Lobachevskii Journal of Mathematics Vol.9, 2001, 29–35 \begin{matrix}c \\ ◯ \end{matrix}S. Grigoryan, T. Tonev

S. Grigoryan, T. Tonev LINEAR MULTIPLICATIVE FUNCTIONALS OF

ALGEBRAS OF S-ANALYTIC FUNCTIONS ON GROUPS

(submitted by D.Kh.Mustari)

Abstract. Let S be a subsemigroup of a semigroup Σ that generates a group G. We find conditions that assure extendability of linear multiplicative functionals of the algebra A_{S} of S-analytic functions on G with spectra in S to linear multiplicative functionals of the corresponding algebra A_{Σ}. Conditions for existence of dense homeomorphic embeddings of the upper half plane in the maximal ideal space of the algebra of almost periodic functions on ℝ with spectrum in S, and of the open unit disc in the maximal ideal space of certain subalgebras of H^{∞} are obtained as corollaries.

1. Algebras of S-analytic functions on groups. Let Δ = {|z| ≤ 1} be the unit disc, and S^{1} = {|z| = 1} is the unit circle in the complex plane ℂ. Given an additive semigroup S, a semicharacter of S is any multiplicative function φ : S → \overline{Δ}, i.e. |φ(a)| ≤ 1, and φ(a + b) = φ(a)φ(b) for every a, b ∈ S. Denote by H(S) the set of all semicharacters on S, and let H_{+}(S) be the set of all positive semicharacters on S with values are in (0, 1]. The set H_{+}(S) is a subsemigroup of H(S) under the pointwise multiplication (ϱ_{1}ϱ_{2})(a) = ϱ_{1}(a)ϱ_{2}(a), ϱ_{1}, ϱ_{2} ∈ H(S), a ∈ S.

Let S be subsemigroup of a semigroup Σ. In this paper we study the extendability property on Σ of semicharacters in H(S) to semicharacters of Σ. This problem is naturally connected with the extendability property of positive semicharacters of S to positive semicharacters of Σ. Closely related to it is also the extendability property of linear multiplicative functionals of uniform algebras A_{S} of generalized analytic functions on groups G with spectra in subsemigroups S of the dual group Γ = Ĝ (S-analytic functions) to multiplicative functionals on the corresponding algebra A_{Σ}. We obtain necessary and sufficient conditions for these extendability properties to hold. As corollaries we find conditions for existence of dense homeomorphic embeddings of the upper half plane Π in the maximal ideal space of algebras of S-analytic functions on groups G, S ⊂ Ĝ, and together, of algebras of analytic almost periodic functions on ℝ with spectra in S, and obtain a natural corona theorem for sertain particular subalgebras of H^{∞}.

The first author is partially supported by the Russian FBR grant, 99-01-00441. The authors acknowledge with gratitude the support for this paper by an NSF International Cooperative Research Grant between USA and Russia. Thanks are due also to the Mathematisches Forschungsinstitut Oberwolfach, Germany, and the International Banach Center in Warsaw, Poland, where part of the work on this paper was performed, for their hospitality.

Let Γ be an additive abelian group equipped by the discrete topology. A continuous function γ : Γ → S^{1} is called a character of Γ, if γ(a + b) = γ(a)γ(b), a, b ∈ Γ. Denote by G the dual group Γ̂, i.e. the set of all characters of Γ. By Pontrjagin duality theorem, G is a compact group under the multiplication (γ_{1}γ_{2})(a) = γ_{1}(a)γ_{2}(a), γ_{1}, γ_{2} ∈ G, a ∈ Γ. The inverse γ^{−1} of γ ∈ G is the function γ^{−1}(a) = \overline{γ(a)}, a ∈ Γ. Every a ∈ Γ generates a continuous character χ^{a} on G by χ^{a}(g) = g(a). Moreover, all characters χ of G are of the form χ = χ^{a}, a ∈ Γ. The G-disc is the cone Δ_{G} over G with generatrix [0, 1), i.e. Δ_{G} = ([0, 1) × G)/({0} × G).

