PSEUDOCHARACTERS ON FREE SEMIGROUPS INVARIANT WITH ...

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 2, February 1998, Pages 333–342 S 0002-9939(98)04110-0

PSEUDOCHARACTERS ON FREE SEMIGROUPS INVARIANT WITH RESPECT TO THEIR AUTOMORPHISM GROUPS V. A. FA˘IZIEV (Communicated by Palle E. T. Jorgensen) Abstract. Let F be a free semigroup and let A be an automorphism group of F. A description is given of the space of real functions ϕ on semigroup F satisfying the following conditions: 1) the set {ϕ(xy) − ϕ(x) − ϕ(y); x, y ∈ F} is bounded; 2) ϕ(xn ) = nϕ(x) for any x ∈ F and n ∈ N ; 3) ϕ(xτ ) = ϕ(x) ∀x ∈ F, and ∀τ ∈ A.

Introduction The question ‘if we replace a given functional equation by a functional inequality, then under what conditions will the solutions of the inequality be close to the solutions of the equation’ was posed for the functional equation f (xy) = f (x) · f (y) for x, y in a group G in [1] in connection with the results of the papers [2]–[4]. For a mapping f of the group G into the semigroup of linear transformations of a vector space, in the papers [5], [6] sufficient conditions for the coincidence of the solution of the functional inequality kf (xy)−f (x)·f (y)k < c with the solution of the corresponding functional equation f (xy)−f (x)·f (y) = 0 was studied. In the papers [7]–[9], it was independently shown that if a continuous mapping f of a compact group G into the algebra of endomorphisms of a Banach space satisfies the relation kf (xy) − f (x) · f (y)k 6 δ for all x, y ∈ G with a sufficiently small δ > 0, then it is ε-close to a continuous representation g of the same group in the same Banach space (i.e., we have kf (x) − g(x)k < ε for all x ∈ G). In the paper [9], for any positive ε, an example is constructed of a matrix-valued mapping f of the group G with generators a, b, c, and d and defining relation a−1 b−1 abc−1 d−1 cd = 1, such that f satisfies the relation kf (xy) − f (x) · f (y)k < ε but is not δ-close (0 < δ < 1/10) to any representation of the group G. In the papers [10]–[13] the following new algebraic object was introduced. Definition 1. A pseudocharacter of a semigroup S is a real function ϕ on S satisfying the following conditions: 1) the set {ϕ(xy) − ϕ(x) − ϕ(y); x, y ∈ S} is bounded; 2) ϕ(xn ) = nϕ(x) for any x ∈ S and n ∈ N . Received by the editors December 19, 1995 and, in revised form, June 26, 1996 and July 26, 1996. 1991 Mathematics Subject Classification. Primary 20M15, 20M30. Key words and phrases. Semigroup, free semigroup, group, semidirect product, function, endomorphism, automorphism, pseudocharacter, character, linear space. c

