A Class of Distal Functions on Semitopological Semigroups
The norm closure of the algebra generated by the set {n→λ^nk : λ belongs T and k belongs N} of functions on (Z,+) was studied in [11] (and was named as the Weyl algebra). In this paper, by a fruitful result of Namioka, this algebra is generalized for a general semitopological semigroup and, among ot...
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| Цитувати: | A class of distal functions on semitopological semigroups / A. Jabbari, H.R.E. Vishki // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 2. — С. 188-194. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-57062010-02-03T12:01:11Z A Class of Distal Functions on Semitopological Semigroups Jabbari, A. Vishki, H.R.E. The norm closure of the algebra generated by the set {n→λ^nk : λ belongs T and k belongs N} of functions on (Z,+) was studied in [11] (and was named as the Weyl algebra). In this paper, by a fruitful result of Namioka, this algebra is generalized for a general semitopological semigroup and, among other things, it is shown that the elements of the involved algebra are distal. In particular, we examine this algebra for (Z,+) and (more generally) for the discrete (additive) group of any countable ring. Finally, our results are treated for a bicyclic semigroup. 2009 Article A class of distal functions on semitopological semigroups / A. Jabbari, H.R.E. Vishki // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 2. — С. 188-194. — Бібліогр.: 11 назв. — англ. 1029-3531 http://dspace.nbuv.gov.ua/handle/123456789/5706 en Інститут математики НАН України |
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The norm closure of the algebra generated by the set {n→λ^nk : λ belongs T and k belongs N} of functions on (Z,+) was studied in [11] (and was named as the Weyl algebra). In this paper, by a fruitful result of Namioka, this algebra is generalized for a general semitopological semigroup and, among other things, it is shown that the elements of the involved algebra are distal. In particular, we examine this algebra for (Z,+) and (more generally) for the discrete (additive) group of any countable ring. Finally, our results are treated for a bicyclic semigroup. |
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Jabbari, A. Vishki, H.R.E. |
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Jabbari, A. Vishki, H.R.E. A Class of Distal Functions on Semitopological Semigroups |
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Jabbari, A. Vishki, H.R.E. |
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Jabbari, A. |
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A Class of Distal Functions on Semitopological Semigroups |
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A Class of Distal Functions on Semitopological Semigroups |
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A Class of Distal Functions on Semitopological Semigroups |
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A Class of Distal Functions on Semitopological Semigroups |
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A Class of Distal Functions on Semitopological Semigroups |
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class of distal functions on semitopological semigroups |
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Інститут математики НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/5706 |
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A class of distal functions on semitopological semigroups / A. Jabbari, H.R.E. Vishki // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 2. — С. 188-194. — Бібліогр.: 11 назв. — англ. |
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2025-07-02T08:47:10Z |
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Methods of Functional Analysis and Topology
Vol. 15 (2009), no. 2, pp. 188–194
A CLASS OF DISTAL FUNCTIONS ON SEMITOPOLOGICAL
SEMIGROUPS
A. JABBARI AND H. R. E. VISHKI
Abstract. The norm closure of the algebra generated by the set
�
n �→ λn
k
: λ ∈ � and k ∈ �
�
of functions on (�,+) was studied in [11] (and was named as the Weyl algebra). In
this paper, by a fruitful result of Namioka, this algebra is generalized for a general
semitopological semigroup and, among other things, it is shown that the elements of
the involved algebra are distal. In particular, we examine this algebra for (�,+) and
(more generally) for the discrete (additive) group of any countable ring. Finally, our
results are treated for a bicyclic semigroup.
1. Introduction
Distal functions on topological groups were extensively studied by A. W. Knapp [7].
The norm closure of the algebra generated by the set F = {n �→ λnk
: λ ∈ T and k ∈ N}
of functions on (Z, +) was called the Weyl algebra by E. Salehi in [11]. A. W. Knapp [7],
showed that all of the elements of F are distal on (Z, +). Also I. Namioka [9, Theorem 3.6]
proved the same result by using a very fruitful result ([9, Theorem 3.5]) which played
an important role for the construction of this paper. By the above mentioned results of
A. W. Knapp and I. Namioka, all elements of the Weyl algebra are distal, however it dose
not exhaust all distal functions on (Z, +), [11, Theorem 2.14]. In this paper, we generalize
the notion of Weyl algebra to an arbitrary semitopological semigroup and also we show
that all elements of the involved algebra are distal. In particular, our method provides a
convenient way to deduce a result of M. Filali [4] on the distality of the functions χ(q(t)),
where χ is a character on the discrete additive group of a (countable) ring R and q(t) is
a polynomial with coefficients in R.
