Blaschke inductive limits of uniform algebras

Uniform algebras that can be expressed as inductive limits of standard simpler algebras are of particular interest. For instance, some G-disc algebras are inductive limits of sequences of disc algebras, connected with finite Blaschke products, called also Blaschke inductive limit algebras. Here we show, among others, that only G-disc algebras, and the spaces \( H^\infty (\mathbb{D}_G ) \) with G ⊂ ℚ, can be expressed as limits of countable inductive sequences of algebras of type \( A(\mathbb{D}) \) and H∞ correspondingly. We study also inductive limits of sequences of spaces of type H∞ and prove corresponding corona theorems. Further, we establish relationships between Bourgain algebras of coordinate algebras, and the Bourgain algebra of their inductive limit, and also between H∞-spaces on \( \mathbb{D} \), as coordinate algebras in an inductive sequence, and their inductive limit. While we state all results for general shift-invariant algebras AS, they apply automatically to the particular cases of algebras APS of almost periodic functions, and of H S ∞ -algebras.

Let Hp(m), 0 < p ^ oo, be the Hardy spaces on a quotient K of the Bohr group. In this paper we completely determine the isometries of Hp{m), p Φ 2, onto itself. Our result is a generalization of a recent work of Muhly who determined the isometries of Hp(m) onto itself under the assumption that the dual group of K is countable, and it may be regarded as a partial answer to a question posed by Muhly. 1* Introduction* Many results have been obtained concerning isometries of Hardy spaces in the theory of uniform algebras. The most fundamental result in this direction is due to de Leeuw, Rudin, and Wermer [2], which states that an automorphism of the classical Hardy space H°°{T) is induced via composition with the unit circle T of a fractional linear transformation of the unit disc onto itself. Their work was carried on independent of Nagasawa [13], who also described the isometries of ίί oo(Tr) onto itself. On the other hand, Arens [1] completely determined the automorphisms of the uniform algebra of analytic functions on a compact abelian group K whose dual group Γ is archimedean ordered (cf. [11]). This result was extended by Muhly [11] to the uniform algebra of analytic functions induced by a flow which has no periodic orbits. Moreover Muhly [12] has recently given, among other things, the following interesting generalization of this result of Arens to the case of isometries of Hardy spaces Hp(m), p Φ 2, on K: Under the assumption that Γ is countable, every isometry of Hp(m), p Φ 2, is induced via composition with an affine map of K such that the adjoint of the additive factor of this map preserves the order of Γ. The purpose of this paper is to remove the assumption on Γ. This result provide a partial positive answer to the following question posed by Muhly in [12; §5]: Is it possible to describe the isometries of ergodic Hardy spaces

In Pontrjagin's theory of duality for compact abelian groups, the following theorem is known : Let G be a compact abelian group, G* the dual group. Then the topological dimension of G, in the sense of Lebesgue, is equal to the rank of discrete abelian group G*. It was Prof. T. Tannaka who has called my attention to the lack of corresponding theorem in non-commutative case. I intend to give, in this note, a theorem of this kind in the following form: THEOREM A. Let G be an arbitrary compact group, G the aggregate of continuous finite dimensional representstions of G, C[ G ] the algebra over the complex numbers C generated by the coefficients of representations in G, i.e., the representative ring of G in the sense of C. Chevalley^. Then the topological dimension of G, in the sense of Lebesgue, is equal to the transcendental degree of C[G] over C.

Let G denote a locally compact abelian topological group (an l.c.a.group) with dual group C\ We will denote by A7(G) the Banach algebra of bounded regular Borel measures on G under convolution multiplication and by L(G) the algebra of bounded measures absolutely continuous with respect to Haar measure on G (for discussions of these Banach algebras cf.[1], [2], and [5]).In this paper we shall be concerned with closed subalgebras 501 of M(G) with the following two properties:(1) if p e 50c and v is absolutely continuous with respect to p, then vs;(2) the maximal ideal space of 501 is G~, where the Gelfand transform p~ of P e 501 is given by p~(x)=J x dp for ^Ga; i.e., the Gelfand transform coincides with the Fourier-Stieltjes transform on 50.Any closed subalgebra of M(G) satisfying (1) will be called an L-subalgebra of M(G).It is well known thatL(G) satisfies ( 1) and (2) (cf.[5, Chapter 1]).In Theorem 1 we show that any L-subalgebra 50c of M(G), with L(G)<=50cc(L(G))1,2) also satisfies (2), where (L(G))112 is the intersection of all maximal ideals of M(G) containing L(G).We conjecture that the converse is also true; i.e., if 501 satisfies (1) and (2) then L(G)c 50c c(L(G))1/2.In Theorem 2 we prove that this is true provided G contains no copy of 7?" for > 1.The problem arises in the following way : in [6] we define the concept of abstract convolution measure algebra and prove that any such algebra 501, provided it is commutative and semisimple, may be represented as an L-subalgebra of M(S), where S is a compact topological semigroup called the structure semigroup of 5D.M(G) and L(G) are convolution measure algebras as is any L-subalgebra of the measure algebra on a semigroup.The map p-> ps which embeds 50c in M(S) is an isometry as well as an algebraic isomorphism and it preserves the order theoretic properties of 50 as a space of measures.The maximal ideal space of 50c may be identified as S~, the set of all semicharacters of S, where the Gelfand transform of p e 50c is given by p~(f) = $fdp.sfor fe S~ (a semicharacter of S is a continuous homomorphism of S into the unit disc which is not identically zero).Under pointwise multiplication S~ is a semigroup provided 50c has an approximate identity.