1998 American Mathematical Society

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V. A. FA˘IZIEV

The set of pseudocharacters of a semigroup S is a vector space (with respect to the usual operations of addition of functions and their multiplication by numbers), which will be denoted by P X(S). The subspace of P X(S) consisting of real additive characters of the semigroup S will be denoted by X(S). The notion of pseudocharacter permits us to understand the nature of the existence of mappings satisfying the relation of the form kf (xy) − f (x) · f (y)k < ε without being close to any representation: if ϕ is a pseudocharacter and |ϕ(xy) − ϕ(x) − ϕ(y)| < c, then for any α ∈ R the function f (x) = exp(iαϕ(x)), x ∈ G, satisfies the relation |f (xy) − f (x) · f (y)| < | exp(iαc) − 1| for |αc| < π. Thus, for any c > 0 and for any ε > 0, the value |f (xy) − f (x) · f (y)| can be considered to be uniformly smaller than ε for a suitable choice of α > 0. In [12]–[17] a description of the spaces of pseudocharacters on free groups and semigroups, semidirect and free products of semigroups was given. The referee of this paper pointed out to the following connection of pseudocharacters with the theory of Banach algebra cohomology. The condition (1) of the definition of pseudocharacter exactly gives a bounded 2−cocycle on S. Hence, if semigroup S has a nontrivial pseudocharacter, i.e., P X(S) \ X(S) 6= ∅ , then arguing as [18, Proposition 2.8] we obtain H 2 (S, C) 6= 0. Definition 2. A pseudocharacter f of a semigroup S is said to be invariant with respect to a semigroup K of endomorphisms of S if for any elements x ∈ S and any τ ∈ K the relation f (xτ ) = f (x) holds. Let us denote the space of pseudocharacters of semigroup S that are invariant with respect to a semigroup K of its endomorphisms by P X(S, K). Denote by X(S, K) the subspace of P X(S, K) consisting of real additive characters of S. In [15] the following result was obtained. ˙ X(S, K), where K ·S denotes the semidirect Theorem 1. P X(K ·S) = P X(K)+P product of the semigroups K and S and K acts on S by endomorphisms. In connection with this result, the problem of describing the space of pseudocharacters of some semigroup S which are invariant with respect of some semigroup K of its endomorphisms arises. §1. The main results The aim of this article is to give a description of the space of pseudocharacters of a free semigroup invariant with respect to some group of its automorphisms. From Definition 1 it follows that if F is a free semigroup of rank 1, then P X(F ) = X(F ). Therefore, in the sequel we shall assume that F is a free semigroup with the set X of free generators, and that the cardinality of X is greater than 1. Let A be a subgroup of Aut F . Definition 3. Two elements u, v of F are said to be conjugate if there are a and b in F such that u = ab, v = ba. The relation of conjugacy is denoted by ∼. We will say that elements u and v of F are A-conjugate if there is an a ∈ A such that ua ∼ v. Definition 4. An element of F is called simple if it is not a nontrivial power of another element. The set of the simple elements of the semigroup F will be denoted by P.


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It is easy to see that the relation of A-conjugacy is an equivalence relation on F and on P. We denote the relation of A-conjugacy by ∼A . Definition 5. Let v = x1 · x2 . . . xn , length of the element v.

xi ∈ X. The number |v| = n is called the

Now for each word v in the semigroup F we introduce the set of “beginnings” H(v) and the set of “ends” K(v) as follows. If v ∈ X, we put H(v) = K(v) = ∅. If v = xi1 · xi2 · · · · · xin , n > 1, where xij ∈ X, we set H(v) = {xi1 , xi1 xi2 , . . . , xi1 xi2 · · · xin−1 }, K(v) = {xi2 · · · xin , xi3 · · · xin , . . . , xin−1 xin , xin }. Lemma 1. There is a set P of representatives of conjugacy classes in P such that for each w ∈ P H(w) ∩ K(w) = ∅. Proof. See Lemma 8 in [17]. For any element w of F such that H(w) ∩ K(w) = ∅ in [17] the functions ηw and ew were defined as follows: if v ∈ F , then ηw (v) is equal to the number of occurrences of w in the word v; ew = max{ηw (v 0 ), v 0 ∼ v} . Lemma 2. Let w ∈ P . Then the function ew is a pseudocharacter and we have the following relations: 1) |ew (uv) − ew (u) − ew (v)| 6 2 for any elements u and v of F ; 2) if |u| < |w|, then ew (u) = 0; 3) if |u| = |w|, then ew (u) = 1 for u ∼ w and ew (u) = 0 for u not conjugate to w. Proof. See Lemma 13 in [17]. Let u ∼ v. It is easy to verify that for any a ∈ A the relation ua ∼ v a holds. For an arbitrary element w of F we define the sets St(w), B(w), C(w), A(w), D(w) as follows. Let St(w) denote the stabilizer of element w in group A. Let B(w) denote the stabilizer of the class of conjugacy elements containing w in group A, i.e., b ∈ B(w) if and only if wb ∼ w. It is evidently that St(w) and B(w) are subgroups of group A and St(w) ⊆ B(w). Let C(w) (respectively A(w)) be a system of representatives of left cosets of the group B(w) (respectively A) by subgroup St(w), i.e., B(w) = St(w) · C(w), A = St(w) · A(w). By D(w) we denote the system of representatives of left cosets of the group A by the subgroup B(w). Then we have A = B(w) · D(w). For any v ∈ F denote by M (v) the set of values of the function a → v a , a ∈ A. For each element w of the semigroup F such P that H(w) ∩ K(w) = ∅ let us define functions δw and δ w as follows: 1 δ w (v) = |C(w)| δw (v). δw (v) = t∈M(w) et (v); From [17, Theorem 1] we obtain that the functions δw and δw are pseudocharacters of the semigroup F . Lemma 4. For each v in F and a in A the equalities δw (v a ) = δw (v); δ w (v a ) = δ w (v) is valid, i.e., the pseudocharacters δw and δ w are invariant relative to the action of the group A. Let x ∈ F ; we set H(x) = H(x) ∪ {x} and K(x) = K(x) ∪ {x}. For each pair of elements x, y of F we define measures µx,y , µx,y,x , and νx,y on P as follows.


