A New Triviality Theorem for Group Pseudorepresentations
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c Pleiades Publishing, Ltd., 2020.
A New Triviality Theorem for Group Pseudorepresentations A. I. Shtern∗,∗∗,∗∗∗1 ∗
Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991 Russia, ∗∗ Department of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia, ∗∗∗ Scientific Research Institute for System Analysis of the Russian Academy of Sciences (FGU FNTs NIISI RAN), Moscow, 117312 Russia E-mail: 1 [email protected] Received May 12, 2020; Revised June 10, 2020; Accepted September 11, 2020
Abstract. It is proved that if G is a group and π is a pseudorepresentation of G in a Banach space E with a sufficiently small defect and if π is a sufficiently small perturbation of the identity representation of G in E, then π(g) = 1E for all g ∈ G. DOI 10.1134/S1061920820040123
1. INTRODUCTION For the definitions, notation, and generalities concerning pseudorepresentations, see [1–4]. Recall that a mapping π of a given group G into the family of invertible operators in the algebra L(E) of bounded linear operators on a Banach space E is said to be a quasirepresentation of G on E if π(eG ) = 1E , where eG stands for the identity element of G and 1E for the identity operator on E, and if π(g1 g2 ) − π(g1 )π(g2 )L(E) ε,
g1 , g2 ∈ G,
for some ε, which is usually assumed to be sufficiently small and its greatest lower bound for π is referred to as the defect of π; a quasirepresentation π of G is said to be a pseudorepresentation of G if π(g n ) is conjugate to π(g)n , n ∈ Z, with the help of an operator sufficiently close to the identity operator. Below we omit the subscript L(E) for the operator norms. 2. PRELIMINARIES A version of the desired triviality theorem was given in [5] for the case in which E is a Hilbert space. In this note, we present a triviality theorem for a general Banach space E. Recall that, by Theorem 5.3 of [1], if π(g)L(E) C and π(g)−1 L(E) C for all g ∈ G, then an operator implementing the conjugacy of π(g n ) and π(g)n can be chosen in such a way that def
Q − 1E L(E) F (C, ε) = 2ε(C + ε)2 ,
(1)
where ε stands for the defect of π. Note that F is positive and monotone increasing with respect to ε. 3. MAIN RESULT Theorem 1. Let G be a group, let π be a pseudorepresentation of G in a Banach space E with defect ε, let · be the operator norm in the space L(E) of the bounded linear operators on E and let π(g) − 1E δ, g ∈ G, 0 δ < 1. There is an ε0 > 0 such that π(g) = 1E for all g ∈ G if 0 ε < ε0 . In particular, if ε0 is the smallest positive root of the following equation with respect to ε (for the given δ) 2
1 − F (C, ε) 1 + F (C, ε) −δ = 1, 1 + F (C, ε) 1 − F (C, ε)
then π(g) = 1E for all g ∈ G and for ε ∈ [0, ε0 ). 535
536
SHTERN
Proof. Let us replace δ by inf{δ} so that π(g) − 1E δ, g ∈ G. If δ = 0, then there is nothing to prove. Let 0 < r < δ. Let π(g) = 1 + t(g), g ∈ G; then t(g) δ, and there is a gr ∈ G such that t(gr ) > r. In this case, −1 2 −1 π(gr2 ) = Q2 (1E + 2t(gr ) + t(gr )2 )Q−1 2 = 1E + Q2 t(gr )Q2 + Q2 t(gr ) Q2 , and it fo
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