2014
DOI: 10.1016/j.jfa.2013.09.014
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Anlp-version of von Neumann dimension for Banach space representations of sofic groups

Abstract: Following the methods of [13], we introduce an extended version of von Neumann dimension for representations of a discrete, measure-preserving, sofic equivalence relation. Similar to [13], this dimension is decreasing under equivariant maps with dense image, and in particular is an isomorphism invariant. We compute dimensions of L p (R, µ) ⊕n for 1 ≤ p ≤ 2. We also define an analogue of the first l 2 -Betti number for l p -cohomology of equivalence relations, provided the equivalence relations satisfies a cert… Show more

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Cited by 9 publications
(26 citation statements)
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“…Consequences of the control on vol(Ball(l p (n, µ n )) 1/n are already present in the proof of Theorem 4.7 in [14], as well as Theorem 6.2 in [16]. Lastly the notion of "microstates rank" is an alternate version of the entropic formula for von Neumann dimension developed in Theorem 6.6 in [14], and Theorem 6.4 in [16]. Additionally, certain portions of the paper are adapting the techniques in [20] (some of which can be traced back to [25]) particularly in subsection 2.2.…”
mentioning
confidence: 68%
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“…Consequences of the control on vol(Ball(l p (n, µ n )) 1/n are already present in the proof of Theorem 4.7 in [14], as well as Theorem 6.2 in [16]. Lastly the notion of "microstates rank" is an alternate version of the entropic formula for von Neumann dimension developed in Theorem 6.6 in [14], and Theorem 6.4 in [16]. Additionally, certain portions of the paper are adapting the techniques in [20] (some of which can be traced back to [25]) particularly in subsection 2.2.…”
mentioning
confidence: 68%
“…Similarly, we will use that vol(Ball(l p (n, µ n ))) 1/n (here µ n is the uniform probability measure) is bounded to prove that p-metric mean dimension is bounded below by mean dimension. Consequences of the control on vol(Ball(l p (n, µ n )) 1/n are already present in the proof of Theorem 4.7 in [14], as well as Theorem 6.2 in [16]. Lastly the notion of "microstates rank" is an alternate version of the entropic formula for von Neumann dimension developed in Theorem 6.6 in [14], and Theorem 6.4 in [16].…”
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confidence: 81%
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“…We will need to extend a sofic approximation to an embedding sequence of L(Γ) as in Lemma 5.5 in [17], however we will also want σ i (L R (Γ)) ⊆ M di (R).…”
Section: 3mentioning
confidence: 99%