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KAZHDAN-MILMAN PROBLEM:
SKETCH OF THE PROOF 1
A. I. Shtern
M. V. Lomonosov Moscow State University, Russia;
Research Institute for Systems Research, Russian Academy of Sciences
The following question was attributed to Milman by Kazhdan [1] in 1982: “Let p: 0(n) —> 0(N) be a map which is “almost” a representation, that is, |p(gg') — p(g)p(g')\ is small for all g,g' 6 0(n).2 Is it true that p is near to an actual representation of O(n)?” As is known for a long time, the answer is “yes” for continuous mappings p for any n and in the general case if n = 1. The main result of the present note claims that the answer is “yes” for any n > 1 for any (not necessarily continuous) mapping p of the above type. Some applications of this result are also indicated.
§ 1. Introduction
The natural mathematical interest related to combining the ideas of symmetry and nearness has a physical sense which can be sketched as follows. If the symmetry group of some physical system admits nontrivial quasi-representations, i.e., mappings into the group of invertible continuous linear operators in some topological vector space that have uniformly small difference (say, regulated by the level of accuracy of the measurements) between the image of any product and the product of the corresponding images and for which there are no “sufficiently close” ordinary representations in the same topological vector space, then the interpretation of any experiment can turn to be much more complicated than that for the case in which a close representation really exists, which can force a careful distinction between true symmetries (related to “laws of nature”) and “quasi-symmetries.” From this point of view, one of the main results of the present note claims that finite-dimensional bounded realizations of simple compact Lie groups have physical sense.
The study of nontrivial quasi-representations turned out to be fruitful already at the level of one-dimensional mappings. Historically, the first (and the simplest) objects of this kind were one-dimensional mappings, the so-called quasi-characters and pseudo characters introduced in 1983 [2]—[4]. Recall that a real-valued function / on a group G is said to be {real) quasi-character on G [3, 4] if the numerical set
if{gh) -f(g)~ f(h) | g,h e G}
is bounded, and a quasi-character / is said to be a pseudocharacter on G if
f(gn) = nf(g) for any g G G and any n 6 Z.
The notion of pseudocharacter (used later on in some papers under the title of homogeneous quasi-morphism [5]) turned out to be very fruitful in the theory of bounded cohomology ([6]-[10]), in the theory of diffeomorphism groups [11], in symplectic geometry ([12], [13]), in combinatorial group theory ([5], [7], [10]), and in the theory of group representations ([14], [15]), and gained a popularity which caused a separate explanatory publication in Notices of AMS [16].
Note that characters (=pseudocharacters) on commutative and/or solvable Lie groups need not be continuous. However, as was shown in [17], any pseudocharacter on a semisimple Lie group is automatically continuous.
Martially supported by the RBRF under grant no. 02-01-00574 and by the Program of Supporting Leading Scientific Schools under grant no. NSh-619.2003.1 ^3
2N instead of n in this formula in [1] is an obvious misprint.
Higher-dimensional mappings satisfying the above conditions are now often called almost representations or quasi-representations (if the condition that the product of images is close to the image of the product holds on a generating set or is valid uniformly on the entire group, respectively). In the present report we discuss quasi-representations of compact Lie groups.
Recall that Grove, Karcher, and Ruh [18] had proved already in 1974 that continuous representations of any compact topological group in a Banach space are “stable” in the sense of Definition 5 below. In 1977, de la Harpe and Karoubi [19] obtained this result in another way (with a better estimate). However, the methods of these papers, as well as some other methods (including those using invariant means on groups), cannot be immediately extended beyond the limits of compact groups (and beyond the limits of right uniformly continuous mappings), and the solution of the problem for amenable groups was related to use of the language of group algebras. Namely, in 1972 B. E. Johnson introduced amenable Banach algebras and proved that a locally compact group is amenable if and only if the group algebra Ll{G) is [20] and, in 1986-87, he introduced “approximatively multiplicative” functionals and mappings of Banach algebras and proved in particular that any approximatively multiplicative mapping of an amenable Banach algebra into a Banach algebra which is a dual Banach space admits a close homomorphism ([21]—[24], see also [25]). The translation of the apparatus of amenable algebras to the language of groups enabled one to extend the existence theorem for representations close to a given quasi-representation of any amenable locally compact group in a dual Banach space to some unbounded quasi-representations [26].
