ON SOME DISTAL FUNCTIONS

Wojciech Chojnacki

1. Introduction

Let S be a discrete Abelian semigroup, l^{∞}(S) the space of all complex bounded functions on S endowed with the topology of pointwise convergence, and L(l^{∞}(S)) the space of all linear operators in l^{∞}(S) equipped with the corresponding strong operator topology. For each s ∈ S, let T_{s} be the operator of translation by s, defined by

(T_{s}f)(t) = f(s + t) (f ∈ l^{∞}(S), t ∈ S).

Let Σ(S) be the closure of {T_{s}: s ∈ S} in L(l^{∞}(S)).

An element f of l^{∞}(S) is called a distal function on S if the following conditions are satisfied:

(i) f = σ(f) for some σ in Σ(S); (ii) if στ_{1}(f) = στ_{2}(f) for σ, τ_{1}, τ_{2} in Σ(S), then

τ_{1}(f) = τ_{2}(f).

Namioka [4] established a criterion for distality with the use of which he proved the following generalization of a result of Knapp [3]:

If S is a subsemigroup of the group of integers and p is a real polynomial, then the function f on S defined by

f(n) = e^{ip(n)} (n ∈ S)

is distal.

In this note, we establish one more theorem from which the latter result may easily be deduced. Its proof will be straightforward and will make no appeal to rather involved arguments of Knapp and Namioka.

2. Preliminaries

If A is a subset of the domain of a function f, we denote by f|_{A} the restriction of f to A.

A complex function with values of unit modulus will be called unitary.

Let f be a unitary function on S. For each s ∈ S, put

δ_{s}f = f̄· T_{s}f

and, for any s_{1}, …, s_{n} ∈ S, set inductively

δ_{s_{1}...s_{n}}f = δ_{s_{n}}(δ_{s_{1}...s_{n−1}}f).

A Banach mean on l^{∞}(S) is a linear functional m on l^{∞}(S), continuous under the supremum norm on l^{∞}(S), satisfying the following conditions:

(a) ∥m∥ = 1 = m(1);

(b) m(T_{s}f) = m(f) for each f ∈ l^{∞}(S) and each s ∈ S.

The existence of the Banach mean on l^{∞}(S) is ensured by a theorem of Day [1]. Let G be a discrete Abelian group. An element of l^{∞}(G) is called an almost

periodic function on G if the set {T_{g}f: g ∈ G} is relatively compact under the supremum norm on l^{∞}(G). The set AP(G) of all almost periodic cost functions on G is a subalgebra of l^{∞}(G), closed in norm and under conjugation of functions. If f is an almost periodic function on G and m is a Banach mean on l^{∞}(G), then m(f) does not depend on the particular choice of m, and setting

(f|g) = m(fḡ) (f, g ∈ AP(G))

defines a scalar product on AP(G). We shall denote by |||·||| the norm derived from that scalar product.

3. A distality condition

We shall find it convenient to reformulate the distality condition given in the Introduction.

Let I be the identity operator in l^{∞}(S) and let

Σ^{∗}(S) = Σ(S) ∪ {I}.

Proposition. An element f of l^{∞}(S) is a distal function on S if and only if the following condition is satisfied:

(∗) if στ_{1}(f) = στ_{2}(f) for σ in Σ(S) and τ_{1}, τ_{2} in Σ^{∗}(S), then

τ_{1}(f) = τ_{2}(f).

Proof. That (i) and (ii) imply (∗) is evident. Also the implication (∗) ⇒ (ii) is clear. To establish (∗) ⇒ (i), note first that Σ(S) is a right topological semigroup, that is, for each τ ∈ Σ(S), the mapping

Σ(S) ∋ σ ↦ στ ∈ Σ(S)

is continuous. Moreover, by Tikhonov’s theorem, Σ(S) is compact. Now a lemma of Ellis [2] ensures the existence of an idempotent ϵ in Σ(S). The identity ϵ(f) = ϵ(ϵ(f)) jointly with (∗) implies that f = ϵ(f). The proof is complete.