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We set µx,y (w) = 1 if there exist a and b such that a ∈ K(x), b ∈ H(y), and w = ab; otherwise we set µx,y (w) = 0. We set µx,y,x (w) = 1 if there exist c and d such that c ∈ K(x), d ∈ H(x), and w = cyd; otherwise we set µx,y,x (w) = 0. Finally, we set νx,y (w) = µx,y (w) + µy,x (w) + µx,y,x (w) + µy,x,y (w) − µx,x (w) − µy,y (w). A (w) and ω A Definition 6. For any pair u, v ∈ F we define measures ωu,v u,v (w) on P 1 A A A · ωu,v (w). the set T as follows: ωu,v (w) = t∈M(w) νu,v (t), ω u,v (w) = |C(w)|

Let us fix some system of representatives T of the classes of A-conjugate elements lying in P such that H(w) ∩ K(w) = ∅ for any w ∈ T . Denote by Tn the subset of T consisting of elements of length n . Denote by ZP X(F , A) the subspace of P X(F , A) consisting of the pseudocharacters of semigroup F vanishing on T1 . Denote by L0 (T ) the space of real functions α on T satisfying the following conditions : 1) α T1 ≡ 0; 2) α T is bounded for each positive integer n; n R A 3) the set x, y ∈ F is bounded. T α dω x,y , Now we can give the statements of the main results. Theorem 2. Let γ be a bounded function on the set Tn ; then the function ψγ = P w∈Tn γ(w)δ w belongs to the space P X(F , A) and ψγ (w) = γ(w) ∀w ∈ Tn . Theorem 3. P Each element ϕ of the space ZP X(F , A) is uniquely representable in the form ϕ = w∈T α(w)δ w , α ∈ L0 (T ). P Lemma 9. The expression t∈T1 γ(t)δ t is the general form of an element of the space X(F , A), where γ is an arbitrary function from T1 to R. Corollary 1. Each P element f of the space P X(F , A) is uniquely representable in the form f = χ + w∈T α(w)δw , χ ∈ X(F , A), α ∈ L0 (T ). §2. Proofs of the main results Lemma 3. For any element a ∈ A the following equalities hold: a−1 B(w)a = B(wa ) ; a−1 H(w)a = H(w)a . Proof. The proof is straightforward. This lemma implies that the mapping b a : b → a−1 ba establishes a one-to-one a correspondence between B(w) and B(w ), and between St(w) and St(wa ) too. Hence, the sets C(w) and C(wa ) have equal cardinality. For each γ ∈ C(w) the word wγ is a cyclic permutation of word w. Let γ1 and γ2 be distinct elements of C(w) ; then wγ1 6= wγ2 . In fact if wγ1 = wγ2 , then γ2−1 γ1 ∈ S(w) and there is h ∈ St(w) such that γ1 = γ2 · h. Hence, the elements γ1 , γ2 belong to the same coset of B(w) by S(w). This means that γ1 = γ2 and we obtain a contradiction to the assumption. Thus the mapping w → wγ is an embedding of C(w) into the set of distinct elements of F which are conjugated with w. Therefore, |C(w)| ≤ |w|. Let a1 , a2 be elements of A(w) such that a1 6= a2 . It can easily be checked that wa1 6= wa2 . Let M (w) = {wa ; a ∈ A(w)}. It is clear that M (w) coincides with the set of values of the function a → wa , a ∈ A. It is evident that for any a ∈ A and any w ∈ F the equality (M (w))a = M (w) holds.