2. Preliminaries
For the background materials and notations we follow J. F. Berglund et al. [1] as much
as possible. For a semigroup S, the right translation rt and the left translation ls on S are
defined by rt(s) = st = ls(t), (s, t ∈ S). The semigroup S, equipped with a topology, is
said to be right topological if all of the right translations are continuous, semitopological
if all of the left and right translations are continuous. If S is a right topological semigroup
then the set Λ(S) = {s ∈ S : ls is continuous} is called the topological centre of S.
Throughout this paper, unless otherwise stated, S is a semitopological semigroup.
The space of all bounded continuous complex valued functions on S is denoted by C(S).
For f ∈ C(S) and s ∈ S the right (respectively, left) translation of f by s is the function
Rsf = f ◦ rs (respectively, Lsf = f ◦ ls).
A left translation invariant unital C∗-subalgebra F of C(S) (i.e., Lsf ∈ F for all s ∈ S
and f ∈ F ) is called m-admissible if the function s �→ (Tµf)(s) = µ(Lsf) belongs to F
2000 Mathematics Subject Classification. 54H20, 54C35, 43A60.
Key words and phrases. Semitopological semigroup, distal function, strongly almost periodic func-
tion, semigroup compactification, m-admissible subalgebra.
188
A CLASS OF DISTAL FUNCTIONS . . . 189
for all f ∈ F and µ ∈ SF (=the spectrum of F). If F is m-admissible then SF under the
multiplication µν = µ◦Tν (µ, ν ∈ SF ), furnished with the Gelfand topology is a compact
Hausdorff right topological semigroup and it makes SF a compactification (called the
F -compactification) of S.
Some of the usual m-admissible subalgebras of C(S), that are needed in the sequel, are
the left multiplicatively continuous, strongly almost periodic and distal functions on S.
These are denoted by LMC(S), SAP (S) and D(S), respectively. Here and also for other
emerging spaces when there is no risk of confusion, we have suppressed the letter S from
the notation.
The interested reader may refer to [1] for ample information about these m-admissible
subalgebras and the properties of their corresponding compactifications.
3. Main results
The idea of defining our new algebras Wk and W , in the form given below, came from
a nice result of I. Namioka [9, Theorem 3.5].
Let Σ = {Tµ : LMC(S) → LMC(S); µ ∈ SLMC}. Let F0 be the set of all constant
functions of modulus 1. For every k ∈ N assume that we have defined Fi for i =
1, 2, . . . , k − 1 and define Fk by
Fk = {f ∈ LMC : |f | = 1 and for every σ ∈ Σ, σ(f) = fσf, for some fσ ∈ Fk−1}.
It is clear from definitions that Fk ⊆ Fk+1, for all k ∈ N ∪ {0}. Let Wk and W be the
norm closure of the algebras generated by Fk and
⋃
k∈N
Fk in C(S), respectively; then
trivially Wk ⊆ Wk+1 ⊆ W . Hence, W is the uniform closure of the algebra
⋃
k∈N
Wk.
It is also readily verified that W is the direct limit of the family {Wi : i ∈ N} (ordered
by inclusion, and with the inclusion maps as morphisms). From now on, we assume that
k ∈ N is arbitrary. We leave the following simple observations without proof.
Proposition 3.1. (i) For every f ∈ Fk and σ ∈ Σ the function fσ with the property
σ(f) = fσf is unique.
(ii) For every f, g ∈ Fk and σ ∈ Σ, (fg)σ = fσgσ. In particular, Fk is a multiplicative
subsemigroup of LMC.
(iii) Fk is conjugate closed; in other words, if f ∈ Fk then f ∈ Fk.
(iv) Fk contains the constant functions.
Lemma 3.2. The set Fk is left translation invariant and it is also invariant under Σ;
in other words, LS(Fk) ⊆ Fk and Σ(Fk) ⊆ Fk.
