Convolution functions on compact Abelian groups

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.

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.

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.

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.

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.

Given a locally compact group G, G L U C is the largest semigroup compactification of G and G * = G L U C ∖ G . We show that (i) for every locally compact compactly generated Abelian group G and for every p , q ∈ G * , the multiplication in G L U C is discontinuous at ( p , q ) , (ii) there is a locally compact σ-compact torsion-free Abelian group G for which, assuming Martin's Axiom, there are p , q ∈ G * such that the multiplication in G L U C is continuous at ( p , q ) , and (iii) it is consistent with ZFC that for every locally compact Abelian group G and for every p , q ∈ G * , the left translation by p in G * is discontinuous at q.

We prove that the category M of abelian groups with a Polish cover introduced in collaboration with Bergfalk and Panagiotopoulos is the left heart of (the derived category of) the quasi-abelian category A of abelian Polish groups in the sense of Beilinson–Bernstein–Deligne and Schneiders. Thus, M is an abelian category containing A as a full subcategory such that the inclusion functor A→M is exact and finitely continuous. Furthermore, M is uniquely characterized up to equivalence by the following universal property: for every abelian category B, a functor A→B is exact and finitely continuous if and only if it extends to an exact and finitely continuous functor M→B. In particular, this provides a description of the left heart of A as a concrete category.We provide similar descriptions of the left heart of a number of categories of algebraic structures endowed with a topology, including: non-Archimedean abelian Polish groups; locally compact abelian Polish groups; totally disconnected locally compact abelian Polish groups; Polish R-modules, for a given Polish group or Polish ring R; and separable Banach spaces and separable Fréchet spaces over a separable complete non-Archimedean valued field.

Let G be a compact abelian group and <p a complex-valued function defined on the dual I\ The main result of this paper is that y is a compact multiplier of type (p, q), 1 -p < oo and 1 <? oo, if and only if it satisfies the following condition : Given >0 there corresponds a finite set XCT such that |2 ayby<p(y)\ <e whenever P=1ayy and Q = 2 byy are trigonometric polynomials satisfying ||.P||pi=l, ||||,'el iq' the conjugate index of q) and by = 0 for y e K. Using the above characterization we obtain the following necessary and sufficient condition for <p to be the Fourier transform of a continuous complex-valued function on G: Given c>0 there corresponds a finite set #<= T such that |2 byy(y)\ <e whenever 0 = 2 byy isa trigonometric polynomial satisfying || II i = 1 and by=0 for y e K.Throughout the paper G is a compact abelian group, <p a complex-valued function defined on the dual F and V(G) (1 ^pf^co) the usual Lebesgue space of index p formed with respect to Haar measure on G. Let M(G) denote the convolution algebra of complex-valued regular measures which are bounded on G, and C(G) the class of all continuous complex-valued functions defined on G.The Fourier transform / of a function feL1(G) is defined by f(y)=\aAx)(-x,y)dx (yeT)and the Fourier-Stieltjes transform fi of a measure p e M(G) by p(y)=^(-x,y)dp(x) (yeT).The function <p is said to be a multiplier of type (/?, q) if given fe LP(G) there corresponds age LQ(G) such that <pf=g.Now any (p, q) multiplier induces a bounded linear operator from W(G) into L"(G), T9, where (Tvfy = <pf and Tv commutes with translation.Conversely for p<co, there corresponds to any such bounded linear operator T mapping LV(G) into L"(G), a unique multiplier 93 of type (p, q) such that T= T0 (see [5, pp.249-250]).We say that <p is a compact multiplier if T9 is a compact operator.Let M^(T) denote the set of all multipliers of type (p, q) and mqp(Y) the set of all 93 e M(T) which are compact.Then M^(Y) is a Banach space where the norm || ||(P_", of the multiplier 95 is defined to be the norm of the multiplier operator T0.Let 3(G)

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

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

Let be an arbitrary locally compact Abelian group and let be the space of all continuous complex-valued functions on . A closed linear subspace is referred to as an invariant subspace if it is invariant with respect to the shifts , . By definition, an invariant subspace admits strict spectral synthesis if coincides with the closure in of the linear span of all characters of belonging to . We say that strict spectral synthesis holds in the space on if every invariant subspace admits strict spectral synthesis. An element of a topological group is said to be compact if is contained in some compact subgroup of . A group is said to be element-wise compact if all elements of are compact. The main result of the paper is the proof of the fact that strict spectral synthesis holds in for a locally compact Abelian group if and only if is element-wise compact. Bibliography: 14 titles.

In this paper, two theorems about the compactness of almost invariant operators on homogeneous Banach spaces of distributions (in the sense of Feichtinger [11]) defined on a locally compact abelian group are proved. Our theorems generalize the corresponding results of K. de Leeuw [3] and Tewari and Madan [18] for operators on homogeneous Banach spaces on the circle group and Segal algebras on a compact abelian groups, respectively.

0. Introduction. In [4], [5], [6] the structures A1 0 ) A2, where A1 and A2 are Banach algebras, are discussed. Actually, a proper parallel to the algebraic situation is: three commutative Banach algebras A, B, C, where A and B are C-bimodules (in the sense described below), and some Banach-algebraic version of A 03cB. In the first part of the following, we shall give a general discussion of A ?cB, a natural Banach-algebraic version of the (algebraic) tensor product of A and B over C. Thereafter, we shall discuss a special case in which A, B, C are group algebras of locally compact abelian groups. Finally we shall handle the even more special problem in which A and B are group algebras of locally compact abelian groups and C is a group algebra of a compact abelian group. In the last two parts, particularly the last, a connection will be established between the theory of tensor products and the theory of group extensions. In this connection, the author thanks L. Auslander for indicating the point in question. The author is also indebted to Dr. B. Natzitz for pointing out errors and the need for clarifying several items of an earlier draft of this paper.

Generalised sk-Spline Interpolation on Compact Abelian Groups

Extensions of the Nehari theorem and of the Sarason commutation theorem are given for compact abelian groups whose dual have a complete linear order compatible with the group structure. As a special case a version of the classical interpolation theorem due to Caratheodory — Fejer is obtained.