For nonamenable groups, the existence conditions for homomorphisms close to a given quasihomomorphism were studied in many papers from different points of view (see, e.g., [22], [25], [27]—
[29] and the bibliography therein). In 1994, the author expressed a conjecture claiming that the amenability condition is not only sufficient but also necessary for the stable unitary representabil-ity of groups in Hilbert spaces and verified the validity of the conjecture for almost connected locally compact groups [30]. A similar question on the relationship between the stable repre-sentability and amenability was posed later on (in somewhat other terms) by Gromov 1995 [6].
In the present report we sketch the proof of an assertion which can be regarded as a theorem on automatic nearness to a continuous homomorphic mapping of any (not necessarily continuous) unitary quasi-representation of an arbitrary semisimple compact Lie group. Namely, let G be a connected compact n-dimensional linear Lie group coinciding with its commutator subgroup. For any e > 0, there is a 8 > 0 such that, for any finite-dimensional unitary ¿-quasi-representation /: G —> U(n), there is a continuous homomorphism F: G —> H which is e-close to /. This gives a complete description of the finite-dimensional e-quasi-representations (with small defect e) of the semisimple compact Lie groups regarded in discrete topology (this gives the first example of a nonamenable group for which the e-quasi-representations with small e are completely described). In particular, this shows that, when solving the existence problem for a nonamenable group all of whose unitary e-quasi-representations with small e are perturbations of ordinary representations, one must use infinite-dimensional quasi-representations.
The paper is organized as follows. In Section 2 we give a preliminary information concerning “almost multiplicative” mappings of Banach algebras, measurable bounded quasi-representations of a locally compact group, properties of commutators in semisimple compact Lie groups, and continuity conditions for representations of an arbitrary connected locally compact group which are not assumed to be measurable. The proof of the main theorem is sketched in Section 3.
For the references concerning group representations, see [31] and [32], for invariant means, see [33] and [34], for the theory of semisimple Lie groups, see [35], for harmonic analysis, see [36], and for the theory of automatic continuity of mappings between Banach algebras, see [37].
Some results of the paper were reported on the conference on geometry on Usedom (Germany) in 1999 and on the XIV JMS conference in Mysore (India) in 2003. Some other results of the author presented here were reported at the seminar of A. S. Mishchenko, V. M. Manuilov, and E. V. Troitskii at the Moscow State University. The author is indebted to the leaders and
participants of the seminar for friendly attention.
§ 2. Preliminaries
2.1. Mappings close to representations of algebras.
Amenable Banach algebras.
All Banach algebras below are assumed to be defined over the complex field.
Definition 1. Let A be a Banach algebra. A Banach space X is said to be a Banach A-bimodule if X is a two-sided A-bimodule in the ordinary algebraic sense and there is a constant C > 0 such that
||a;a|| < C7||a||||aj||, ||aa;|| < C||a||||a:|| for any a G A and x £ X.
A Banach A-bimodule X is said to be dual if X is a Banach space dual to the underlying Banach space of some Banach A-bimodule X* and the operations in X are dual to the anticorresponding operations in X*.
The following definition is due to Johnson [20].
Definition 2. A Banach algebra A is said to be amenable if any derivation D from A to a dual Banach A-bimodule in X* (i.e., a mapping D: A —> X* such that D(ab) = D(a)b + aD(b) for any a,b G A) is of the form D(a) = ax — xa, a £ A, for some x in X*.
The role of amenable Banach algebras is clarified by the following assertion.