Proof. A direct verification reveals that Fk is left translation invariant. For the invariance
under Σ let f ∈ Fk and σ ∈ Σ, the equality σ(f) = fσf for some fσ ∈ Fk−1 implies that
|σ(f)| = 1 and so for every τ ∈ Σ,
τ(σ(f)) = τ(fσf) = τ(fσ)τ(f) = ((fσ)τfσ)(fτf) = ((fσ)τfτ )(fσf) = ((fσ)τfτ )σ(f).
Since (fσ)τfτ ∈ Fk−1 we have σ(f) ∈ Fk; in other words Σ(Fk) ⊆ Fk, as required. �
Lemma 3.3. All elements of Fk remain fixed under the idempotents of Σ.
Proof. It is easily seen that the result holds for k = 1. Assume that k > 1 and that the
result holds for k − 1. Let f ∈ Fk and let ε ∈ Σ be an idempotent, then ε(f) = fεf for
some fε ∈ Fk−1. Therefore
fεf = ε(f) = ε2(f) = ε(ε(f)) = ε(fεf) = ε(fε)ε(f) = fε(fεf) = fε
2f,
hence fε = 1 and ε(f) = f , as claimed. �
Lemma 3.4. Fk ⊆ D.
Proof. Let f ∈ Fk. To show that f ∈ D, using [1, Lemma 4.6.2], it is enough to show that
εσ(f) = σ(f) for each σ ∈ Σ and each idempotent ε in Σ. By Lemma 3.2, σ(f) ∈ Fk, so
that Lemma 3.3 implies that ε(σ(f)) = σ(f), as required. �
190 A. JABBARI AND H. R. E. VISHKI
Using parts (iii) and (iv) of Proposition 3.1, Wk and W are unital C∗-subalgebras of
C(S). The following result shows that these are indeed m-admissible subalgebras of D.
Theorem 3.5. For every semitopological semigroup S, Wk and W are m-admissible
subalgebras of D(S).
Proof. For the m-admissibility of Wk it is enough to show that it is left translation
invariant and also invariant under Σ. Let 〈Fk〉 be the algebra generated by Fk. Lemma 3.2
implies that LS(〈Fk〉) ⊆ 〈Fk〉 and also Σ(〈Fk〉) ⊆ 〈Fk〉. For every f ∈ Wk there exists a
sequence {fn} ⊆ 〈Fk〉 which converges (in the norm of C(S)) to f . Let σ ∈ Σ and s ∈ S,
then the inequalities ‖Ls(fn)−Ls(f)‖ ≤ ‖fn − f‖ and ‖σ(fn)− σ(f)‖ ≤ ‖fn − f‖ imply
that Ls(fn) → Ls(f) and σ(fn) → σ(f), respectively. Since for each n ∈ N, Ls(fn)
and σ(fn) lie in 〈Fk〉, we have Ls(f) ∈ Wk and also σ(f) ∈ Wk. It follows that Wk is
m-admissible. A similar argument may apply for the m-admissibility of W . The fact
that Wk and W are contained in D follows trivially from Lemma 3.4. �
The next result gives SW in terms of the subdirect product of the family {SWk : k ∈
N}. For a full discussion of the subdirect product of compactifications one may refer to
[1, Section 3.2].
Proposition 3.6. The compactification SW is the subdirect product of the family {SWk :
k ∈ N}; in symbols, SW =
∨
{SWk : k ∈ N}.
Proof. The family (of homomorphisms) {πk : SW → SWk ; k ∈ N}, where for each µ ∈
SW , πk(µ) = µ|Wk
, separates the points of SW , because for given µ, ν ∈ SW with µ �= ν
(on W ) one has µ �= ν of F =
⋃
k∈N
Fk, hence there exists a natural number j and an
element f ∈ Fj such that µ(f) �= ν(f). Therefore µ|Wj
�= ν|Wj
, that is πj(µ) �= πj(ν).
Now the conclusion follows from [1, Theorem 3.2.5]. �
Proposition 3.7. (i) For every abelian semitopological semigroup S, SAP (S) ⊆ Wk(S).
(ii) For every abelian semitopological semigroup S with a left identity, W1(S) =
SAP (S).