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Lemma 4. For each v in F and each a in A the equalities δw (v a ) = δw (v), δ w (v a ) = δ w (v) are valid; i.e., the peudocharacters δw and δ w are invariant relative to the action of the group A. Proof. In [17, Lemma 11] it was shown that for each w ∈ F such that H(w) ∩ K(w) = ∅ we have the equality ew (v) = ηw (v) + µv,v (w).

(1)

It is easy to verify that for any u, v from F the equalities ηw (v a ) = ηwa−1 (v) and µv,u (wa ) = µva−1 ,ua−1 (w) hold. Now from (1) we obtain the equality ew (v a ) = ewa−1 (v).

(2) Further, we have δw (v a ) =

X t∈M(w)

(3) =

X

et (v a ) =

X

t∈M(w)

X

eu (v) =

u∈M a−1 (w)

eta−1 (v)

(by

(2))

et (v) = δw (v).

t∈M(w)

The lemma is proved. Lemma 2 implies that δw (w) is equal to the number of elements in the set M (w) which conjugate with w. Therefore, δw (w) = |C(w)| and 1 ≤ δw (w) ≤ |w| and we have δ w (w) = 1. It is clear that a class of A-conjugate elements containing t consists of a union of classes of conjugate elements such that their representatives are elements tγ , γ ∈ D(t). Let γ1 , γ2 be different elements of D(t). It is clear that wγ1 6∼ wγ2 . By Lemma 1 in every class of A-conjugate elements belonging to P there is an element w such that the relation H(w) ∩ K(w) = ∅ holds. Let us fix some system of representatives T of classes of A-conjugate elements lying in P and having the property H(w) ∩ K(w) = ∅. Evidently, every element ϕ from the space P X(F , A) is defined by its restriction to T . Lemma 5. Let γ be an arbitrary function on T1 . Then the function X γ(w)δ w ψγ = w∈T1

belongs to the space X(F , A) and ψ T1 = γ. Proof. The proof is by direct calculation. Let P X(FT1 , A) be the subspace of P X(F , A), consisting of pseudocharacters of the semigroup F vanishing on the set T1 . Suppose ϕ ∈ P X(F , A) and γ = ϕ T1 . P Then the function f = ϕ − w∈T1 γ(w)δw belongs to the space P X(F , A) and f T ≡ 0. 1 Thus the problem of a description of space P X(F , A) is reduced to the problem of a description of the space P X(FT1 , A). A Definition 6. For any pair u, v ∈ F we define measures ωu,v (w) and ω A u,v (w) on P 1 A A A · ωu,v (w). the set T as follows : ωu,v (w) = t∈M(w) νu,v (t), ωu,v (w) = |C(w)|

From Lemma 2 we obtain Lemma 6. For any v ∈ F and any w ∈ T the following relations hold: 1) δ w (v) = 0 if |v| < |w|; 2) δ w (v) = 0 if |v| < |w| and v 6∼A w; 3) δ w (w) = 1.