Lemma 1 [20]. A locally compact group is amenable if and only if its group algebra is amenable as a Banach algebra.
The following definition introduces /4MiVM-pairs of Banach algebras, i.e., pairs for which every mapping of A into B for which the image of any product of two elements is close to the product of the corresponding images is close to an ordinary homomorphism.
Definition 3 [24]. Let A and B be Banach algebras. A pair (A, B) is said to be an AMNM-pair if, for any e > 0 and C > 0, there is a 8 > 0 such that, for any bounded linear operator T: A -¥ B satisfying the conditions \\T\\ < C and
||r(aai) — T(a)T(ai)|| < <5||a||||ai|| for any (a,ai€A), (1)
there is a bounded homomorphism S'. A^ B such that || T — 5|| < e.
For instance, an arbitrary finite-dimensional pair of Banach algebras defines an AMNM-pair [24].
The following theorem proved by Johnson [24] is one of the main tools used below in Section 3. Theorem 1 [24]. If A is an amenable Banach algebra and B is a Banach algebra and a dual Banach B-bimodule, then (A, B) is an AMNM-pair.
For the continuity conditions concerning mappings T satisfying condition (1), see, e.g., [23],
[25], [29].
2.2. Quasi-representations and their simplest properties.
Definition 4. Let S be a semigroup and let E be a topological vector space. A mapping T of the semigroup S into the multiplicative semigroup of the Banach algebra L(E) of all continuous linear operators on E is said to be a quasi-representation (to be more exact, a [/-quasi-representation, where U is a given equicontinuous family in L(E)) of the semigroup S in E if the family U(T) defined by the relation
U(T) = {T(sis2) -T(Sl)T(S2), sus2 G S}
is equicontinuous (ifU(T) C U, respectively).
435
Let E be a metrizable locally convex space, d is an admissible metric on E, and e > 0. A mapping T : G —> L(E) is said to be an e-quasi-representation with respect to the metric d if
d(T(sis2)x, T{si)T{s2)x) < ed(x, 0), si,s2£S,x€E, (2)
and the number e is referred to as a defect of the quasi-representation T.
Definition 5. Let G be a topological group, let e be the identity element of G, let B be a Banach space, let C(B) be the algebra of all bounded linear operators on B, and let 1b be the identity operator on B. The group G is said to be stably representable in B if, for any e > 0 and C > 0, there is a number 8 > 0 with the following property, if T is a weakly continuous 8-quasi-representation of G in B by invertible operators such that
T(e) = 1B; T(g)-1 = Tig-1) and ||T(s)|| < C for any g G G,
then there is a weakly* continuous representation R of G in B* such that ||T(g)* — i?(g)|| < e for any g G G, i.e., the mapping T* is an e-perturbation of the ordinary (weakly* continuous) representation of G.
The following assertion holds.
Theorem 2 [28]. Every amenable locally compact group is stably representable in any Banach space.
2.3. Almost homomorphisms of group algebras and bounded £-quasi-repre-sentations.
The class of bounded e-quasi-representations of a locally compact group is related to the class (introduced by Johnson) formed by almost homomorphisms (or “generalized homomorphisms”) ([22]—[25]) of the group algebra of the group. Indeed, if T is a scalarly measurable and essentially bounded essential e-quasi-representation of a locally compact group G in a separable normed space E (i.e., there is a C > 0 such that |)T(g)|| < C for Haar-almost-all elements g G G and the relation ||T(gi<72) — T(gi)T(g2)\\ < £ holds for Haar-almost-all pairs of elements g\,g2 G G) and if C(£) is the algebra of all bounded linear operators on E, then the formula
T(f)= [ fig)T{g)dg, feLx(G),
Jg
where dg is a left invariant Haar measure on G, defines an LMNM-mapping of the group algebra L\(G) to £(£), i.e., a linear mapping T: L\(G) —> C{£), close to a multiplicative mapping, that is,
\\T{h*f2)-T{fl)T{f2)\\<£\\h\\\\f2\\ for any h,f2eL1(G)
(this follows from an estimate for the norm of the difference T{f\ * f2) — T(fi)T(f2) by using the inequality between the norm of an operator integral and the integral of the norm of the integrand [38]). Since the space E is assumed to be separable, the weak and strong measurability conditions for the mapping T are equivalent and imply the condition that the function g >->• 11^(9)Hi 9 ^ G, is measurable [38].
