Unitarizability implies amenability for discrete groups

1 Introduction

A representation \pi:G\to\mathrm{GL}(H) of a discrete group on a complex Hilbert space is uniformly bounded if

Here \mathrm{GL}(H) consists of the bounded invertible operators. The representation is unitarizable if some bounded invertible S makes every S\pi(g)S^{-1} unitary. Equivalently, the Hilbert space admits an equivalent inner product invariant under the representation. A group is unitarizable when every uniformly bounded representation on every complex Hilbert space is unitarizable. The similarity may depend on the representation; its condition number \norm{S}\norm{S^{-1}} is not assumed to have a common bound.

A discrete group G is amenable if there is a positive linear functional m:\ell^\infty(G)\to\C with m(1)=1 and m(L_gf)=m(f), where (L_gf)(x)=f(g^{-1}x). The one-operator case goes back to Sz.-Nagy: an invertible operator whose integer powers are uniformly bounded is similar to a unitary operator [19, Theorem I]. His generalized-limit construction of an invariant inner product preceded the invariant-mean argument of Day and Dixmier [3, 4]. Thus amenability implies unitarizability. Dixmier explicitly asked whether this property characterizes groups admitting an invariant mean [4, §5, p. 221]. The question asks whether changing the Hilbert norm separately for every uniformly bounded representation already forces the existence of such a mean.

We work in ZFC. All Hilbert spaces below are complex, and \N=\{1,2,\ldots\}. The main point is the absence of a similarity for one representation, without any a priori control on possible similarities. For an uncountable group the induced witness may be nonseparable, as is allowed by the definition of unitarizability used here.

Prior work and the remaining obstruction

Early counterexamples showed that uniform boundedness alone does not force unitarizability. Ehrenpreis and Mautner constructed examples for the locally compact group \mathrm{SL}_2(\mathbb R) [5]; explicit families for free groups include those of Pytlik and Szwarc [18]. Nonunitarizability passes from a subgroup to the ambient group by induction. The free-group examples therefore cover groups containing nonabelian free subgroups, but leave nonamenable groups without such subgroups to be understood.

Quantitative unitarizability gives one route to amenability. Pisier proved that a discrete group G is amenable exactly when there are constants C and \alpha<3, depending only on G, such that every uniformly bounded representation admits a unitarizer with condition number at most C|\pi|^\alpha [15, Theorem 0.1]. The estimate is uniform over all Hilbert spaces and representations of the fixed group. Unitarizability itself implies a polynomial bound of this form for some finite exponent [16, Theorem 4.1]; the additional content of the amenability criterion is the threshold \alpha<3.

Random forests gave a route beyond free subgroups. Epstein and Monod used invariant random forests to relate unitarizability to \ell^2-Betti numbers and cost. Their estimates imply nonunitarizability for residually finite groups with positive first \ell^2-Betti number, and for finitely generated residually finite groups of cost greater than one [6, Theorems 1.1–1.3]. Combining their criterion with group constructions, Osin obtained finitely generated torsion and torsion-free nonunitarizable groups containing no nonabelian free subgroup [14, Theorems 1.2–1.3].

A different approach uses measurable actions. Gaboriau and Lyons showed that a Bernoulli action of a countable nonamenable group contains, within its orbits, those of an essentially free probability-preserving action of the free group F_2 [8, Theorem 1]. Monod and Ozawa used this to show that, for an infinite abelian group A, the restricted wreath product A\wr G=(\bigoplus_G A)\rtimes G is unitarizable exactly when G is amenable [12, Theorem 1]. This places the obstruction on an extension of G. Alpeev subsequently allowed every nontrivial countable lamp group A when G is countable and nonamenable, so the conclusion also holds for finite lamps [1, Theorem A]. The distinction remains between a witness for an extension and one for G itself.

Approximation properties give a complementary route. Vergara proved that a discrete unitarizable group satisfying the M_d-approximation property for every d\ge2 is amenable. These conditions hold for countable groups acting properly on finite-dimensional CAT(0) cube complexes [20, Theorems 1.2–1.3]. His later extension theorem preserves M_d-AP when both a normal subgroup and the quotient possess it [21, Theorem 1.3].