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Lemma 7. For each u, v ∈ F and each w ∈ T we have ω A u,v (w) = δ w (uv)−δ w (u)− δ w (v). Proof. By Lemma 12 from [17] we have νu,v (t) = et (uv) − et (u) − et (v). 1 P Therefore, ω A u,v (w) = C(w) t∈M(w) [et (uv)−et (u)−et (v)] = δ w (uv)−δ w (u)−δ w (v). The lemma is proved. (4)

Theorem 2. Let γ be a bounded function on the set Tn ; then the function X ψγ = (5) γ(w)δ w w∈Tn

belongs to the space P X(F , A) and ψγ (w) = γ(w)

∀w ∈ Tn .

Proof. Consider the measures νu,v . Let Mn be the subset of F consisting of the words of length n in the alphabet X. Then for each w ∈ Tn M (w) is a subset of M|w| . From the definition of the measures νu,v it follows that for any words u and v and nS ≥ 2 the estimate |Mn ∩ supp νu,v | ≤ 4(n − 1) holds. Note that the collection w∈Tn {t; t ∈ M (w)} consists of pairwise distinct elements. In fact, it is clear for elements belonging to {t; t ∈ M (w)}. Now suppose that t1 = t2 , t1 ∈ M (w1 ), t2 ∈ M (w2 ), w1 6∼A w2 . Since t1 ∼A w1 , and t2 ∼A w2 we obtain that w1 ∼A w2 , and we come to a contradiction to the assumption. Hence, A there exist at most 4(n − 1) elements w of Mn such that ωu,v (w) 6= 0. Therefore, |Tn ∩ supp ωu,v | ≤ 4(n − 1) and |Tn ∩ supp ωA u,v | ≤ 4(n − 1).

(6)

From (4) and Lemma 2 we obtain |νu,v (w)| ≤ 2; therefore, (7)

A (w)| ≤ |ωu,v

X

|νu,v (t)| ≤ 2 · |Tn ∩ supp νu,v | ≤ 2 · 4(n − 1),

t∈M(w)

and |ωA (w)| ≤ 8(n − 1). Let c > 0 be such that |γ(w)| ≤ c for all w ∈ Tn ; then γ(w) u,v ≤ c. Hence, we have C(w)

|ψγ (uv) − ψγ (u) − ψγ (v)| = | =|

X

X

γ(w)[δ w (uv) − δ w (u) − δ w (v)]|

w∈Tn

γ(w)ω A u,v (w)|

w∈Tn

≤c·

X

2 |ω A u,v (w)| ≤ c · 8(n − 1) · 4(n − 1) = 32c(n − 1)

(by

(6), (7)).

w∈Tn

Now from Lemma 6 it follows that ψγ (w) = γ(w) proved.

∀w ∈ Tn . The theorem is

Denote by K the set of functions ϕ on the semigroup F satisfying the following conditions: 1) ϕ(xn ) = nϕ(x) ∀n ∈ N, ∀x ∈ F ; 2) ϕ(xy) = ϕ(yx) ∀x, y ∈ F ;



339

3) ϕ P1 ≡ 0, ϕ Pi is bounded ∀i ∈ N ; 4) ϕ(xa ) = ϕ(x) ∀x ∈ F , ∀a ∈ A. It is evident that K is a linear space relative to the ordinary operations. Denote by ZP X(F , A) the subspace of P X(F , A) consisting of pseudocharacters of the semigroup F vanishing on T1 (and hence on the set P1 = X too). It is clear that ZP X(F , A) is a subspace of K. Lemma 8. Let ϕ ∈ P X(F , A) and ϕ T1 ≡ 0. Then the function ϕ is bounded on the set Tn ∀n ∈ N . Proof. The proof is by induction on n. Denote by L(T ) the space of real functions α on T satisfying the following con ditions: α T ≡ 0, and α T is bounded for each positive integer n. It is obvious 1 n that L0 (T ) is a subspace of L(T ). Let us construct an isomorphism θ between the spaces K and L(T ). Let ϕ ∈ K. For each i ∈ N we define the function αi : Ti → R by induction as follows: let α1 ≡ 0, and if the values α1 , α2 , . . . , αn have already been defined, then we set (8)

n X αn+1 (w) = ϕ − ϕαi i=1

Tn+1

(w),

w ∈ Tn+1 .

