Cohomological obstructions to lifting properties for full C$$^*$$-algebras of property (T) groups

Ioana, Adrian; Spaas, Pieter; Wiersma, Matthew

doi:10.1007/s00039-020-00550-4

Cited by 9 publications

(10 citation statements)

References 45 publications

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“…For instance, while the arithmetic groups SL n (Z), n ≥ 3, are not HS-stable by [BL18], it is open whether they are flexibly HS-stable. The first examples of residually finite groups which are not flexibly HS-stable were found only recently in [ISW20], where certain groups with the relative property (T), including Z 2 ⋊ SL 2 (Z), were shown to have this property.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Almost commuting matrices and stability for product groups

Ioana

2021

Preprint

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We prove that any product of two non-abelian free groups, Γ = Fm ×F k , for m, k ≥ 2, is not Hilbert-Schmidt stable. This means that there exist asymptotic representations πn : Γ → U(dn) with respect to the normalized Hilbert-Schmidt norm which are not close to actual representations. As a consequence, we prove the existence of contraction matrices A, B such that A almost commutes with B and B * , with respect to the normalized Hilbert-Schmidt norm, but A, B are not close to any matrices A ′ , B ′ such that A ′ commutes with B ′ and B ′ * . This settles in the negative a natural version of a question concerning almost commuting matrices posed by Rosenthal in 1969.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Almost commuting matrices and stability for product groups

Ioana

2021

Preprint

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“…Later on, Thom [29] gave an explicit example of G for which C * (G) fails the LLP. More recently, Ioana, Spaas and Wiersma [9] proved that SL n (Z) for n ≥ 3 and many other similar property (T) groups fail the LLP. See §9 for more on this theme.…”

Section: Lp and Ultraproductsmentioning

confidence: 99%

“…To illustrate the focus our paper gives to the property in (0.1) and its variants, we turn to Kazhdan's property (T), following [21,29,9]. The proof of Theorem 9.1 below is merely a reformulation of an argument from [9, Th.…”

Section: Illustration: Property (T) Groupsmentioning

confidence: 99%

On the Lifting Property for $C^*$-algebras

Pisier¹

2020

Preprint

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“…Switching to non-examples of HS-stable groups, we recall some of the known obstructions to HS-stability. In the non-amenable case, besides the "ad-hoc" F 2 × F 2 ( [22]), all other non-examples use property T or property τ groups [3], [21]. For amenable groups, the situation is quite different.…”

Section: Introductionmentioning

confidence: 99%

On amenable Hilbert-Schmidt stable groups

Eckhardt¹,

Shulman²

2022

Preprint

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We examine Hilbert-Schmidt stability (HS-stability) of discrete amenable groups from several angles. We give a short, elementary proof that finitely generated nilpotent groups are HS-stable. We investigate the permanence of HS-stability under central extensions by showing HS-stability is preserved by finite central quotients, but is not preserved in general. We give a characterization of HS-stability for semidirect products G ⋊ γ Z with G abelian. We use it to construct the first example of a finitely generated amenable HS-stable group which is not permutation stable. Finally, it is proved that for amenable groups flexible HS-stability is equivalent to HS-stability, and very flexible HS stability is equivalent to maximal almost periodicity. There is some overlap of our work with the very recent and very nice preprint [24] of Levit and Vigdorovich. We detail this overlap in the introduction.Where our work overlaps it appears that we take different approaches to the proofs and we feel the two works compliment each other.

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Cohomological obstructions to lifting properties for full C$$^*$$-algebras of property (T) groups

Cited by 9 publications

References 45 publications

Almost commuting matrices and stability for product groups

Almost commuting matrices and stability for product groups

On the Lifting Property for $C^*$-algebras

On amenable Hilbert-Schmidt stable groups

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