The p-approximation property for simple Lie groups with finite center

Abstract

We prove that, for any 1 < p < ∞, the groups SL(3, R) and Sp(2, R) do not have the p -approximation property of An, Lee and Ruan, which implies in particular that they are not p -weakly amenable. It follows that the same holds for any connected simple Lie group with finite center and real rank greater than 1, as well as for any lattice in it. This extends Haagerup and de Laat's result for the AP, which in this language corresponds to the case p = 2.

On convoluters on $L^p$-spaces

“…A certain p-approximation property, which is implied by the approximation property was defined by An, Lee and Ruan [1], and studied vigorously by Vergara [23]. Using this, Vergara strengthens our Theorem 1.1, but relies heavily on our methods.…”

“…Using this, An, Lee and Ruan [1] defined the p-approximation property (p-AP), which is having 1 in the weak* closure of A p (G) in M p (G). As verified by Vergara [23], p-AP is equivalent to p ′ -AP, and if 2 ≤ p ≤ q < ∞, then p-AP implies q-AP. Hence p-AP for p = 2 is ostensibly more general than approximation property (2-AP).…”

“…Hence p-AP for p = 2 is ostensibly more general than approximation property (2-AP). However, it is shown in [23] that certain canonical higher rank simple Lie groups fail p-AP for any p -the same ones known to Lafforgue and de la Salle [18] and Haagerup and de Laat [11] -giving convincing evidence that, at least for connected groups and lattices within, that p-AP is equivalent to 2-AP. No examples are known of groups admitting p-AP for some p, but not 2-AP.…”

“…Theorem 1.1 was used in recent work of Oztop and the second named author [20] to obtain, for a pair of groups G and H with approximation property, a tensor product description of A p (G × H). Given recent developments related to this work and interest in it, in particular [21,23], we have elected to update our results and submit them for publication.…”

Strong Property (T), weak amenability and $\ell^p$-cohomology in $\tilde{A}_2$-buildings

“…We cannot resist to give a more direct derivation. This also provides a simpler proof of the results of [LDlS11], in the spirit of [Ver17] (see also [dlS16a]):…”