The space C(G) of all continuous functions on G is a Banach space under the uniform norm ∥f∥ = sup_{g∈G} |f(γ)|. By the Stone-Weierstrass theorem the set of linear combinations of characters of G is dense in C(G). Every function f ∈ C(G) has a formal Fourier series

f ∼ ∑_{a∈Γ} c_{a}^{f}χ^{a}

with Fourier coefficients

c_{a}^{f} = ∫_{G} f(g)\overline{χ}^{a}(g)dσ(g),

where σ is the Haar measure on G. The set sp(f) = {a ∈ Γ : c_{a}^{f} ≠ 0} is called the spectrum of f.

Let S be a semigroup of Γ containing 0 and generating Γ, i.e. Γ = S + (−S). The set A_{S} = {f ∈ C(G) : sp(f) ⊂ S} of S-analytic functions on G (in the sense of Arens-Singer) is an uniform algebra on G, i.e. A_{S} is a closed subalgebra of C(G) containing the constants and separating the points of G. Algebras of this type were studied extensively in the recent past (e.g. [1], [2], [5]). As shown by Arens and Singer [1], the maximal ideal space M_{S} of A_{S} is homeomorphic to the set H(S) of all semicharacters φ of S with φ(0) = 1.

Algebras of type A_{S} have been stidied primarily in the case when Γ is a semigroup of ℝ. If Γ is dense in ℝ, and S = Γ_{+} = Γ ∩ [0, ∞), then the set M_{S} = H(S) is homeomorphic to the closed G-disc (the big disc) \overline{Δ}_{G} = ([0, 1] × G)/(\{0\} × G) over G (e.g. Gamelin [2]). As shown in [5], the same is true if the semigroup S is of type S = Γ ∩ [ν, ∞) ∪ {0}, ν < 0. The algebras A_{S} in these cases are called big disc algebras, and their elements, the S-analytic functions on G, are known as generalized analytic functions on G. It is well known that under the above mentioned conditions, there is a dense homomorphic and homeomorphic embedding of ℝ into G, which extends to a dense homeomorphic embedding of the upper half plane Π = {z ∈ ℂ : Im z ≥ 0\} into \overline{Δ}_{G} (e.g. Gamelin [2]). The converse is also true, i.e. if there is a dense homeomorphic embedding of the upper half plane Π into M_{S}, such that its restriction on ℝ is a dense homomorphic and homeomorphic embedding of ℝ into G, then M_{S} = H(S) = \overline{Δ}_{G}. If Γ is not dense in ℝ, then the set M_{S} = H(S) is homeomorphic to the unit disc (i.e. the S^{1}-disc) \overline{Δ} = \overline{Δ}_{S^{1}} ⊂ ℂ.

2. Extendability property of semicharacters and linear multiplicative functionals. Let S be an additive subsemigroup which contains 0, and has the cancellation property, i.e. a = c whenever a + b = c + b for some b ∈ S. In this case S is a semigroup of some group. Denote by Γ = S − S the group generated by S. Consider a subsemigroup P ⊃ S of Γ such that P + (−P) = Γ, and P ∩ (−P) = {0}. P generates a partial order on Γ. Namely, b ≻ a if and only if b − a ∈ P. Denote by Γ_{+} = Γ ∩ P the set of all P -non-negative elements of Γ. Clearly, Γ_{+} ⊃ S.

Proposition 2.1. A positive semicharacter ρ ∈ H_{+}(S) is extendable on Γ_{+} as positive semicharacter in H_{+}(Γ_{+}) if and only if ρ is monotone decreasing with respect to the order generated by P.

Proof: Let first ρ ∈ H_{+}(S). If b ∈ Γ_{+}, then b = a − c for some a, c ∈ S, a ≻ c. The function ρ̃(b) = ρ(a)/ρ(c) is a natural extension of ρ to an element in H_{+}(Γ_{+}). Clearly, ρ̃(b) ≤ 1 if and only if ρ(a) ≤ ρ(c), i.e. if and only if ρ is a decreasing function. □

Observe, that actually every semicharacter ρ ∈ H(S) that is non-negative and extendable on Γ_{+} to a non-negative semicharacter of H(Γ_{+}) is necessarily monotone decreasing. Indeed, let ρ(c) = 0 for some c ∈ S, and let a ≻ c, i.e. b = a − c ∈ P. If ρ extends on Γ_{+}, then ρ(a) = ρ(b)ρ(c) = 0, since b + c = a. Together with the argument from the proof of Proposition 2.1, this implies that ρ is decreasing on S.