Littlewood functions give the analytic formulation closest to the present proof. Their scalar convolution kernels split into a part with uniformly absolutely summable rows and a part with uniformly absolutely summable columns. Epstein and Monod, following Bożejko and Fendler, use this splitting to obtain bounded commutators and triangular representations [6, Propositions 2.3–2.4]. Gerasimova, Gruber, Monod and Thom relate the corresponding norms to Cheeger constants and study the resulting restrictions on unitarizable groups [9, §2.B]. The passage from bounded non-inner operator cocycles to nonunitarizable representations is classical [12, §2], [17, Remark 4 and Proposition 5(ii)]. The present construction retains that mechanism while allowing finite-dimensional matrix coefficients. Positive rank-one weights preserve the row and column bounds while increasing the kernel’s Hilbert–Schmidt energy relative to the fibre dimension. This gives a direct obstruction with no prescribed bound on possible unitarizers.

The passage from countable groups to arbitrary discrete groups uses standard facts. Følner’s finite-set criterion implies that every nonamenable discrete group has a finitely generated nonamenable subgroup [7, Main Theorem, p. 245]. Induction carries a nonunitarizable representation of that subgroup to the full group with the same uniform bound [16, Proposition 0.5]; Pisier’s Corollaries 0.10–0.11 also show that unitarizability of discrete groups is countably determined.

The operator obstruction and the proof

Let U be a unitary representation of G on a Hilbert space \mathcal H, and write \mathcal B(\mathcal H) for its bounded operators. A bounded operator cocycle is a map D:G\to\mathcal B(\mathcal H) with \sup_g\norm{D(g)}<\infty and

It is inner if one bounded operator B satisfies D(g)=B-U(g)BU(g)^{-1} for all g. The cocycle gives the uniformly bounded triangular representation

A similarity making this representation unitary would produce an invariant complement to \mathcal H\oplus0. Writing that complement as the graph of a bounded operator makes D inner. Thus it suffices to construct a bounded cocycle with no bounded implementer.

The criterion in Section 2 reduces this task to operators T_j on \ell^2(G;\C^{k_j}). Their conjugation differences under left translation must be bounded uniformly in both j and g, while their distance from the operators commuting with left translation must tend to infinity. A translation-invariant comparison operator A_j, with blocks A_j[x,y]=W_j(x^{-1}y), detects that distance: if the block row at e of T_j and the block column at e of A_j-T_j have norm at most K, then

Here U_j is left translation, U_j(G)' is its commutant, and \norm{W}_{\HS}^2=\operatorname{Tr}(W^*W) uses the unnormalized trace. This estimate compares one finite-dimensional row with one finite-dimensional column; it imposes no bound on \norm{T_j}.

Nonamenability supplies the operators in two steps. In Section 3, a contracting finite average gives lists s_1,\ldots,s_n of group elements, with multiplicity. For x\in G and 1\le i\le n, the labelled edge (x,i) joins row x to column xs_i. These edges can be assigned to their endpoints so that each endpoint receives at most r edges, where (r/n)\log n\to0. Compare an assignment with any left translate. At a row, the edges whose assignment changes lie in the two sets assigned to that row; at a column, they lie in the two sets assigned to that column. Thus at most 2r labels change at either endpoint. In Section 4, we choose n unit vectors in \C^k, with k of order r\log n, whose synthesis operators have bounded norm on every subfamily of at most 2r vectors. This is the concentration, finite-net and union-bound method familiar from random matrix estimates; compare [2, Section 5]. Only its upper norm bound is needed here.

