Let X be a normed space and the group of all linear surjective isometries of X that are finite-dimensional perturbations of the identity. We prove that if acts transitively on the unit sphere then X must be an inner product space.
We study the reflexivity of the automorphism (and the isometry) group of the Banach algebras for various measures μ. We prove that if μ is a non-atomic σ-finite measure, then the automorphism group (or the isometry group) of is [algebraically] reflexive if and only if is *-isomorphic to . For purely atomic measures, we show that the group of automorphisms (or isometries) of is reflexive if and only if Γ has non-measurable cardinal. So, for most “practical” purposes, the automorphism group...
It is a famous result of Alexandroff and Urysohn that every compact metric space is a continuous image of a Cantor set ∆. In this short note we complement this result by showing that a certain uniqueness property holds. Namely, if (K,d) is a compact metric space and f and g are two continuous mappings from ∆ onto K, the, for every e > 0 there exists a homeomorphism phi of ∆ such that d(g(x), f(phi(x))) < e for all x∆.
We study the stability of homomorphisms between topological (abelian) groups. Inspired by the "singular" case in the stability of Cauchy's equation and the technique of quasi-linear maps we introduce quasi-homomorphisms between topological groups, that is, maps ω: 𝒢 → ℋ such that ω(0) = 0 and ω(x+y) - ω(x) - ω(y) → 0 (in ℋ) as x,y → 0 in 𝒢. The basic question here is whether ω is approximable by a true homomorphism a in the sense that ω(x)-a(x) → 0 in ℋ as x →...
It is shown that if F is a topological vector space containing a complete, locally pseudo-convex subspace E such that F/E = L₀ then E is complemented in F and so F = E⊕ L₀. This generalizes results by Kalton and Peck and Faber.
This note is to report some of the advances obtained as a follow-up of the book [2] on the topic of twisted sums of Banach spaces. Since this announcement is no longer enough to contain the theory being developed, we submit the interested reader to [2] and to [1], where full details and proofs shall appear.
We investigate the reflexivity of the isometry group and the automorphism group of some important metric linear spaces and a1gebras. The paper consists of the following sections: 1. Preliminaries. 2. Sequence spaces. 3. Spaces of measurable functions. Hardy spaces. 5. Banach algebras of holomorphic functions. 6. Fréchet algebras of holomorphic functions. 7. Spaces of continuous functions.
This paper studies the relationship between the bidual of the (projective) tensor product of Banach spaces and the tensor product of their biduals.
Sobczyk's theorem is usually stated as: . Nevertheless, our understanding is not complete until we also recall: . Now the limits of the phenomenon are set: although c is complemented in separable superspaces, it is not necessarily complemented in a non-separable superspace, such as l.
These notes deal with the extension of multilinear operators on Banach spaces. The organization of the paper is as follows. In the first section we study the extension of the product on a Banach algebra to the bidual and some related structures including modules and derivations. Tha approach is elementary and uses the classical Arens' technique. Actually most of the results of section 1 can be easily derived from section 2. In section 2 we consider the problem of extending multilinear forms on a...
We prove that no ultraproduct of Banach spaces via a countably incomplete ultrafilter can contain c₀ complemented. This shows that a "result" widely used in the theory of ultraproducts is wrong. We then amend a number of results whose proofs have been infected by that statement. In particular we provide proofs for the following statements: (i) All M-spaces, in particular all C(K)-spaces, have ultrapowers isomorphic to ultrapowers of c₀, as also do all their complemented subspaces isomorphic to their...
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