Theorem 2.2. The elements φ ∈ H(S), φ ≠ 0 can be extended as elements of H(Γ_{+}) if and only if every ρ ∈ H_{+}(S) is monotone decreasing on S with respect to the order generated by P.

Proof: Let φ ∈ H(S), φ ≠ 0. Define the following function on Γ = S ∪ (−S):

γ(a) = \{ \begin{matrix}amp; φ(a)/|φ(a)| for a ∈ S \\ amp; \overline{γ(−a)} = 1/γ(a) for a ∈ (−S) \end{matrix}

It is clear that |γ| = 1 and that γ is multiplicative on both semigroups S and (−S). γ can be extended naturally on the group Γ as a character of Γ by

γ̃(b) = \frac{γ(a)}{γ(c)} whenever b = a − c ∈ Γ, a, c ∈ S.

The extended function is again of modulus 1. In addition, γ̃ is multiplicative on Γ. Indeed, if b, b_{1} ∈ Γ and b = a − c, b_{1} = a_{1} − c_{1}, where a, a_{1}, c, c_{1} ∈ S, then

γ̃(b + b_{1}) = γ̃((a + a_{1}) − (c + c_{1})) = \frac{γ(a + a_{1})}{γ(c + c_{1})}

= \frac{γ(a)γ(a_{1})}{γ(c)γ(c_{1})} = (\frac{γ(a)}{γ(c)})(\frac{γ(a_{1})}{γ(c_{1})}) = γ̃(b)γ̃(b_{1}).

Thus, the extended function γ̃ is a character of the group Γ ⊃ Γ_{+}, as claimed.

The function ρ(a) = |φ(a)| is a positive semicharacter of S. Indeed, it is straightforward to see that 0 < ρ(a) ≤ 1, and ρ(a + b) = ρ(a)ρ(b), a, b ∈ S. Therefore, ρ ∈ H_{+}(S).

Thus, if φ ≠ 0, then φ = |φ|\frac{φ}{|φ|} = ργ extends on Γ_{+} (as an element of H(Γ_{+})) if and only if ρ = |φ| does. By Proposition 2.1 this happens if and only if ρ is monotone decreasing on S with respect to the order generated by P. □

Let ρ_{0}^{S} ∈ H_{+}(S) be the positive semicharacter on S, defined by: ρ_{0}^{S}(0) = 1, ρ_{0}^{S}(a) = 0 for every a ∈ S \ {0}. Clearly, ρ_{0}^{S} is extendable to the positive semicharacter ρ_{0}^{Γ_{+}} on Γ_{+}. In the case when Γ ⊂ ℝ, and P = ℝ_{+}, the only vanishing semicharacter on Γ_{+} is ρ_{0}^{Γ_{+}}. Consequently, if S is an additive subsemigroup of ℝ_{+} containing 0, then a vanishing semicharacter ρ ∈ H(S) is extendable on Γ_{+} if and only if ρ = ρ_{0}^{S}.

Corollary 2.3. Let S be an additive subsemigroup of ℝ_{+} containing 0, and let the group Γ = S −S ⊂ ℝ generated by S is dense in ℝ. The maximal ideal space M_{S} of the algebra A_{S} of S-analytic functions on G = Γ̂ with spectrum in S is homeomorphic to the G-disc \overline{Δ}_{G} if and only if all positive semicharacters on S are monotone decreasing.

Proof: By Theorem 2.2 and the remark from the above, the given conditions are necessary and sufficient for every semicharacter φ ∈ H(S) to be uniquely extendable as a semicharacter in H(Γ_{+}), which yields M_{S} ≅ H(S) ≅ H(Γ_{+}) ≅ M_{Γ_{+}} ≅ \overline{Δ}_{G}. □

Consider be the set AP (ℝ) of almost periodic functions on ℝ. Every f ∈ AP (ℝ) has a formal Fourier series of type

f(t) ∼ ∑_{a∈ℝ} c_{a}^{f}e^{ita},

the spectrum of f being the set sp(f) = {a ∈ ℝ : c_{a}^{f} ≠ 0}. For a given subset F ⊂ ℝ denote by AP_{F} (ℝ) the set of almost periodic functions with spectrum in F, i.e. AP_{F} (ℝ) = {f ∈ AP (ℝ) : sp(f) ⊂ F }. Let S be an additive subsemigroup of ℝ, and Γ = S − S is the subgroup of ℝ generated by S. In this case AP_{S}(ℝ) is an uniform algebra that is isometrically isomorphic to the algebra A_{S} of S-analytic functions on G = Γ̂ with spectrum in S. Corollary 2.3 now implies the following

