Wyniki wyszukiwania

1

A theorem on isotropic spaces

100%

Cabello Sánchez F.

Studia Mathematica

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1999

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tom 133

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nr 3

257-260

EN

Let X be a normed space and $G_F(X)$ the group of all linear surjective isometries of X that are finite-dimensional perturbations of the identity. We prove that if $G_F(X)$ acts transitively on the unit sphere then X must be an inner product space.

2

The group of automorphisms of $L_{∞}$ is algebraically reflexive

100%

Cabello Sánchez F.

Studia Mathematica

|

2004

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tom 161

|

nr 1

19-32

EN

We study the reflexivity of the automorphism (and the isometry) group of the Banach algebras $L_{∞}(μ)$ 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 $L_{∞}(μ)$ is [algebraically] reflexive if and only if $L_{∞}(μ)$ is *-isomorphic to $L_{∞}[0,1]$. For purely atomic measures, we show that the group of automorphisms (or isometries) of $ℓ_{∞}(Γ)$ is reflexive if and only if Γ has non-measurable cardinal. So, for most "practical" purposes, the automorphism group of $ℓ_{∞}(Γ)$ is reflexive.

3

On ultrapowers of Banach spaces of type $ℒ_{∞}$

45%

Avilés A., Cabello Sánchez F., Castillo J., González M., Moreno Y.

Fundamenta Mathematicae

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2013

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tom 222

|

nr 3

195-212

EN

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 square. (ii) No ultrapower of the Gurariĭ space can be complemented in any M-space. (iii) There exist Banach spaces not complemented in any C(K)-space having ultrapowers isomorphic to a C(K)-space.

Ograniczanie wyników

2 Studia Mathematica

1 Fundamenta Mathematicae

3 Cabello Sánchez F.

1 Avilés A.

1 Castillo J.

1 González M.

1 Moreno Y.

1 2013

1 2004

1 1999

A theorem on isotropic spaces

The group of automorphisms of $L_{∞}$ is algebraically reflexive

On ultrapowers of Banach spaces of type $ℒ_{∞}$