Conversely, let S be an e-almost homomorphism of the group algebra L\{G) of a separable locally compact group G into the algebra C(£) of all bounded linear operators on a Banach space E, where E is dual to some Banach space E* (which is not uniquely defined in general). In other words, suppose that the mapping S is linear and continuous and satisfies the inequality \\S(h * f2) - S(h)S(f2)\\ < ellMIII/sll for some e > 0 and all /1;/2 G L\(G).
Since S is continuous, it follows that the mapping S has separable image. Therefore ([38], Corollary VI.8.7), there is a mapping T of G to the space L(E) such that the function g (T(g)£,r]), g G G, is measurable and essentially bounded for any £,77 G E, the matrix elements are
436 {S(f)£,v)= [ f{g){T{g)i,r])dg,
Jg
and the norm of the mapping T is equal to the essential least upper bound of the function g i-> ||T(g,)||, g £ G (this follows from [38], Theorem VI.8.2, because the Banach space £{£) is isometrically isomorphic to the Banach space dual to the projective tensor product of E* and E, cf. [39], Theorem IV.2.3). Therefore, it follows from the relation ||S(/i */2) — S(fi)S(f2)\\ < e 11 /1111 [ /211 that the essential least upper bound of the function
{g,h}*\((T(gh)-T(g)T(h))Z,ri)\
does not exceed the number e||£|| \\r]\\ for any £, 7/ £ E, and this proves the inequality
ess snpgMG ||T(gh) - T(g)T(h)\\ < e.
2.4. Commutators in compact Lie groups.
Let us recall some properties of commutators in Banach algebras and compact Lie groups.
Definition 6. Let A be a Banach algebra with unit e, let A~l be the set of invertible elements of A, let B i(e) be the ball (of unit radius) centered at e, lei exp: A —>• A"1 be the mapping defined by the formula
OO
expx = y^(n!)~1-xn, x £ A,
n=0
let log: B\(e) -> A be the mapping defined by the formula
OO
log(l + x) = ^(-lr^n-V, x e -81(e),
n=1
and let D be the set of all pairs (x,y) £ Ax A such that exp x exp y £ B i(e). The mapping taking D to A and given by the formula
(x, y) x * y = log (exp x exp y), (x,y) £ D,
is referred to as the CH-multiplication (or the Baker Campbell-Hausdorff Dynkin multiplication) on A.
Definition 7. (1) Let g be a Lie algebra and let a, b C 0. Denote by [a, b] the linear hull of the elements [a, 6], a £ a, b £ b and by g' = [3,0] the so-called commutator ideal of 0.
(2) Let G be a group, let A,BcG, and let a £ A and b £ B. Write comm (a, b) = aba~1b~1 and denote by comm(A, B) the subgroup generated by all elements of the form comm(a,6), a £ A, b£B. Let G’ = comm(G, G).
Note that the construction (A,B) -> comm(^4,5) does not involve closure.
Lemma 2 (Proposition 5.59 in [40]). Let A be a Banach algebra, let g be a closed finitedimensional Lie subalgebra of A, and let B be an open ball in A centered at the origin and such that the product B * B * B * B is defined and is contained in the open ball of radius 7r.3 Let the dimension of the commutator algebra [0,0] be n. Let comm*(x,y) — x * y * (—x) * (—y), x, y £ g n B. Then
(1) comm* (2:, y) £ [0, g] for any x,y £ gCi B.