The rank-one projections onto these vectors weight the convolution edges, and T_j retains the row-assigned edges. Factoring through the scalar Hilbert space indexed by the edges bounds the relevant rows, columns and all conjugation differences. Keeping the edge labels is essential: distinct products may coincide in G, yet correspond to distinct coordinates in the factorization. Positivity ensures that such coincidences do not reduce the kernel energy, which is at least n. Since k/n\to0, the distance estimate diverges. Finally, Section 5 rescales these operators and applies the obstruction to one direct sum of their cocycles, giving any prescribed uniform bound 1+\varepsilon. Induction passes the witness from a finitely generated nonamenable subgroup to an arbitrary discrete group. The Day–Dixmier averaging argument completes the equivalence. Section 6 then uses the separate companion similarity theorem to show that the resulting \ell^1(G)-representations have no bounded algebra-homomorphism extension to either group C^*-algebra.

2 An invariant-kernel obstruction

We first prove the criterion announced in the introduction. A large Hilbert–Schmidt energy will force an operator far from the translation commutant; a sequence of such operators with uniformly bounded conjugation differences will then give one bounded non-inner cocycle. Write \mathcal B(H) for the bounded operators on H and \norm{X}_{\HS}^2=\Tr(X^*X) for the Hilbert–Schmidt norm of a finite matrix, with the unnormalized trace \Tr. All operator norms below are ordinary Hilbert-space operator norms. The finite-dimensional fibres will enter through Hilbert–Schmidt estimates for one block row and one block column.

For a countable group G and a positive integer k, put

For x\in G, let \iota_x:\C^k\to\mathcal H_k be the isometric inclusion into the coordinate indexed by x. For F\in\mathcal B(\mathcal H_k), write

Thus a block row has codomain \C^k and a block column has domain \C^k. Write U_k(G)' for the bounded operators commuting with every U_k(g). If F\in U_k(G)', then F[x,y]=F[g^{-1}x,g^{-1}y], because U_k(g)\iota_y=\iota_{gy}. Taking g=x gives

This statement concerns the blocks of a bounded operator and requires no summability assumption on a convolution formula.

The next proposition turns an unbounded sequence of these distances into one uniformly bounded representation that cannot be unitarized. The relation between bounded operator cocycles and triangular representations is standard; see [6, Proposition 2.4], [12, Section 2], and [17, Remark 4 and Proposition 5(ii)]. We include the argument, in particular the bounded-complement step and the direct-sum quantifiers.

For clarity, each individual D_j in this construction is already implemented by the bounded operator T_j. The obstruction is to a bounded implementer on the whole direct sum. Indeed, Remark 2.1 shows that any operator B_j satisfying D_j(g)=B_j-U_j(g)B_jU_j(g)^{-1} for all g must obey

Thus the proof requires only the existence of a similarity for one representation in order to obtain a contradiction; it assumes no uniform bound on similarity operators across different representations.

3 Sparse assignments from nonamenability

The obstruction requires uniform operator bounds together with increasing energy per fibre dimension. This section supplies its combinatorial input: a list of convolution edges that can be assigned to their endpoints with small loads. The matrix weights in Section 4 will convert those loads into operator bounds.

Write (R_tf)(x)=f(xt) for right translation on \ell^2(G). Thus R_sR_t=R_{st}. We first obtain the finite averaging operator needed for the construction. This is the elementary averaging-norm form of the random-walk criterion for nonamenability; compare [11]. We give the implication at the precise strength used here.

The next argument converts an operator-norm estimate into an assignment of edges with bounded loads at their endpoints. It is related to the rectangular kernel estimates for Littlewood functions [9, Proposition 2.6]; we include the complete assignment argument and retain all multiplicities. The conversion from endpoint assignments to row–column bounds also parallels the forest orientation in [6, Propositions 2.3–2.4].

4 Finite-dimensional weights and sparse block operators

The sparse assignment from Section 3 will now produce the three bounds required by Remark 2.2: a bound on each block row of T, each block column of A-T, and every conjugation difference T-U(g)TU(g)^{-1}. All three bounds will be independent of the number of labels. To make the energy grow faster than the fibre dimension, we use rank-one projections onto many unit vectors in a smaller space.

We first construct vectors whose synthesis operators are bounded on every small subfamily. The proof uses the standard concentration, finite-net and union-bound method for random matrices; compare [2, Section 5]. Here only an upper bound is needed, and the argument below supplies its constants directly.