Proposition 2.4. Let S be an additive subsemigroup of ℝ_{+} containing 0, and let the group Γ = S −S ⊂ ℝ generated by S is dense in ℝ. The maximal ideal space M_{S} of the algebra AP_{S}(ℝ) of almost periodic functions with spectrum in S is homeomorphic to the G-disc \overline{Δ}_{G}, G = Γ̂, if and only if all positive semicharacters on S are monotone decreasing.

Since the upper half plane Π = {z ∈ ℂ : Im z ≥ 0} is dense in the maximal ideal space M_{S} of the algebra A_{S} (and, together, of AP_{S}(ℝ)) if and only if M_{S} = \overline{Δ}_{G}, we have

Corollary 2.5. Let S be an additive subsemigroup of ℝ_{+} containing 0, and let the group Γ = S − S ⊂ ℝ generated by S is dense in ℝ. The upper half plane Π is dense in the maximal ideal space M_{S} of the algebra AP_{S}(ℝ) of almost periodic functions with spectrum in S if and only if all positive semicharacters on S are monotone decreasing.

For every ρ ∈ H_{+}(S) the function θ = ln ρ is an additive negative function on S. It is straightforward to see that the necessary and sufficient condition in Corollary 2.5 is equivalent to the following one: Every additive positive function θ on S is of type θ(a) = y_{θ}a for some y_{θ} ∈ [0, ∞), or θ(a) = ∞, for a ≠ 0. This is exactly the condition for dense embedding of Π proposed previously by Boettcher [2].

For an a ∈ S let φ_{a} ∈ H^{∞} be the singular function

φ_{a}(z) = e^{ia \frac{1+z}{1−z}}

on the unit disc Δ. Denote by H_{S}^{∞} the Banach algebra on Δ generated by the functions φ_{a}(z), a ∈ S equipped by the sup-norm on Δ. Note that the M&quot;obius map ψ(z) = i\frac{1+z}{1−z} maps the unit disc \overline{Δ} onto the upper half

plane Π. Moreover, e^{ψ(z)} = e^{i \frac{1+z}{1−z}}, and (e^{ψ(z)})^{a} = e^{ia \frac{1+z}{1−z}} for every a ∈ S. Therefore, H_{S}^{∞} is a subalgebra of H^{∞}, which is isometrically isomorphic to the algebra A_{S} of S-analytic functions on G = (S − S) . Corollary 2.5 now implies the following corona theorem for the algebra H_{S}^{∞}.

Proposition 2.6. Let S be an additive subsemigroup of ℝ_{+} containing 0, and let the group Γ = S − S ⊂ ℝ generated by S is dense in ℝ. The unit disc Δ is dense in the maximal ideal space of the algebra H_{S}^{∞} if and only if all positive semicharacters on S are monotone decreasing.

Let P be a semigroup of Γ that generates a partial order on Γ. If S is a semigroup of P such that for every a ∈ Γ_{+} there is an n ∈ ℕ such that na ∈ S, then every nonnegative semicharacter ρ ∈ H(S) can be extended naturally on Γ_{+} as a monotone decreasing semicharacter, namely by ρ̃(a) = [ρ(na)]^{1/n}. Moreover, in this case there are no vanishing on Γ_{+} nonnegative semicharacters in H(Γ_{+}) besides ρ_{0}^{Γ_{+}}. Theorem 2.2 implies the following

Proposition 2.7. If S is a subsemigroup of P such that for every a ∈ Γ_{+} = Γ ∩P there is an n ∈ ℕ such that na ∈ S, then every semicharacter φ ∈ H(S) on S is uniquely extendable on Γ_{+} as a semicharacter in H(Γ_{+}).

