On Mazur rotations problem and its multidimensional versions

Sánchez, Félix Cabello; Ferenczi, Valentin; Randrianantoanina, Beata

doi:10.1007/s40863-021-00234-3

Cited by 5 publications

(2 citation statements)

References 107 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the literature, especially from the previous century, there are only vague indications as to why this is so. Since a recent article [6] admits that this proposition is "not completely evident from the definition" and since it is instrumental for Lusky's proof, I'll sketch a proof. To begin with, we have to recall a characterisation of L 1 -preduals from Lindenstrauss's memoir; see [20,Th.…”

Section: The Lazar-lindenstrauss Approachmentioning

confidence: 99%

“…In an addendum to [L], dated January 10, 1976, Lusky applies his methods to Mazur's rotation problem that asks whether a separable transitive space is isometric to a Hilbert space; a Banach space X is called transitive if whenever ∥x∥ = ∥y∥ = 1, there is an isometric automorphism T : X → X mapping x to y, i.e., T x = y. This problem is open to this day, and recent papers on the subject include [4] and [6].…”

Section: Lusky's Uniqueness Proofmentioning

confidence: 99%

See 1 more Smart Citation

On Wolfgang Lusky’s paper “The Gurarij spaces are unique”

Werner

2023

Arch. Math.

View full text Add to dashboard Cite

show abstract

Section: The Lazar-lindenstrauss Approachmentioning

confidence: 99%

Section: Lusky's Uniqueness Proofmentioning

confidence: 99%

On Wolfgang Lusky’s paper “The Gurarij spaces are unique”

Werner

2023

Arch. Math.

View full text Add to dashboard Cite

show abstract

Group Actions on Twisted Sums of Banach Spaces

Castillo

Ferenczi

2023

Bull. Malays. Math. Sci. Soc.

View full text Add to dashboard Cite

We study bounded actions of groups and semigroups G on exact sequences of Banach spaces from the point of view of (generalized) quasilinear maps, characterize the actions on the twisted sum space by commutator estimates and introduce the associated notions of G-centralizer and G-equivariant map. We will show that when (A) G is an amenable group and (U) the target space is complemented in its bidual by a G-equivariant projection, then uniformly bounded compatible families of operators generate bounded actions on the twisted sum space; that compatible quasilinear maps are linear perturbations of G-centralizers; and that, under (A) and (U), G-centralizers are bounded perturbations of G-equivariant maps. The previous results are optimal. Several examples and counterexamples are presented involving the action of the isometry group of $$L_p(0,1), p\ne 2$$ L p ( 0 , 1 ) , p ≠ 2 on the Kalton–Peck space $$Z_p$$ Z p , certain non-unitarizable triangular representations of the free group $${\mathbb {F}}_\infty $$ F ∞ on the Hilbert space, the compatibility of complex structures on twisted sums, or bounded actions on the interpolation scale of $$L_p$$ L p -spaces. In the penultimate section we consider the category of G-Banach spaces and study its exact sequences, showing that, under (A) and (U), G-splitting and usual splitting coincide. The purpose of the final section is to present some applications, showing that several previous result are optimal and to suggest further open lines of research.

show abstract

Envelopes in Banach spaces

Ferenczi,

Lopez-Abad

2024

Banach J. Math. Anal.

View full text Add to dashboard Cite

We introduce the notion of isometric envelope of a subspace in a Banach space, establishing its connections with several key elements: (a) we explore its relation to the mean ergodic projection on fixed points within a semigroup of contractions, (b) we draw parallels with Korovkin sets from the 1970s, (c) we investigate its impact on the extension properties of linear isometric embeddings. We use this concept to address the recent conjecture that the Gurarij space and the spaces $$L_p$$ L p , $$p \notin 2{\mathbb {N}}+4$$ p ∉ 2 N + 4 are the only separable approximately ultrahomogeneous Banach spaces (a certain multidimensional transitivity of the action of the linear isometry group). The similar conjecture for Fraïssé Banach spaces (a strengthening of the approximately homogeneous property) is also considered. We characterize the Hilbert space as the only separable reflexive space in which any closed subspace coincides with its envelope; and we show that the Gurarij space satisfies the same property. We compute some envelopes in the case of Lebesgue spaces, showing that the reflexive $$L_p$$ L p -spaces are the only reflexive rearrangement invariant spaces on [0, 1] for which all 1-complemented subspaces are envelopes. We also identify the isometrically unique “full” quotient space of $$L_p$$ L p by a Hilbertian subspace, for appropriate values of p, as well as the associated topological group embedding of the unitary group into the isometry group of $$L_p$$ L p .

show abstract

On Mazur rotations problem and its multidimensional versions

Cited by 5 publications

References 107 publications

On Wolfgang Lusky’s paper “The Gurarij spaces are unique”

On Wolfgang Lusky’s paper “The Gurarij spaces are unique”

Group Actions on Twisted Sums of Banach Spaces

Envelopes in Banach spaces

Contact Info

Product

Resources

About