(2) there are elements Xj,Yj £ g, j = /,..., n = dimg' such that, for any sets (ri,..., rn) of reals such that 0 < \rj\ < 1 for any j = I,... ,n, there is an e > 0 such that the function ip: (—e, e)n —» BD g defined by the formula
ip(s 1,... ,sn) = (rvX 1 * srYi * -ryX 1 * -si-Fi) * ■ ■ ■ * (rn-Xn * sn-Yn * -rn-Xn * -sn-Yn)
= comm(rr Xi,sr Yx) * • • • * comm(rn-Xn, sn-Yn),
is a homeomorphism onto some neighborhood of zero in g1.
—------------------------------------ ---- 43'
As is known, it suffices to assume that ||x|| < (1/2) ln(2 — ^/(1/2)) for any x G B.
(3) Every element of some neighborhood of zero in g' is a *-product of at most n *-commuta-tors. In particular, there is an open ball B' (centered at zero) in g' such that B'Og' is the smallest local subgroup with respect to B1 that contains all *-commutators of the form comm*(X, Y) £ B X,Y € B'.
2.5. Continuity conditions for finite-dimensional representations of a connected locally compact group.
Definition 8 [41]. Let G be a topological group, let S be a (not necessarily continuous) representation of G in a normed space E over the field C of complex numbers. Let E* be the space dual to E. Introduce the weak variation (or weak oscillation) uj(S) > 0 of the representation S at the identity element e of G (or, briefly, the weak variation of the representation S) by setting
u{S)= sup inf sup \f{S(g)£ — £)|.
{ÇeE,\\t\\<l;feE*,\\f\\<l}u3eg€u
Let us recall the notion of locally uniformly bounded representation.
Definition 9. Let G be a topological group and let S be a representation of G in a normed space E. We say that S is locally uniformly bounded if there is a neighborhood V of the identity element e in G and a constant C such that
ll-S'ig)!! < C for any g£V.
The following criterion for the weak continuity of representations of topological groups by bounded linear operators in an arbitrary Banach space holds.
Theorem 3 [41]. Let G be a topological group, let E be a Banach space, and let S be a locally bounded representation of G in a Banach space E by bounded linear operators. The representation S is continuous in the weak operator topology if and only if u>(S) < 1.
Thus, if the weak oscillation w(S) of a representation S of a topological group in a Banach space is less than one, then this oscillation vanishes and the representation is weakly continuous. Thus, in the class of group representations one has a finite verification of the infinitesimal continuity condition. In the finite-dimensional case, the assertion can be refined as follows.
Theorem 4 [42]. Let G be a connected locally compact group. Every finite-dimensional representation S of G such that w(S) <2 is continuous.
§ 3. Sketch of the proof of the main theorem
Theorem 5. Let G be a connected n-dimensional linear Lie group coinciding with the commutator subgroup G = [G,G]. For any n £ N and any £ > 0, there is a & > 0 such that, for any 5-quasi-representation T: G —> U(n), there is a continuous homomorphism S: G —> H which is e-close to T.
Proof. Let e and 8 be positive numbers. The conditions used below mean that the numbers e and 8 are sufficiently small, and we shall impose these conditions in the course of the proof. Let E be a finite-dimensional Hilbert space, let U(i?) be the unitary group of E, and let T: G —> U(E) be a finite-dimensional unitary <5-quasi-representation of G. Let V be a neighborhood
of the identity element in U(E). Since the multiplication operation is continuous, there is a
neighborhood W of the identity element in U(E) such that
comm({/ii} x U(.E)) ■ • ■ comm({/in} x U(E)) C V (3)
j?_„ __ 7_ 7_ - tt r t_t________ Tr . ii.. t:. ...r ii. . t . . s~i
ior any fix,.. . ,/•¿n t w (see .ueimiici o.uo in 11 £ is me ijie ¿ugeura, 01 me .uie group u,
then, by Lemma 2, there are vectors Xj G fl, j = 1,... ,n, such that the set
438
comm({exp(ri-Xx)} x G) ■ ■ ■ comm({exp(r„- Xn)} x G)
is a neighborhood of the identity element in G for any family r\,..., rn of nonzero real numbers such that 0 < \rj\ < 1. It is clear that this property is preserved under a small perturbation of vectors Xj, j = 1,. .., n, and, after an arbitrarily small perturbation of these vectors if needed one can assume that the one-parameter subgroups of G generated by Xj be closed.
