_{\infty }Y$ has the weak BPBp-nu, then (X,Y) has the BPBp. Examples of strongly lush spaces are C(K) spaces, L₁(μ)...

Let $X$ denote a specific space of the class of $X_{\alpha ,p}$ Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily ${\ell }_p$ Banach spaces. We show that for $p>1$ the Banach space $X$ contains asymptotically isometric copies of ${\ell }_p$. It is known that any member of the class is a dual space. We show that the predual of $X$ contains isometric copies of ${\ell }_q$ where $\frac{1}{p}+\frac{1}{q}=1$. For $p=1$ it is known that the predual of the Banach space $X$ contains asymptotically isometric copies of $c_0$. Here we...

Let X and Y be Banach spaces and let 𝓐(X,Y) be a closed subspace of 𝓛(X,Y), the Banach space of bounded linear operators from X to Y, containing the subspace 𝒦(X,Y) of compact operators. We prove that if Y has the metric compact approximation property and a certain geometric property M*(a,B,c), where a,c ≥ 0 and B is a compact set of scalars (Kalton's property (M*) = M*(1, {-1}, 1)), and if 𝓐(X,Y) ≠ 𝒦(X,Y), then there is no projection from 𝓐(X,Y) onto 𝒦(X,Y) with norm less than max|B| + c....

We generalize an important class of Banach spaces, the M-embedded Banach spaces, to the non-commutative setting of operator spaces. The one-sided M-embedded operator spaces are the operator spaces which are one-sided M-ideals in their second dual. We show that several properties from the classical setting, like the stability under taking subspaces and quotients, unique extension property, Radon-Nikodým property and many more, are retained in the non-commutative setting. We also discuss the dual...

We prove a very general theorem concerning the estimation of the expression ||T((a+b)/2) - (Ta+Tb)/2|| for different kinds of maps T satisfying some general perturbed isometry condition. It can be seen as a quantitative generalization of the classical Mazur-Ulam theorem. The estimates improve the existing ones for bi-Lipschitz maps. As a consequence we also obtain a very simple proof of the result of Gevirtz which answers the Hyers-Ulam problem and we prove a non-linear generalization of the Banach-Stone...

We study the Gromov-Hausdorff and Kadets distances between C(K)-spaces and their quotients. We prove that if the Gromov-Hausdorff distance between C(K) and C(L) is less than 1/16 then K and L are homeomorphic. If the Kadets distance is less than one, and K and L are metrizable, then C(K) and C(L) are linearly isomorphic. For K and L countable, if C(L) has a subquotient which is close enough to C(K) in the Gromov-Hausdorff sense then K is homeomorphic to a clopen subset of L.

2000 Mathematics Subject Classification: Primary 46B20. Secondary 47A99, 46B42.It was shown in [2] that the most natural equalities valid for every rank-one operator T in real Banach spaces lead either to the Daugavet equation ||I+T|| = 1 + ||T|| or to the equation ||I − T|| = ||I+T||. We study if the spaces where the latter condition is satisfied for every finite-rank operator inherit the properties of Daugavet spaces.

The polynomial functions on a projective space over a field = ℝ, ℂ or ℍ come from the corresponding sphere via the Hopf fibration. The main theorem states that every polynomial function ϕ(x) of degree d is a linear combination of “elementary” functions ${|⟨x,·⟩|}^d$.