Here ϕαi are pseudocharacters introduced by equality (5) (note that ϕα 1 ≡ 0). Now we define the function α = θ(ϕ) via its restriction to Ti by setting α Ti = αi . Let us show that θ(ϕ) belongs to L(T ). Indeed, we have α T1 = α1 ≡ 0. Furthermore, suppose that we have already established that the functions α T , . . . , α T are 1 n bounded. Let us prove that α Tn+1 is also bounded. Since the restrictions of the pseudocharacters ϕ, ϕα2 , . . . , ϕαn to T1 are zero functions it follows from Lemma 8 that all these pseudocharacters are bounded functions on Tn+1 ; therefore, formula (8) implies that the function α T is bounded too. n+1

Theorem 3. 1) The mapping θ is an isomorphism between the linear spaces K and L(T ). The restriction of θ to ZP X(F , A) is an isomorphism between the linear spaces ZP X(F , A) and L0 (T ). 2) Each element ϕ of the space ZP X(F , A) is uniquely representable in the form X α(w)δ w , α ∈ L0 (T ). ϕ= w∈T

Proof. Let us show that the mapping θ is linear. First we note that if τ1 , τ2 , τ are bounded functions on Tn+1 and λ1 and λ2 are reals such that τ = λ1 τ1 + λ2 τ2 , then we have (9)

ϕτ = λ1 ϕτ1 + λ2 ϕτ2 ,

where ϕτ , ϕτ1 , ϕτ2 are the pseudocharacters defined by equality (5). Suppose that the function ϕ ∈ K satisfies formula (8). Let λ ∈ R. Assume that we have already established that the restrictions of the functions θ(λϕ) to T1 , . . . , Tn are , respectively. Then from P formulas (8) equal to λα1 , . . . , λαnP and (9) we obtain n n θ(λϕ) Tn+1 = λϕ − i=2 ϕλαi Tn+1 = λϕ − i=2 λϕαi Tn+1 = λαn+1 , i.e., θ(λϕ) = λθ(ϕ).


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Let ψ ∈ K and θ(ψ) Tn = βn , n ∈ N . Then it is clear that (ϕ + ψ) T1 ≡ 0 and (ϕ + ψ) T2 = ϕ T2 + ψ T2 = α2 + β2 . Suppose that we have already established that (ϕ + ψ) Ti = ϕ Ti + ψ Ti , i = 1, 2, . . . , n. Then for n + 1 formulas (8) and (9) imply (ϕ + ψ) T

n+1

n X = (ϕ + ψ) − ϕ(αi +βi ) i=2

Tn+1

n X = (ϕ + ψ) − (ϕαi + ϕβi )

= ϕ−

n X

i=2

ϕαi

i=2

= αn+1 + βn+1

Tn+1

Tn+1

n X + ψ− ψβi i=2

= ϕ Tn+1 + ψ Tn+1 .

Tn+1

Thus, the mapping θ is linear. Let us show that θ maps K onto L(T ). Indeed, let α ∈ L(T ) and α Ti = αi . Since for each t ∈ F therePis only a finite set of nonzero numbers of the form ϕαn (t), ∞ n ∈ N, the function ϕ = i=2 ϕαi on F is well defined and belongs to the space K. Let us show that θ(ϕ) = α. We set θ(ϕ) Ti = βi ; let us verify that βi = αi , ∀i ∈ N . Since β1 = ϕ T1 ≡ 0, we have β1 = α1 . Suppose that we have already established then for all w ∈ the relations βi = Pnαi for i ≤ n; P PTnn+1 by formula P∞(8) we have ∞ βn+1 (w) = ϕ − i=2 ϕ (w) = ϕ (w) − ϕ (w) = α α α i i i=2 i=2 i=n+1 ϕαi (w). i ≡ 0, ∀i > n + 1, we obtain Furthermore, since ϕαi T n+1