Consider a closed one-parameter subgroup Gj of G generated by the elements Xj of g. The subgroup Gj is compact. Moreover, it is commutative, and hence amenable as a discrete group. The restriction Tj of the mapping T to the subgroup Gj is a quasi-representation of the subgroup Gj in the space E.
By Theorem 2, there is a (not necessarily continuous) ordinary representation Sj of G3 in the same space E such that Sj is uniformly close to Tj. Let a > 0 be a number such that
IIS^G?) - Tj(g) || < cr for any g G Gj and any j = 1,..., n. (4)
By the same theorem, Theorem 2, we may assume that the number a is small together with Consider the composition of the mapping T with the operation of taking the determinant of an operator. Since the determinants of close unitary matrices are close and any continuous func-
firm rm fViP mmnar'i- TlY T?,} ic iinifrvrrnlv rnn+irmmis t."his rnrrmnsitinn a nna«i-rPnrPQPTitfltiAn
of G. Recall that every element of G is a product of at most n commutators and the determinant of every commutator is equal to one. If the number 5 is sufficiently small, then the determinant of every operator Sj(g), g G Gj, differs from 1 (as a complex number) by less than a/3. Therefore, the mapping Sj takes the values in SU(.E). Replacing the mapping T by a uniformly close mapping, which we again denote by T, we can assume that the quasi-representation T takes the values in SU(jB). Thus, we can restrict the consideration to the case in which the quasi-representation T takes the values in the special unitary group S\J(E).
Let Hj be the image of the representation Sj. Let us show that, for any j, j = 1,..., n, there is a real number rj, 0 < \tj\ < 1, such that the element Sj(exp(rj-Xj)) belongs to the set W. If Hj = {e}, then Sj(ex.p(rj-Xj)) = 1 G W for rj = 1/2. Let Hj / {e}. By construction, the group Hj is a homomorphic image of the divisible group R and hence the group Hj is divisible by itself. Thus, Hj is infinite and nondiscrete in H. Therefore, the image of any neighborhood of zero in R cannot be finite because the image of any punctured neighborhood is nonempty and the intersection of all these images is empty. Thus, the set M1/2 = {Sj(exp(r-Xj)), 0 < |r| < 1/2}, is infinite and thus nondiscrete, i.e., it has an accumulation point. Considering the ratios of close distinct elements of the commutative family MXj2, we see that the set M\ = {Sj(exp(r- Xj)), 0 < |r| < 1} contains nonidentity elements as close to the identity element e G U(E) as desired. Thus, anyway, the set of elements of the form
S(exp (rrXi))/iiS'(exp (-ri-Xi))/^1 ■ • • 5*(exp (rn-Xn))hnS(exp (-rn-Xn))h~l,
where h\,..., hn are arbitrary elements of U(£J), is contained in the above neighborhood h of the identity element. In particular, the set of all possible elements of U(E) representable in the form
S(exp {rv Xi))T(gi)S(exp (~rv Xi))T{gi)~l ■ ■ ■ x ¿'(exp (rn- Xn))T(gn)S{exp (-rn-Xn))T(gn)~\
where <7i,... ,gn are arbitrary elements of G, is contained in the same neighborhood h chosen above.
