Weakly almost periodic functionals on the Fourier algebra

We consider various algebras of functions on a semitopological left-group S = X × G S = X \times G , the direct product of a left-zero semigroup X and a group G. In §1 we examine various analogues to the theorem of Eberlein that a weakly almost periodic function on a locally compact abelian group is uniformly continuous. Several appealing conjectures are shown by example to be false. In the second section we look at compactifications of products S × T S \times T of semitopological semigroups with right identity and left identity, respectively. We show that the almost periodic compactification of the product is the product of the almost periodic compactifications, thus generalizing a result of deLeeuw and Glicksberg. The weakly almost periodic compactification of the product is not the product of the weakly almost periodic compactifications except in restrictive circumstances; for instance, when T is a compact group. Finally, as an application, we define and study analytic weakly almost periodic functions and derive the theorem, analogous to a classical theorem about almost periodic functions, that an analytic function which is weakly almost periodic on a single line is analytic weakly almost periodic on a whole strip.

Fourier–Stieltjes algebras and measure algebras on compact right topological 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.

The canonical mapping on the product of a LCA group with its dual is shown to extend uniquely in a homomorphic and analytic way to the product of the respective complexifications.

Orthogonal ℓ1-sets and extreme non-Arens regularity of preduals of von Neumann 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.

In this note we point out two properties (Theorems 1 and 2) of the group algebra of the direct product of a compact group and a locally compact abelian group and indicate briefly how these properties enable one to extend to these group algebras theorems which are known in the compact and abelian cases. Our interest in these properties is two-fold; they establish the applicability of known results in [4]1 to the group algebras in question, and they provide an interesting example of a type of B-algebra, a sort of Kroliecker product algebra, which may very well deserve independenlt systematic study. The reader is referred to results of Kaplansky [1 ] and Segal [2 ] of which this note is essentially a continuation. Let C be a compact group, A be a locally compact abelian group, and G = CXA be their direct product group. We are interested in the group algebras L1(C), L1(A), and L1(G). In particular, we ask how the latter is related to the first two. We will show (1) how the B-algebra L1(G) is formed out of the B-algebras L1(C) and L1(A) and (2) how the structure space of L1(G) is formed from the structure spaces of L1(C) and L1(A). DEFINITION 1. The complex algebra R is the Kronecker product of the complex algebras R1 and R2 (written R = R1 XR2) if there exists a map (xl, X2)>X1 X X2 of (R1, R2) into R such that

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.

In this paper the structure of all open semigroups in any compact topological group is determined. We can then extend this result to a classification of the angular semigroups of any abelian topological group H which contains a compact open subgroup K. Coupling this with the results for Euclidean n-space, we obtain an essentially complete description of the angular semigroups of an arbitrary locally compact group. For compact groups, the result is remarkably simple.

We study the existence of many nonclosed pure subgroups of nondiscrete locally compact abelian groups. It is shown that every nondiscrete locally compact abelian group has uncountably many nonclosed pure subgroups. This in particular solves a question of Armacost. It is also shown that, if G G is a nondiscrete locally compact abelian group and if either G G is a compact group or the torsion part t ( G ) t\left ( G \right ) of G G is nonopen, then G G has 2 c {2^c} proper dense pure subgroups, where c c denotes the power of the continuum. This in particular gives a partial answer to another question of Armacost.

Fourier–Stieltjes Algebras of Locally Compact Groupoids

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.

Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group and 1-injectivity of as an operator -module. In particular, a locally compact group G is amenable if and only if its group von Neumann algebra VN(G) is 1-injective as an operator module over the Fourier algebra A(G). As an application, we provide a decomposability result for completely bounded -module maps on , and give a simpliûed proof that amenable discrete quantum groups have co-amenable compact duals, which avoids the use of modular theory and the Powers-Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.

In this paper, we introduce new notions of $I$-biflatness and $I$-biprojectivity, for a Banach algebra $A$, where $I$ is a closed ideal of $A$. We show that $M(G)$ is $L^{1}(G)$-biprojective ($I$-biflat) if and only if $G$ is a compact group (an amenable group), respectively. Also, we show that, for a non-zero ideal $I$, if the Fourier algebra $A(G)$ is $I$-biprojective, then $G$ is a discrete group. Some examples are given to show the differences between these new notions and the classical ones.

In this chapter we introduce shift-invariant algebras, the main objects of this book. These are uniform algebras on a compact connected group G, consisting of continuous functions on G, whose spectrum is contained in a semigroup S of the dual group Ĝ. If Ĝ is a subgroup of ℝ, and S ⊂ ℝ+, then the maximal ideal space of the corresponding shift-invariant algebra is the G-disc, or, big disc over G. In this chapter we describe two important models of shift-invariant algebras, namely, by the means of almost periodic functions on ℝ, and by the means of H∞-functions on the unit circle. The set of automorphisms, and the peak groups of shift-invariant algebras are also characterized. Extensions on G-discs and groups of several classical theorems of Complex Analysis, such as Radó’s theorem for null-sets and the Riemann theorem for removable singularities of analytic functions, are stated and proved. It is shown that these extensions hold for some semigroups, while in general they fail. In principle we state all results for general shift-invariant algebras AS, though they apply automatically to the particular cases of algebras APS of almost periodic functions, and of H S ∞ -algebras. Asymptotically almost periodic functions combine the properties of classical almost periodic functions on ℝ, and of continuous functions on ℝ that vanish at infinity.