Noncommutative Lp-spaces without the completely bounded approximation property

Abstract

For any 1 ≤ p ≤ ∞ different from 2, we give examples of noncommutative L p spaces without the completely bounded approximation property. Let F be a non-archimedian local field. If p > 4 or p < 4/3 and r ≥ 3 these examples are the non-commutative L p

Approximation properties for noncommutative Lp -spaces of high rank lattices and nonembeddability of expanders

“…Note that this result extends the result of Lafforgue and the second named author mentioned above. An analogue of Theorem 1.1 in the non-Archimedean setting was already known from [20]. Theorem 1.1 was therefore expected.…”

“…The first explicit examples of such spaces were given by Lafforgue and the second named author in [20]. They proved that for n ≥ 3 and p ∈ [1, 3 ) ∪ (4, ∞], the noncommutative L p -spaces L p (L(SL(n, Z))) do not have the CBAP.…”

Towards strong Banach property (T) for SL(3,ℝ)

“…Haagerup and de Laat [9] were able to deduce from this, with a very short proof, that SL(3, R) does not have the Approximation Property. This was already known as a consequence of [16], where we also used Lafforgue's machinery to prove that the non-commutative L p spaces of the von Neumann algebras of lattices in SL(3, R) fail the completely bounded approximation property for all p ∈ (4, ∞]. If one looks at the proofs, one remarks that the 4 here is the same 4 as in the definition of the class E 4 .…”

“…As we will explain at the end of this introduction, our original contribution to Theorem 1.2 is a result on the representations of SO(3) on spaces in E 4 (Theorem 1.6) relying on a computation (Lemma 2.13) that was made in [16].…”

“…The fact that all the analysis is done on the level of compact groups is what allows to consider also representations with small growth, but it also has a remarkable consequence in harmonic analysis/operator algebra ( [16], [9], [7]). Indeed, since for a compact group K, the space of completely bounded multipliers of the Fourier algebra 5 A(K) coincides with A(K), the Hölder continuity of U -biinvariant matrix coefficients of K can be rephrased as Hölder continuity of U -biinvariant completely bounded multipliers of the Fourier algebra A(K), and the proof sketched above gives that not only the C 0 K-biinvariant matrix coefficients of unitary representations of SL(3, R), but also the C 0 K-biinvariant completely bounded multipliers of A(SL(3, R)) have an explicit decay at infinity.…”