Proof. (i) Since S is abelian SAP (S) is the closed linear span of the set of all continuous
characters on S; see [1, Corollary 4.3.8]. Let f be any continuous character on S and let
σ ∈ Σ. Then there exists a net {sα} in S such that σ(f) = limα Rsα
f . By passing to a
subnet, if necessary, we may assume that f(sα) converges to some λσ ∈ T. Therefore for
every s ∈ S, σ(f)(s) = limα Rsα
f(s) = limα f(ssα) = f(s) limα f(sα) = f(s)λσ. Hence
σ(f) = λσf or equivalently f ∈ F1.
(ii) By part (i) it is enough to show that W1 ⊂ SAP . Indeed we are going to show
that F1 ⊂ SAP ; for this end, let f ∈ F1 and let s ∈ S, then Rsf = λsf for some λs
in T. Let e be a left identity of S, then for each s in S, f(s) = Rsf(e) = λsf(e). Let
h = f/f(e), then h is a continuous character on S. But f = f(e)h, now using the fact
that SAP is the closed linear span of continuous characters of S we have f ∈ SAP , as
required. �
As a consequence of the latter result we have
Corollary 3.8. For any compact abelian topological group G, Wk(G) = W (G) = C(G).
Proof. Since for every compact topological group G, SAP (G) = C(G), [1, Theorem 4.3.5],
the result follows from the last proposition. �
4. Examples
Example (a). Here we restrict our discussion to the discrete group (Z, +) and examine
W and Wk for this particular case, which were studied extensively by E. Salehi in [11].
Note that although we would deal with countable discrete rings in part (b), but the
A CLASS OF DISTAL FUNCTIONS . . . 191
proofs on Z are more interesting and characterizations of Fk(Z) are more explicit. We
commence with the next key lemma which characterizes Fk in l∞(Z).
Lemma 4.1. The set Fk(Z, +) is the (multiplicative) sub-semigroup of l∞(Z) generated
by the set {n �→ λni
: λ ∈ T, i = 0, 1, . . . , k}.
Proof. For each k ∈ N, let Ak denote the multiplicative sub-semigroup of l∞(Z) generated
by the set {n �→ λni
, λ ∈ T and i = 0, 1, . . . , k}. For k = 1 a direct verification reveals
that A1 ⊆ F1; for the reverse inclusion let f ∈ F1. Then for some λ ∈ T, R1f = λf , hence
f(1) = R1f(0) = λf(0) = λλ1, in which λ1 = f(0). Also f(2) = R1f(1) = λf(1) = λ2λ1,
by induction it is easily proved that for each n ∈ N, f(n) = λnλ1. Let R−1f = βf ,
where β ∈ T, then f(−1) = R−1f(0) = βf(0). But f(1) = R−1f(2) = βf(2) = βλ2λ1,
therefore λλ1 = βλ2λ1, hence β = λ−1 and for all n ∈ Z, f(n) = λnλ1. Thus f ∈ A1,
and so F1 = A1.
Let k ≥ 2 and assume that Ak−1 = Fk−1. Let n ∈ Z and λ ∈ T and let f ∈ Ak
and assume (without loss of generality) that f(n) = λnk
, then for given σ = limαmα ∈
Σ we have σ(f)(n) = limαRmα
f(n) = limαλ(n+mα)k
= f(n)fσ(n), in which fσ(n) =
µnk−1
1 µnk−2
2 . . . µn
k−1µk, where (by going through a sub-net of {mα}, if necessary) µi =
limαλ(k
i )mα
i
, for i = 1, 2, . . . , k. But fσ ∈ Ak−1 = Fk−1, so f ∈ Fk. Hence Ak ⊆ Fk. Now
let f ∈ Fk, we have to show that f ∈ Ak. We have R1f = f1f , for some f1 ∈ Fk−1 =
Ak−1. Since f1 ∈ Ak−1 we may assume that f1(n) = λnk−1
1 λnk−2
2 . . . λn
k−1λk, where
λ1, λ2, . . . , λk ∈ T. Then f(1) = R1f(0) = f1(0)f(0) and f(2) = R1f(1) = f1(1)f(1) =
f1(1)f1(0)f(0), and by induction,
f(n) = (λ
(n−1)k−1
1 λ
(n−1)k−2
2 . . . λn−1
k−1λk)(λ
(n−2)k−1
1 λ
(n−2)k−2
2 . . . λn−2
k−1λk)
. . . (λ1λ2 . . . λk)(λk)f(0)
= λ
�n−1
j=1
jk−1
1 λ
�n−1
j=1
jk−2
2 . . . λ
�n−1
j=1
j
k−1 λn
kf(0).