4. The result

Our major result is the following

Theorem. Let S be a subsemigroup of a discrete Abelian group G. Suppose that f is a unitary function on G such that, for some non-negative n and all s_{1}, …, s_{n} ∈ S, the function δ_{s_{1}...s_{n}}f is almost periodic (when n = 0, we assume that the latter function coincides with f). Then the restriction of f to S is distal.

Proof. Since the restriction to a subgroup of an almost periodic function defined on a group is almost periodic, without loss of generality we may assume that G coincides with the group

S − S = {g ∈ G: g = s_{1} − s_{2} for s_{1}, s_{2} ∈ S}.

Suppose that

(1)  στ_{1}(f|_{S}) = στ_{2}(f|_{S})

for σ ∈ Σ(S) and τ_{1}, τ_{2} ∈ Σ^{∗}(S). Let μ be a Banach mean on l^{∞}(S). Define a linear functional m on l^{∞}(G) by setting

m(h) = μ(σ(h|_{S})) (h ∈ l^{∞}(G)).

We claim that m is a Banach mean on l^{∞}(G).

Indeed, we clearly have ∥m∥ = 1 = m(1). If g = s_{1} − s_{2} with s_{1} and s_{2} in S, and h ∈ l^{∞}(G), then

m(T_{g}h) = μ(T_{s_{2}}(σ(T_{g}h|_{S}))) = μ(T_{s_{1}}(σ(h|_{S}))) = m(h),

which establishes the claim.

Given s_{1}, …, s_{n} ∈ S, we have

(2) στ_{1}(δ_{s_{1}...s_{n}}f|_{S}) = δ_{s_{1}...s_{n}}στ_{1}(f|_{S}) = δ_{s_{1}...s_{n}}στ_{2}(f|_{S}) = στ_{2}(δ_{s_{1}...s_{n}}f|_{S}).

A moment’s reflection shows that there exist τ̄_{1} and τ̄_{2} in Σ^{∗}(G) such that

τ̄_{i}(δ_{s_{1}...s_{n}}f)|_{S} = τ_{i}(δ_{s_{1}...s_{n}}f|_{S}) (i = 1, 2).

The functions τ̄_{i}(δ_{s_{1}...s_{n}}f) (i = 1, 2) are clearly unitary almost periodic, so the function

u = τ̄_{1}(δ_{s_{1}...s_{n}}f)[τ̄_{2}(δ_{s_{1}...s_{n}}f)]^{−1}

is also unitary almost periodic. In view of (2),

(3)  m(u) = 1.

Rewriting the latter identity in the form

(u|1) = |||u||| |||1|||,

we see that u is constant. Now it results from (3) that actually u = 1, whence

(4)  δ_{s_{1}...s_{n}}τ_{1}(f|_{S}) = δ_{s_{1}...s_{n}}τ_{2}(f|_{S}).

Put

v = τ_{1}(f|_{S})[τ_{2}(f|_{S})]^{−1}.

The function v is unitary and, by (4), we have

δ_{s_{1}...s_{n}}v = 1 for any s_{1}, …, s_{n} ∈ S.

Since any unitary function e on S with δ_{s}e = 1 for each s ∈ S is constant, the function δ_{s_{1}...s_{n−1}}v is constant for any s_{1}, …, s_{n−1} ∈ S. By (1), we have σ(v) = 1, therefore

δ_{s_{1}...s_{n−1}}v = σ(δ_{s_{1}...s_{n−1}}v) = δ_{s_{1}...s_{n−1}}σ(v) = 1.

The repeated use of this argument shows that v = 1, whence

τ_{1}(f|_{S}) = τ_{2}(f|_{S}).

To complete the proof, it suffices now to apply the Proposition.

References

1. M. M. Day, Ergodic theorems for abelian semigroups, Trans. Amer. Math. Soc. 51 (1942), 399–412.

2. R. Ellis, Distal transformation groups, Pacific J. Math. 8 (1958), 401–405.

3. A. W. Knapp, Distal functions on groups, Trans. Amer. Math. Soc. 128 (1967), 1–40.

4. I. Namioka, Affine flows and distal points, Math. Z. 184 (1983), 259–269.

Institute of Applied Mathematics and Mechanics University of Warsaw Pałac Kultury i Nauki, IX p. 00-901 Warszawa, Poland