Non p -norm approximated Groups
- PDF / 204,989 Bytes
- 17 Pages / 459.36 x 707.04 pts Page_size
- 25 Downloads / 205 Views
ALEXANDER LUBOTZKY AND IZHAR OPPENHEIM To Larry Zalcman with gratitude Abstract. It was shown in a previous work of the first-named author with De Chiffre, Glebsky and Thom that there exists a finitely presented group which cannot be approximated by almost-homomorphisms to the unitary groups U(n) equipped with the Frobenius norms (a.k.a. as L2 norm, or the Schatten-2-norm). In his ICM18 lecture, Andreas Thom asks if this result can be extended to general Schatten-p-norms. We show that this is indeed the case for 1 < p < ∞.
1
Introduction
Let U(n) be the group of unitary n×n matrices equipped with a bi-invariant metric dn induced by a Banach norm . on Mn (C), as dn (g, h) = g−h. Examples of special interest are: (1) The Hilbert–Schmidt norm: TH.S. = 1n tr(T ∗ T). √ 1 (2) For 1 ≤ p < ∞, the Schatten p-norm: Tp = (tr |T|p ) p , where |T| = T ∗ T. When p = 2, this is usually called the Frobenius norm: T2 = TFrob =
√
nTH.S..
(3) The operator norm, Top = max{Tv : v = 1}, also known as the Schatten ∞-norm. Whatever {dn }∞ n=1 are, define for G = (U(n), dn ) the following: Definition 1.1. A finitely presented group is called G-approximated if there exists an infinite sequence {nk }∞ k=1 of integers and (set-theoretic) maps φ = (φnk ), φnk : → U(nk ) such that: (1) ∀g, h ∈ , lim dnk (φnk (gh), φnk (g)φnk (h)) = 0. (2) ∀g ∈ , g = 1, there is ε(g) = ε > 0 such that lim sup dnk (φnk (g), idU(nk ) ) ≥ ε, where idU(nk ) is the nk × nk identity matrix. 305 ´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 141 (2020) DOI 10.1007/s11854-020-0119-2
306
A. LUBOTZKY AND I. OPPENHEIM
There are two long-standing questions regarding whether there exist groups which are not (U(n), dn)-approximated with respect to the dn ’s defined in cases (1) and (3). The question for case (1), where dn is defined by the Hilbert–Schmidt norm, is equivalent to Alain Connes’ problem whether every group is Connes-embeddable (see [7] and [19] for details), while case (3) is related to Kirchberg’s question whether any stably finite C∗ -algebra is embeddable into a norm-ultraproduct of matrix algebras (see [5] for details), which implies that any group is (U(n), dn )approximated with respect to the distance induced by the operator norm. In this paper, a group will be called p-norm approximated if it is approximated with respect to G = (U(n), .p). A recent breakthrough [8] shows that there exist groups that are not Frobenius approximated (i.e., groups that are not 2-norm approximated). Following this, Andreas Thom asks in his ICM 2018 talk [24], if that result can be extended to all Schatten p-norms. We answer this affirmatively in the case where 1 < p < ∞, and in fact we prove a somewhat stronger result: Theorem 1.2. There exists a finitely presented group which is not p-norm approximated for any 1 < p < ∞. The case of p = 1 is left open, as well as the cases of the Hilbert–Schmidt and the operator norms. The method of proof follows the one implemented in [8] for p = 2, but some further cohomology vanishing results are need
Data Loading...