So for each i = 0, 1, 2, . . . , k − 1 the power of λk−i is a polynomial in n of degree i + 1.
Hence the power of λ1 (which has the maximum degree) is a polynomial of degree k. It
follows that f ∈ Ak and the proof is complete by induction. �
As an immediate consequence of the latter lemma we have the next theorem which
characterizes our algebras for the additive group Z.
Theorem 4.2. (i) Wk(Z, +) coincides with the norm closure of the algebra generated by
the set of functions {n �→ λni
: λ ∈ T, i = 0, 1, . . . , k}.
(ii) W (Z, +) coincides with the norm closure of the algebra generated by the set of
functions {n �→ λnk
: λ ∈ T, and k ∈ N}, that is, W (Z, +) coincides with the Weyl
algebra.
Example (b). Let R be a countable discrete ring. Let χ be an arbitrary character on
the additive group (Rd, +), where Rd denotes R with the discrete topology. Without loss
of generality, assume that R is abelian. We are going to show that for each s in R the
function f(t) = χ(st3) belongs to F3(Rd, +). To this end, let σ ∈ Σ. Thus there exists a
sequence sn in R such that for each h ∈ l∞(Rd), σ(h)(t) = limn Rsn
h(t) = limn h(t+sn).
Thus σf(t) = limn f(t + sn) = limn χ(s(t + sn)
3
) = f(t)fσ(t), in which fσ(t) =
limn χ(ss3
n + 3ssnt2 + 3ss2
nt). Since R is countable, by the diagonal process, there exists
a subsequence, say sn, of the sequence sn such that, for all t in R, all of the limits
limn χ(ss3
n), limn χ(ssnt2) and limn χ(ss2
nt) exist. (In fact, one may first choose a subse-
quence of sn, if necessary, such that limn χ(ss3
n) exist. Let R = {x1, x2, x3, . . .}. Choose
a subsequence of sn, say s1,n such that both limits limn χ(ss1,nx2
1) and limn χ(ss2
1,nx1)
exist. This time, choose a subsequence of s1,n, say s2,n, such that both limits
192 A. JABBARI AND H. R. E. VISHKI
limn χ(ss2,nx2
2) and limn χ(ss2
2,nx2) exist. Continue this process and choose the result-
ing sequence sn,n on the diagonal, which is eventually our desired subsequence, say sn).
Hence for each t ∈ R,
fσ(t) = lim
n
χ(ss3
n) lim
n
χ3(ss2
nt) lim
n
χ3(ssnt2).
By definition, to prove that f ∈ F3 it is enough to show that fσ ∈ F2. To see this, let
τ ∈ Σ be arbitrary. Then there exists a sequence tm in R such that for each h ∈ l∞(Rd),
τ(h)(t) = limm Rtm
h(t) = limm h(t + tm).
Thus τfσ(t) = limm fσ(t + tm) = fσ(t)fστ (t), where
fστ (t) = (fσ)τ (t) = lim
m
lim
n
χ3(ss2
ntm) lim
m
lim
n
χ3(ssnt2m + 2ssntmt).
R is countable, hence by going through a subsequence of tm (by using the diagonal
process) one may assume that for all t in R the limits limm limn χ(ssnt2m) and
limm limn χ(ssntmt) exist. Therefore
fστ (t) = lim
m
lim
n
χ3(ss2
ntm) lim
m
lim
n
χ3(ssnt2m) lim
m
lim
n
χ6(ssntmt).
Again by definition, to prove that fσ ∈ F2 it is enough to show that fστ ∈ F1. Let
ξ = liml ul ∈ Σ, then it follows from the above equation that
ξ(fστ )(t) = fστ (t) lim
l
lim
m
lim
n
χ6(ssntmul).
That is, ξ(fστ ) = λfστ where λ = liml limm limn χ6(ssntmul) ∈ F0 = T. Hence fστ ∈ F1
and so fσ ∈ F2 and this implies that f ∈ F3. Our claim is now established. By using the
above method, one may prove that for each k ∈ N and s ∈ R the function t → χ(stk) is
an element of Fk.