Let us now use the inequality
II•S'(exp (rr Xi))T(gi)S(exp (-rr Xi))T(gi)_1 • • •
x 5(exp (r„-Xn))T(3n)5(exp (-7V Xn))T(gn)~l
mi („ Wrni _ \ m (_________________________( \ — 1
~ -L ^pyri^l})± yyi)J. ^l))± yyi)
x T(exp (r„- Xn))T(^n)T(exp (—rn- Xn))T(gn)~~l\\ < 2na,
439
which immediately follows from inequality (4) (because the mappings Sj and T are unitary) by induction on n, and the inequality
||T(exp [rvXi))gi exp (-n-X^gl1 • • ■ exp (rn-Xn))gn exp (-rn-Xn)g~l)
- T(exp (rrX1))T(^1)T(exp {-rv X^Tigx)-1 • • •
x T(exp (rn-Xn))T{gn)T{exp (-rn-Xn))T{gn)-11| < 2n5,
which can be proved in a similar way because the mapping T is unitary. Let the neighborhood V introduced above be of the form {h 6 SU(.E) | \\h — l^H < #}, where the number 6 > 0 is sufficiently small. Combining the inequalities, we see that ||T(g) — l^H < 2nd + 2no + 9 for any element g 6 G admitting a representation (cf. (3))
g = (exp (rr Xi))gi(exp (~rv Xi))(3i)_1 • • • (exp (rn- Xn))gn{exp (-r„- Xn)){gn)~l,
where ri,..., rn 6 M, gi,..., gn € G, and, in particular, this inequality holds in some neighborhood O C W of the identity element of G. Therefore, each representation Sj of the subgroup Gj C G satisfies the condition
\\Sj(g) - 1e\\ < ||Si(3) - T(g)|| + ||T(g) - 1£|| < 2nS + (2n + l)a + 9,
and one can assume that 6, a, and 9 were chosen to be so small that the right-hand side of the last inequality is less than 2.
By Theorem 4, every representation Sj is continuous. A routine consideration (we omit both the construction and the estimate) concerning Borel partitions into subsets defined as products of subgroups enables one to find a locally Borel mapping uniformly close to the original quasirepresentation, i.e., to find a Borel operator-valued function (quasi-representation) R on G such that
\\R(ggi) - ¿*(<7№i)|| < s + Hi?^) - T(g9l)\\ + PG?) - t(9)\\
+ !!-R(oi) — rfoi)!! ^ (3« + 4)<5 + 3no, g, oi £ G.
By Subsec. 2.3, formula
R(f)= [ f{g)R{g)dg, /glx(G),
Jg
defines a continuous quasi-representation of the group algebra Ll(G) of G in E. This algebra is amenable by Lemma 1. Since o is small if <5 is, one can assume that the number (3n+4)£ + 3ner is arbitrarily small together with <5 (the exact meaning of these smallness conditions can be found in the corresponding inequality on p. 299 of [24]). Now it follows from Theorem 1 that, for any e > 0, there is a 5 > 0 for which there is an ordinary continuous representation Q of the group algebra Ll(G) (in the same space E) such that ||-R(/) — Q(/)|| < £||/||> / £ Ll(G).
In turn, the continuous representation Q of Ll{G) in E defines a continuous representation P of the group G in the same space E. The norm of the difference R(g) — P{g), g G G, is essentially bounded by £. Since the topological group G is amenable and the representation P in the finitedimensional space E is continuous, one can assume without loss of generality that E is a Hilbert space and the representation P is unitary with respect to this Hilbert structure. Since the original quasi-representation T is unitary (with respect to the “old” Hilbert structure), the quasi-representation T and the representation R are uniformly close, and R is unitary with respect to the new inner product, it follows that the isomorphism between these Hilbert spaces is close to the identity operator (see, e.g., [19]).
Thus, ess sup?eG ||-R(g) — P(9)\\ < e- Since the difference R — Q is continuous on some neighborhood of the identity element, it follows that the function g i-> ||Z?(g) — jP(5)||,
440
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