∞ X

ϕαi (w) = ϕαn+1 (w) = αn+1 (w),

i=n+1

where the latter equality follows from Theorem 2. Thus, βn+1 ≡ αn+1 , and θ is an epimorphism. Now let us verify that ker θ = 0. Indeed, let ϕ ∈ K and θ(ϕ) ≡ 0. This means αi = θ(ϕ) Ti ≡ 0, ∀i ∈ N . Formula (8) implies ϕ T1 ≡ 0, ϕ T2 ≡ 0, and for each Pn Pn+1 w ∈ Tn+1 we obtain ϕ(w) = i=2 ϕαi (w)+αn+1 (w) = i=2 ϕαi (w). Since αi ≡ 0, ∀i ∈ N, we have ϕαi ≡ 0; therefore, the latter equality implies ϕ Tn+1 ≡ 0. Now from properties 1) and 2) of functions from the space K it follows that ϕ ≡ 0 on F . Therefore, ker θ = 0 and θ is an isomorphism. Hence, as was shown above for a function α from L(T ) such that α Ti = αi , ∀i ∈ N , we have (10)

θ−1 (α) =

∞ X

ϕαi .

i=2

Now let us show that under the isomorphism to the elements of R θ we assign 6 ε for some ε > 0 and ZP X(F , A) functions α in L(T ) such that T α dωA x,y each x and y in F . Indeed, let ϕ ∈ ZP X(F ), α = θ(ϕ), and αi = α Ti , i ∈ N ; then P formula ϕ= ∞ i=2 ϕαi . Now let us show that there exists an ε such (10) implies R that T α dωA x,y 6 ε for all x and y in F .



341

Let us choose ε such that |ϕ(xy) − ϕ(x) − ϕ(y)| 6 ε for each x and y in F . Then we have Z ∞ Z ∞ Z X X A A A = = α dω α dω α dω i x,y x,y x,y T Ti i=2 Ti i=2 ∞ X X A = αi (w) ω x,y (w) i=2

∞ w∈Ti X X = αi (w)[δ w (xy) − δ w (x) − δ w (y)] (by Lemma 7) i=2 w∈Ti X X X X ∞ αi (w)δ w (xy) − αi (w)δ w (x) − αi (w)δ w (y) = i=2

w∈T

w∈T

i i X ∞ = (ϕαi (xy) − ϕαi (x) − ϕαi (y))

w∈Ti

(by Theorem 2)

i=2

= |ϕ(xy) − ϕ(x) − ϕ(y)| 6 ε. R Now let α ∈ L(T ) and | T α dω A some ε > 0 and for each x and y from x,y | 6 ε for P F . If αi = α Ti and ϕ = θ−1 (α), then ϕ = ∞ i=2 ϕαi . Therefore, we have ∞ X [ϕαi (xy) − ϕαi (x) − ϕαi (y)] |ϕ(xy) − ϕ(x) − ϕ(y)| = i=2 ∞ X X X X = αi (w)δ w (xy) − αi (w)δ w (x) − αi (w)δ w (y) i=2 w∈Ti w∈Ti ∞ w∈Ti X X = αi (w)[δw (xy) − δ w (x) − δ w (y)] i=2 w∈T

i X Z ∞ X A 6 ε. = αi (w)ω A (w) = α(w) dω (w) x,y x,y

T

i=2 w∈Ti

The theorem is proved. Denote by RT1 the space of real-valued functions on the set T1 . P Lemma 9. The expression t∈T1 γ(t)δ t is the general form of an element from the space X(F , A). Proof. Follows from Lemmas 5 and 6. It is evident that the mapping γ → linear spaces RT1 and X(F , A).

P t∈T1

γ(t)δ t is an isomorphism between the

Corollary 1. Each element f of the space P X(F , A) is uniquely representable in the form X α(w)δ w , χ ∈ X(F , A), α ∈ L0 (T ). f =χ+ w∈T

Proof. Let χ be an element of X(F ) such that χ(x) = f (x) ∀x ∈ X. It is easy to see that χ ∈ X(F , A). Then we have f − χ ∈ ZP X(F , A). Now from Theorem 3 we obtain the required representation.