Briefly speaking, the above argument and Lemma 3.4 imply that
Corollary 4.3. If R is a countable discrete ring, then for each character χ on the discrete
additive group of R the function χ(q(t)), in which q(t) is a polynomial with coefficients
in R, belongs to W (Rd, +) and is also a distal function.
It should be remarked that the distality of the functions χ(q(t)) was first proved by
M. Filali [4] without the countability condition on R.
Example (c). (i) If S contains a right zero element, i.e. there exists t ∈ S such that
st = t for all s ∈ S, then for f ∈ F1 there exists λt ∈ T such that Rtf = λtf , hence for
all s ∈ S, f(t) = f(st) = Rtf(s) = λtf(s), that is f = f(t)/λt ∈ T. Therefore F1 = T
and so Fk = T for all k ∈ N. It follows that for such a semigroup S, Wk(S) = W (S) =
the set of constant functions.
(ii) If S is a left zero semigroup (i.e. st = s for all s, t ∈ S), then for each function
f ∈ LMC(S) we have σ(f) = f for all σ in Σ, and so if |f | = 1, then f ∈ F1. Hence for
all k ∈ N, W = Wk = W1 = LMC.
Now we examine some of the newly defined algebras on a non-trivial non-group semi-
group.
Example (d). Let S be the bicyclic semigroup of [1, Example 2.10], i.e. S is a semigroup
generated by elements 1, p and q, where 1 is the identity and p and q satisfy pq = 1 �= qp.
The relation pq = 1 implies that any member of S may be written uniquely in the form
qmpn, where m, n ∈ Z
+ and p0 = q0 = 1.
We are going to show that
F1(S) = {qmpn �→ µrν1−r : r = m − n and µ, ν ∈ T}.
To see this, let f ∈ F1, then for each s ∈ S, Rsf = λsf for some λs ∈ T. Hence
f(q) = λqf(1), f(p) = λpf(1) and f(1) = f(pq) = Rqf(p) = λqf(p), therefore λpλq = 1.
It is also readily seen that λqmpn = λm−n
q . One may use induction to simply prove that
for each n ∈ Z+, f(qpn) = f(q)(f(p)/f(1))n, and then use the latter to show (again
by induction on m) that f(qmpn) = f(p)n−mf(1)1−(n−m). But f(p)f(q) = f(1)2, so
A CLASS OF DISTAL FUNCTIONS . . . 193
f(qmpn) = f(q)m−nf(1)1−(m−n). Hence it is enough to take µ = f(q) and ν = f(1).
The converse inclusion is easily verified.
To prove the next theorem, the following lemma is needed.
Lemma 4.4. Let S be the bicyclic semigroup described above. If f ∈ F1(S), then
f(p)f(q) = f(1)2.
Proof. Let f ∈ F1(S), then there exist constants λp and λq ∈ T such that Rpf = λpf
and Rqf = λqf . Hence f(p)f(q) = λpλqf(1)2. Thus it is enough to show that λpλq = 1.
But, f(1) = f(pq) = Rqf(p) = λqf(p) = λqλpf(1), that is λqλp = 1, and the result
follows. �
Theorem 4.5. Let S be the bicyclic semigroup generated by 1, p and q, where 1 is the
identity and pq = 1 �= qp. Then W1(S) and W2(S) are the norm closure of the algebras
generated by the sets
{qmpn �→ µrν1−r : r = m − n and µ, ν ∈ T}
and
{qmpn �→ λ(r2
−r)/2µ(r2+r)/2ν1−r2
, r = m − n and λ, µ, ν ∈ T}
respectively.
Proof. By what we already discussed, the result is clear for W1(S). To complete the
proof, it is enough to show that
F2(S) = {qmpn �→ λ(r2
−r)/2µ(r2+r)/2ν1−r2
, r = m − n and λ, µ, ν ∈ T}.