Under the hypotheses of Proposition 2.7 the maximal ideal space M_{S} is homeomorphic to M_{Γ_{+}}, i.e. to the G-disc \overline{Δ}_{G}, G = Γ̂. If, in addition, S ⊂ ℝ, then the group Γ is dense in ℝ. Since the upper half-plane Π is dense in M_{S} if and only if M_{S} = \overline{Δ}_{G}, we have the following.

Corollary 2.8. If S is a subsemigroup of ℝ such that for every a ∈ Γ_{+} = Γ ∩ [0, ∞) there is an n ∈ ℕ such that na ∈ S, then the upper half plane Π is dense in the maximal ideal space M_{S} of the algebra AP_{S}(ℝ) of almost periodic functions on ℝ with spectrum in S.

By using similar arguments, one can show that if S is a semigroup in P = ℝ_{+}^{k} = [0, ∞)^{k} ⊂ ℝ^{k}, such that for every (x_{1}, x_{2}, …, x_{k}) ∈ P there is an n ∈ ℕ with (nx_{1}, nx_{2}, …, nx_{k}) ∈ S, then the maximal ideal space M(AP_{S}(ℝ^{k})) of the algebra AP_{S}(ℝ^{k}) of almost periodic functions on ℝ^{k} with spectra in S is homeomorphic to M(AP_{P} (ℝ^{k})). Therefore, the Cartesian product Π ×Π ×⋯×Π ⊂ ℝ^{k} can be embedded densely into M(AP_{S}(ℝ^{k})) = M(AP_{P} (ℝ^{k})). Proposition 2.6 implies the following

Proposition 2.9. If S is a subsemigroup of ℝ such that for every a ∈ ℝ_{+} there is an n ∈ ℕ such that na ∈ S, then the unit disc Δ is dense in the maximal ideal space of the algebra H_{S}^{∞}.

The next proposition shows that for every a ∈ Γ_{+} = Γ ∩ [0, ∞) to exist an n ∈ ℕ such that na ∈ S is necessary and sufficient for the semicharacters on all semigroups Σ with S ⊂ Σ ⊂ ℝ_{+} to be uniquely extendable on Γ_{+} as semicharacters on Γ_{+}.

Proposition 2.10. Let S is a subsemigroup of ℝ. Then H(Σ) = H(Γ_{+}) for every semigroup Σ with S ⊂ Σ ⊂ ℝ_{+} if and only if for every a ∈ Γ_{+} = Γ ∩ [0, ∞) there is an n ∈ ℕ such that na ∈ S.

Proof: In the view of Proposition 2.7, it is enough to show that if all semicharacters on all semigroup Σ with S ⊂ Σ ⊂ ℝ_{+} are uniquely extendable as semicharacters on Γ_{+}, then ℕa ∩ S ≠ Ø for every a ∈ Γ_{+} = Γ ∩ [0, ∞). Suppose that ℕa ∩ S = Ø, and consider the semigroup Σ = ℕa + S ⊂ ℝ_{+}. Note that the set Σ\ℕa is a semigroup ideal of Σ. However, the characteristic function of the set ℕa ⊂ Σ is a semicharacter of Σ without a semicharacter extension on Γ_{+}. □

Note that under the hypotheses of Proposition 2.10. M_{Σ} = M_{Γ_{+}} = \overline{Δ}_{G} for every semigroup Σ with S ⊂ Σ ⊂ ℝ_{+}.

References

1. R. Arens and I. Singer, Generalized analytic functions, Trans. Amer. Math. Soc., 81 (1956), 379-393.

2. A. B&quot;ottcher, On the corona theorem for almost periodic functions, Integral Equations and Operator Theory, 33 (1999), 253-272.

3. T.W. Gamelin, Uniform Algebras, 2nd ed., Chelsea, New York, 1984.

4. S. Grigoryan, T. Ponkrateva, and T. Tonev, Inner Automorphisms of Shift-invariant Algebras on Compact Groups, Proc. of the Armenian Acad. Sci., (2000),

5. T. Tonev, Big-planes, Boundaries and Function Algebras, Elsevier – North-Holland, 1992.

Research Institute of Mathematics and Mechanics, Kazan State University, Universitetskaya Str. 17, Kazan, 420008 Russia

E-mail address: sugrigor@green.kcn.ru

Mathematics Department, University of Montana, Missoula, MT 59812, USA E-mail address: ma tt@selway.umt.edu

Received November 15, 2001