Saturation on locally compact abelian groups: An extended theorem

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.

In this paper we prove, for signed or complex Radon measures on completely regular spaces, the analogue of Prokhorov's criterion on the existence of the projective limit of a compatible system of measures. Because of loss of mass under projections this cannot be reduced to the case of positive measures. Countable projective limits are, as in the case of positive measures, particularly simple, the sole condition now being the boundedness of the total variations. It is shown, with the help of the martingale convergence theorem, that the densities of these complex measures with respect to their variations, converge in an appropriate sense. This work is part of an extended project on the mathematical theory of path integrals.

Let G be a compact abelian group with dual group Γ. For 1 ≤ p < ∞, 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.

The spaces of test functions we are going to use are vector spaces of smooth (i.e., sufficiently often continuously differentiable) functions on open nonempty subsets Ω ⊆ ℝ n equipped with a ‘natural’ topology. Accordingly we start with a general method to equip a vector space V with a topology such that the vector space operations of addition and scalar multiplication become continuous, i.e., such that $$ \begin{array}{*{20}c} {_{M\,:\,\,\mathbb{K}\, \times \,V\, \to \,V,}^{A\,:\,\,V\, \times \,V\, \to \,V,} } & {_{M(\lambda ,\,x)\, = \,\lambda x,\,}^{A(x,\,y)\, = \,x\, + \,y,} } & {_{\lambda \, \in \,\mathbb{K},\,\,\,x\, \in \,V}^{x,\,y\, \in \,V} } \end{array} $$ become continuous functions for this topology. This can be done in several different but equivalent ways. The way we describe has the advantage of being the most natural one for the spaces of test functions we want to construct. A vector space V which is equipped with a topology T such that the functions A} and M are continuous is called a topological vector space, usually abbreviated as TVS. The test function spaces used in distribution theory are concrete examples of topological vector spaces where, however, the topology has the additional property that every point has a neighborhood basis consisting of (absolutely) convex sets. These are called locally convex topological vector spaces, abbreviated as LCVTVS.

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

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 give some characterizations of the class of character-pure locally compact abelian groups. As an application we supplement a result of Venkataraman characterizing the class of locally compact abelian groups which are topologically pure in their Bohr compactifications. We also study the class of locally compact abelian groups which are pure closed in their Bohr compactifications.

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

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.

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.

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.

The purpose of this work is to study various geometric properties of Banach spaces, with particular emphasis on their stability with respect to the formation of (infinite) absolute sums and Köthe-Bochner spaces of vector- valued functions. The first three chapters are devoted to the class of acs (``alternatively convex or smooth'') Banach spaces and their relatives, the luacs and uacs spaces. These notions were introduced by Kadets et al. in 2000 (in connection with the so called anti-Daugavet property) and form common generalisations of the usual convexity and smoothness properties of Banach spaces. In the first chapter, these notions (as well as some other related properties introduced by the author) are studied from a general point of view. For example, some equivalent characterisations are established and results on duality and quotient spaces are discussed. In the second chapter, various results on (mainly infinite) absolute sums of acs spaces and their relatives are proved, while in the third chapter, acs properties of Köthe-Bochner spaces of vector-valued functions are discussed. The fourth chapter is devoted to the so called Opial property and its variants, as well as some other geometric notions (WORTH property and Garcia-Falset coefficient) of Banach spaces, which are all related to the fixed point property for nonexpansive mappings. Some stability results for these properties in certain infinite absolute sums are established. Furthemore, some properties analogous to the Opial property in Lebesgue-Bochner spaces and Ces`aro spaces of vector-valued functions are discussed. The fifth chapter concerns Banach spaces with the so called ``ball generated property'' (BGP). The stability of this property with respect to certain sums of two spaces is proved. The last chapter deals with generalised lush (GL) spaces, which were introduced by Huang et al. in 2013 in connection with the Mazur-Ulam extension problem. Some results on GL-spaces are established. In particular, every M-ideal in a GL-space is again a GL-space.

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 consider and study Blaschke inductive limit algebrasA(b), defined as inductive limits of disc algebras A(D) linked by a sequence of finite Blaschke products. It is well known that big G‐disc algebras AG over compact abelian groups G with ordered duals can be expressed as Blaschke inductive limit algebras. Any Blaschke inductive limit algebra A(b) is a maximal and Dirichlet uniform algebra. Its Shilov boundary ∂A(b) is a compact abelian group with dual group that is a subgroup of ℚ. It is shown that a big G‐disc algebra AG over a group G with ordered dual is a Blaschke inductive limit algebra if and only if . The local structure of the maximal ideal space and the set of one‐point Gleason parts of a Blaschke inductive limit algebra differ drastically from the ones of a big G‐disc algebra. These differences are utilized to construct examples of Blaschke inductive limit algebras that are not big G‐disc algebras. A necessary and sufficient condition for a Blaschke inductive limit algebra to be isometrically isomorphic to a big G‐disc algebra is found. We consider also inductive limits H∞(I) of algebras H∞, linked by a sequence of inner functions, and prove a version of the corona theorem with estimates for it. The algebra H∞(I) generalizes the algebra of bounded hyper‐analytic functions on an open big G‐disc, introduced previously by Tonev.

Projective free algebras of continuous functions on compact abelian groups