Light groups of isomorphisms of Banach spaces and invariant LUR renormings

Antunes, Leandro; Ferenczi, Valentin; Grivaux, Sophie; Rosendal, Christian

doi:10.2140/pjm.2019.301.31

Cited by 8 publications

(4 citation statements)

References 11 publications

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“…Example 4.6. In [1] Proposition 6.1, a bounded representation of the group G = 2 <ω on c 0 is defined, with the property of being SOT-discrete but without discrete orbits. The space where it is defined is actually defined as the sum of c 0 with R 2 , but here we choose to simplify the example by using R instead of R 2 , at the cost of losing the property about discrete orbits, which is irrelevant for our present purposes.…”

Section: Compatibilitymentioning

confidence: 99%

“…we have an exact sequence of Hilbert spaces 0 → H 1 → H → H 2 → 0 which does not G-split although it obviously splits in the Banach space category. In the second case it is inspired by an example of [1] providing a certain action of G = 2 <N on c 0 which allows us to obtain that Ext G (R, c 0 ) = {0}. while Ext(R, c 0 ) = {0} in the Banach space category.…”

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confidence: 99%

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Group actions on twisted sums of Banach spaces

Castillo¹,

Ferenczi²

2020

Preprint

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We study bounded actions of groups and semigroups on exact sequences of Banach spaces, characterizing different type of actions in terms of commutator estimates satisfied by the quasi-linear map associated to the exact sequence. As a special and important case, actions on interpolation scales are related to actions on the exact sequence induced by the scale through the Rochberg-Weiss theory [30]. Consequences are presented in the cases of certain non-unitarizable triangular representations of the group F ∞ on the Hilbert space, of the compatibility of complex structures on twisted sums, as well as of bounded actions on the interpolation scale of L p -spaces. As a new fundamental example, the isometry group of L p (0, 1), p = 2 is shown to extend as an isometry group acting on the associated Kalton-Peck space Z p [23]. Finally we define the concept of G-splitting for exact sequences admitting the action of a semigroup G, and give criteria and examples to relate G-splitting and usual splitting of exact sequences: while both are equivalent for amenable groups and, for example, reflexive spaces, counterexamples are provided for the group F ∞ as well as for the space c 0 .

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Section: Compatibilitymentioning

confidence: 99%

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confidence: 99%

Group actions on twisted sums of Banach spaces

Castillo¹,

Ferenczi²

2020

Preprint

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“…we can verify that Isom( p ) is light. Proposition 1.16 was generalized by Grivaux in [AFGR17], proving that if K is an infinite compact connected space, then C(K) is light if K is homeomorphic to a finite union of segments of R, and C(K) is not light if K contains an n-dimensional region for some n 2, C(K).…”

Section: Questions and Commentsmentioning

confidence: 99%

“…The purpose of this thesis is to investigate geometric properties of Banach spaces, especially light groups of isomorphisms, convexity of norms, distinguished points for group actions and polyhedrality. Chapters 1, 3 and 4 are part of a joint work with Valentin Ferenczi, Sophie Grivaux and Christian Rosendal [AFGR17]. The results of Chapters 2 and 5 are part of a joint work with Kevin Beanland and Hùng Chu [ABC19], obtained during the author's doctoral exchange program (Programa de Doutorado Sanduíche no Exterior -PDSE-CAPES) in Washington and Lee University, under the guidance of Prof. Kevin Beanland.…”

Section: Introductionmentioning

confidence: 99%

Light groups of isometries and polyhedrality of Banach spaces

Antunes¹

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Megrelishvili defines light groups of isomorphisms of a Banach space as the groups on which the weak and strong operator topologies coincide and proves that every bounded group of isomorphisms of Banach spaces with the point of continuity property (PCP) is light. We investigate this concept for isomorphism groups G of classical Banach spaces X without the PCP, especially isometry groups, and relate it to the existence of G-invariant LUR or strictly convex renormings of X. We give an example of a Banach space X and an infinite countable group of isomorphisms G GL(X) which is SOT-discrete but such that X does not admit a distinguished point for G, providing a negative answer to a question of Ferenczi and Rosendal. We also prove that every combinatorial Banach space is (V)polyhedral. In particular, the Schreier spaces of countable order provide new solutions to a problem proposed by Lindenstrauss concerning the existence of an infinite-dimensional Banach space whose unit ball is the closed convex hull of its extreme points.

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Group Actions on Twisted Sums of Banach Spaces

Castillo

Ferenczi

2023

Bull. Malays. Math. Sci. Soc.

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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.

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Light groups of isomorphisms of Banach spaces and invariant LUR renormings

Cited by 8 publications

References 11 publications

Group actions on twisted sums of Banach spaces

Group actions on twisted sums of Banach spaces

Light groups of isometries and polyhedrality of Banach spaces

Group Actions on Twisted Sums of Banach Spaces

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