Projective free algebras of continuous functions on compact abelian groups

Let G be a compact abelian group and let Ĝ be its dual group. We shall denote by M(G) the set of all bounded complex valued Radon measures on G and by M(E) the elements of M(G) with support in a compact subset E of G.

Let G be a compact abelian group with dual group Γ. For 1 ≤ p &lt; ∞, denote by Ap(G) the space of integrable functions on G whose Fourier transforms belong to lp(Γ). We investigate several problems related to multipliers from Ap(G) to Aq(G). In particular, we prove that (Ap, Ap) ⊊ (Aq, Aq). For the circle group, we characterise permutation invariant multipliers from Ap to Ar for 1 ≤ r ≤ 2.

Let G be a compact abelian group and E a subset of its dual group Γ. A function ƒ ∈ L1(G) is called an E-function if for all γ ∉ E wheredx is the Haar measure on G. A trigonometric polynomial that is also an E-function is called an E-polynomial.

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

Given a compact group G and r ∈ N, let sr(G) denote the least upper bound of real ϵ > 0 such that for every measurable r-coloring of G, there exists a monochrome symmetric subset of measure ≥ ϵ. A subset A ⊆ G is symmetric if there exists g ∈ G such that gA −1 g = A. We give a general picture of asymptotic behaviour of the function sr(G) for compact Abelian groups.

LetG be a locally compact abelian topological group containing a suitable sequence of compact open subgroups and let Γ be its dual group. LetK (α,p, q; G) andK (α,p, q; Γ) denote the so-called Herz spaces onG and Γ, respectively. In this paper we shall prove that for 1<p≤2 and 0≤α<1/p′=1−1/p, the Fourier transform mapsK (α,p, p; G) continuously intoK (−α,p′, 2; Γ). The proof requires two results that are of independent interest: an extension of the Hausdorff-Young inequality to certain weightedL p-spaces onG and a Littlewood-Paley theorem for certain weightedL p-spaces onG.

We study Pontryagin reflexivity in the class of precompact topological Abelian groups. We find reflexive groups among precompact not pseudocompact and among pseudocompact not compact groups. Making use of Martin's axiom we give an example of a reflexive countably compact not compact Abelian group. We also prove that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.

We prove an arithmetic removal result for all compact abelian groups, generalizing a finitary removal result of Kral’, Serra, and the third author. To this end, we consider infinite measurable hypergraphs that are invariant under certain group actions, and for these hypergraphs we prove a symmetry-preserving removal lemma, which extends a finitary result of the same name by the second author. We deduce our arithmetic removal result by applying this lemma to a specific type of invariant measurable hypergraph. As a direct consequence of our removal result, we obtain the following generalization of Szemeredi’s theorem: for any compact abelian group G, any measurable set \({A \subseteq G}\) with Haar probability \({\mu(A) \geq \alpha > 0}\) satisfies \({\int_{G}\int_{G}1_{A}(x)1_{A}(x+r)...1_{A}(x+(k-1)r)\, {\rm d} \mu(x)\, {\rm d} \mu(r) \geq c}\), where the constant \({c=c(\alpha, k) > 0}\) is valid uniformly for all G. This result is shown to hold more generally for any translationinvariant system of r linear equations given by an integer matrix with coprime \({r \times r}\) minors.

We introduce a vector-valued version of a uniform algebra, called the vector-valued function space over a uniform algebra. The diameter two properties of the vector-valued function space over a uniform algebra on an infinite compact Hausdorff space are investigated. Every nonempty relatively weakly open subset of the unit ball of a vector-valued function space $$A(K, (X, \tau ))$$ over an infinite dimensional uniform algebra has diameter two, where $$\tau $$ is a locally convex Hausdorff topology on a Banach space X compatible to a dual pair. Under the assumption of X equipped with the norm topology being uniformly convex and the additional condition that $$A\otimes X\subset A(K, X)$$ , it is shown that Daugavet points and $$\Delta $$ -points on A(K, X) over a uniform algebra A are the same, and they are characterized by the norm-attainment at a limit point of the Shilov boundary of A. In addition, a sufficient condition for the convex diametral local diameter two property of A(K, X) is also provided. Similar results also hold for an infinite dimensional uniform algebra.

In this note, we give a negative answer to a question raised by A. Arosio in 1974. It concerns the multiplicativity of the inductive limit topology associated to a family of normed algebras.