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Corollary 2. Let F be a nonabelian free semigroup and let S = B · F be the semidirect product of the semigroups B and F , where B acts on F by automorphisms. Then there is a nontrivial pseudocharacter on the semigroup S. (We say that a pseudocharacter ϕ is nontrivial if it is not an additive real character.) Proof. Let w = x1 x2 , where x1 , x2 ∈ X. It is easy to see that δw is not a character and belongs to the space P X(F , AutF ). Taking into account Theorem 1 and the relation P X(F , AutF ) ⊆ P X(F , B) , we obtain δw ∈ P X(S). Acknowledgement The author is indebted to the referee for emphasizing the connection of the results of the paper with the theory of Banach algebra cohomology. References 1. Ulam, S. M., A Collection of Mathematical Problems, Wiley & Sons, Interscience Publ., New York–London, 1960. MR 22:10884 2. Hyers, D. H., On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (2) (1941), 222–224. MR 2:315a 3. Hyers, D. H., Ulam, S. M., On approximate isometry, Bull. Amer. Math. Soc. 51 (1945), 288–292. MR 7:123f 4. Hyers, D. H., Ulam, S. M., Approximate isometries of the space of continuous functions, Ann. of Math. 48 (2) (1947), 285–289. MR 8:588b 5. Baker, J., Lawrence, J., Zorzitto, F., The stability of the equation f (x + y) = f (x) + f (y), Proc. Amer. Math. Soc. 74 (2) (1979), 242–246. MR 80d:39009 6. Shtern, A.I., Stability of homomorphisms into the group R∗ , Vestnik MGU, Ser. matem. mekh. 37 (3) (1982), 33–36. MR 83m:39008 7. Grove, K., Karcher, H., Ruh, E. A., Jacobi fields and Finsler metrics on compact Lie groups with an application to differential pinching problems, Math. Ann. 211 (1) (1974), 7–21. MR 50:8391 8. de la Harpe, P., Karoubi, M., Repres´ entations approch´ ees d’un groupe dans une alg` ebre de Banach, Manuscripta math. 22 (3) (1977), 297–310. MR 58:16948 9. Kazhdan, D., On ε-representations, Israel J. Math. 43 (4) (1982), 315–323. MR 84h:22011 10. Shtern, A. I., A pseudocharacter that is determined by the Rademacher symbol, Uspekhi mat. nauk 45 (3) (1990), 224–226. MR 91j:11027 11. Shtern, A. I., Quasirepresentations and pseudorepresentations, Funkts. analiz i eqo prilozhen. 25 (2) (1991), 70–73. MR 93b:22007 12. Fa˘iziev, V.A., Pseudocharacters of free products of semigroups, Funkts.analiz. i ego priloz. 21 (1) (1987), 86–87. MR 88e:20054 13. Fa˘iziev, V.A., Pseudocharacters on free groups and on some group constructions, Uspehi mat. nauk. 43 (5) (1988), 225–226. MR 90a:20047 14. Fa˘iziev, V.A., Spaces of pseudocharacters of the free product of semigroups, Mathem.Zametki 52 (6) (1992), 119–130. MR 93m:20087 15. Fa˘iziev, V.A., Pseudocharacters on semidirect product of semigroups, Mathem.Zametki 53 (2) (1993), 132–139. MR 94h:20070 16. Fa˘iziev, V.A., Pseudocharacters on free group, Izvestya Russ. Akad. Nauk ser. mathem. 58 (1) (1994), 121–143. MR 95b:20041 17. Fa˘iziev, V.A., Pseudocharacters on free semigroups, Russian Journal of Math. phisics 3 (2) (1995), 191–206. MR 96h:20108 18. Johnson B.E., Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972). MR 51:11130 Institute for Mathematics with Computational Center, Tadjikistan Academy of Sciences, Dushanbe, Tadjikistan Current address: Shirokaia St. 7-3-137, 129282 Moscow, Russia
