An <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math>-version of von Neumann dimension for Banach space representations of sofic groups II

Hayes, Ben

doi:10.1016/j.jfa.2015.07.013

Cited by 3 publications

(4 citation statements)

References 22 publications

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“…This has prompted recent work in pursuit of a more general p -dimension for 1 < p < ∞; see e.g. [9,8,5,6]. The purpose of this short note is to establish a fundamental obstruction to this endeavour.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

See 1 more Smart Citation

An obstruction to \ell ^{p}-dimension

Monod¹,

Petersen²

2014

Ann. inst. Fourier

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…More generally one can also consider this conjecture for sofic groups [9,8]. (iv) For G = Z n , n 2 and p > 2n n−1 the situation in Theorem 0.1 is particularly nice.…”

Section: Remarksmentioning

confidence: 99%

An obstruction to \ell ^{p}-dimension

Monod¹,

Petersen²

2014

Ann. inst. Fourier

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“…Also, the class of groups G for which the canonical trace on C * λ (G) is MF has also been considered in [24]. In particular, see Proposition 3.11 in [24] where a list of permanence properties are given for this class of groups.…”

Section: Theorem 14 Let G Be An Amenable Group and Consider An Action Ofmentioning

confidence: 99%

“…After a preliminary version of this paper was announced, we were informed by Ben Hayes that the hyperlinearity of G ⋊ F r follows from known results obtained in [10]. Also, the class of groups G for which the canonical trace on C * λ (G) is MF has also been considered in [24]. In particular, see Proposition 3.11 in [24] where a list of permanence properties are given for this class of groups.…”

Section: Introductionmentioning

confidence: 99%

Crossed products of nuclear C⁎-algebras and their traces

Rainone

Schafhauser

2019

Advances in Mathematics

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Metric mean dimension for algebraic actions of Sofic groups

Hayes

2017

Trans. Amer. Math. Soc.

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Abstract. We prove that if Γ is a sofic group, and A is a finitely generated Z(Γ)-module, then the metric mean dimension of Γ A, in the sense of Hanfeng Li is equal to the von Neumann-Lück rank of A. This partially extends the results of Hanfeng Li and Bingbing Liang in [22] from the case of amenable groups to the case of sofic groups. Additionally we show that the mean dimension of Γ A is the von Neumann-Lück rank of A, if A is finitely presented and Γ is residually finite. It turns out that our approach naturally leads to a notion of p-metric mean dimension, which is in between mean dimension and the usual metric mean dimension. This can be seen as an obstruction to the equality of mean dimension and metric mean dimension. While we cannot decide if mean dimension is the same as metric mean dimension for algebraic actions, we show that in the metric case that for all p the p-metric mean dimension coincides with the von Neumann-Lück rank of the dual module.

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An lp-version of von Neumann dimension for Banach space representations of sofic groups II

Cited by 3 publications

References 22 publications

An obstruction to \ell ^{p}-dimension

An obstruction to \ell ^{p}-dimension

Crossed products of nuclear C⁎-algebras and their traces

Metric mean dimension for algebraic actions of Sofic groups

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