Let A denote the right hand side of the above equation and let f ∈ A. Then there exist
λ, µ, ν ∈ T such that for all m, n ∈ Z+ ∪ {0}, f(qmpn) = λ(r2
−r)/2µ(r2+r)/2ν1−r2
with
r = m − n. By choosing suitable m, n ∈ Z+ ∪ {0} we derive that λ = f(p), µ = f(q) and
ν = f(1). To prove f ∈ F2(S) is to prove that there exist elements fp and fq in F1(S)
such that Rpf = fpf and Rqf = fqf . In fact it is enough to take
fp(q
mpn) = [f(q)−1f(1)]r[f(p)f(1)−1]1−r
and
fq(q
mpn) = [f(p)f(q)2f(1)−3]r[f(q)f(1)−1]1−r,
where r = m − n. Our discussion preceding the theorem reveals that both fp and fq are
elements of F1(S), therefore f ∈ F2(S).
Conversely, let f ∈ F2(S). To show f ∈ A is to show that for all m, n ∈ Z+ ∪ {0},
(∗) f(qmpn) = f(p)(r
2
−r)/2f(q)(r
2+r)/2f(1)1−r2
, where r = m − n.
Since f ∈ F2, there exists fq ∈ F1(S) such that Rqf = fqf . Therefore (from the above
Lemma)
(I) fq(p)fq(q) = fq(1)2.
On the other hand, fq(1) = f(q)f(1)−1, and also f(1) = f(pq) = Rqf(p) = fq(p)f(p),
thus fq(p) = f(p)−1f(1). Hence it follows from (I) that
(II) fq(q) = f(p)f(q)2f(1)−3.
Fix n ∈ N, then by using induction on m and the fact that f(qm+1pn) = Rqf(qmpn) =
fq(q
mpn)f(qmpn) = fq(q)
m−nfq(1)1−(m−n)f(qmpn), we deduce from (II) and (∗) that
f(qm+1pn) = f(p)(s
2
−s)/2f(q)(s
2+s)/2f(1)1−s2
, where s = (m + 1) − n.
The theorem is now established by induction. �
Corollary 4.6. Let S be the bicyclic semigroup generated by 1, p and q, where 1 is the
identity and pq = 1 �= qp. If either p or q is an idempotent, then W (S) = Wi(S) = C,
for all i.
194 A. JABBARI AND H. R. E. VISHKI
Proof. It is enough to show that F1(S) = T. To this end, let f ∈ F1(S), and assume
that p2 = p. Then there exist λp ∈ T such that Rpf = λpf , hence f(p) = f(p2) =
Rpf(p) = λpf(p) and so λp = 1. That is, Rpf = f and f(p) = f(1). It follows from
Lemma 4.4 that f(q) = f(1). Now the first part of the above theorem implies that for
all m, n ∈ Z+ ∪ {0}, f(qmpn) = f(q)m−nf(1)1−(m−n) = f(1). Hence f = f(1) ∈ T. The
proof for the case where q is an idempotent is similar. �
Remarks. (i) By the results 3.5 and 3.7, for every abelian semitopological semigroup S,
Wk and also W lie between SAP and D. It would be desirable to study the structure
of the (right topological abelian group) compactifications SWk and SW . In particular,
it would be more desirable if one could investigate the size of the topological centres of
SWk and SW . The latter problem is of particular interest among some authors, (the
interested reader is referred to [3, 2, 5, 6, 10]). For an elegant characterization of the
topological centre of the largest compactification of a locally compact group one may
refer to [8] and also [10].
(ii) In [11, Theorem 2.13] and [5, Corollary 3.3.3], (by using different methods) it is
proved that all elements of the Weyl algebra W (Z, +) are uniquely ergodic. One may
seek the same result for W (S), where S is an arbitrary semitopological semigroup.
(iii) It would be quite interesting to find a general formula for Fk(S) in Theorem 4.5.
Acknowledgments. The first author would like to thank Professor Anthony Lau, at
the University of Alberta, for his encouragement and support through his NSERC grant
A7679. The remarks from the kind referee, especially the one to add a non-trivial non-
group example which led to Example (d), are gratefully acknowledged. A research grant
from Ferdowsi University of Mashhad is also acknowledged.
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Department of Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad
91775, Iran
E-mail address: shahzadeh@math.um.ac.ir
Department of Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad
91775, Iran; Centre of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi
University of Mashhad, Iran
E-mail address: vishki@ferdowsi.um.ac.ir
Received 03/03/2008; Revised 20.10.2008
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