The vectors just constructed control each small set of incident labels. We next transfer that finite-dimensional estimate to an operator on the group. The intermediate space keeps one scalar coordinate for each labelled edge, even when several labels describe the same group element.

4.1 A factorization through the indexed edges

Let G be a countable group, let s_1,\ldots,s_n\in G be a finite list, and allow the same group element to occur at several distinct indices. Let v_1,\ldots,v_n\in\C^k be unit vectors and put

Thus P_i is a rank-one orthogonal projection and \norm{P_i}=1. For a function c\colon G\times\{1,\ldots,n\}\to\C with |c(x,i)|\le1, define

For each fixed i, the corresponding summand is the composition of the unitary right shift \xi(x)\mapsto\xi(xs_i), the pointwise application of P_i, and multiplication by the scalar function c(\cdot,i). Its norm is at most one. In particular, Equation (4.4) defines a bounded operator and \norm{T_c}\le n, without any sparsity assumption. The next lemma gives a bound independent of n under two sparsity assumptions.

4.2 Consequences of an assignment to endpoints

We now apply the factorization three times to obtain the hypotheses of Remark 2.2. An endpoint assignment is sparse only for the ones at rows and the zeros at columns. Restricting to a single row or column, and then subtracting a translated assignment, will give masks that are sparse on both sides.

Fix integers n\ge2 and r\ge1, and choose k,p,b_0 and v_1,\ldots,v_n as in Remark 4.1. Keep the preceding notation G,s_i,P_i,\mathcal H,T_c. Suppose a function a\colon G\times\{1,\ldots,n\}\to\{0,1\} satisfies

Thus the ones are sparse at rows and the zeros are sparse at columns, with both counts taken over indexed edges. Set

where \one(x,i)=1. Both A and T are bounded operators, each with norm at most n. For y\in G, let \iota_y:\C^k\to\mathcal H be the coordinate inclusion, and write q_y=\iota_y^* for coordinate evaluation.

5 Proof of the main theorem

We first combine the three uniform bounds with the kernel energy estimate for countable groups. We then pass through a finitely generated subgroup to arbitrary discrete groups and include the classical averaging argument for the reverse direction.

We next recall the induction argument for subgroup permanence of unitarizability [16, Proposition 0.5], keeping track of its norm and of the invariant inner product on the subgroup representation.

For uncountable nonamenable G, the finitely generated subgroup H above has uncountable index, and the induced witness is nonseparable. The conclusion therefore uses the usual definition quantifying over all Hilbert spaces. The argument concerns discrete groups and does not extend the theorem to nondiscrete locally compact groups, where representations are normally required to be strongly continuous.

6 A consequence for group C^*-algebras

The following consequence uses the companion similarity theorem in addition to Remark 1.1.

Let G be any discrete group and let \pi:G\to\mathrm{GL}(H) be a uniformly bounded nonunitarizable representation on a complex Hilbert space. The operator-norm convergent sum

defines a bounded complex-linear unital homomorphism for convolution, with \norm{T_\pi(f)}\le|\pi|\norm{f}_1; absolute summability justifies multiplication of the sums. Nevertheless, its restriction to \C[G] has no bounded complex-linear unital algebra-homomorphism extension to either the full group algebra C^*(G) or the reduced group algebra C_r^*(G). Indeed, a full extension \Phi would be similar to a *-homomorphism by the companion similarity theorem [13, Theorem 1.1]: some bounded invertible S would make S\Phi(\cdot)S^{-1} a unital *-homomorphism. For each canonical group unitary u_g, one has \Phi(u_g)=\pi(g), so every S\pi(g)S^{-1} would be unitary, a contradiction. A reduced extension would compose with the canonical unital *-quotient C^*(G)\to C_r^*(G) to give a full extension. Thus the automatic \ell^1-norm bound does not imply C^*-norm continuity. Applying this observation to the witnesses in Remark 1.1 gives such a failure for every nonamenable discrete group, even with uniform representation bounds arbitrarily close to one.