On the structure of spectral sets

Fourier–Stieltjes Algebras of Locally Compact Groupoids

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

The theory of weakly almost periodic functional on the Fourier algebra is herein developed. It is the extension of the theory of weakly almost periodic functions on locally compact abelian groups to the duals of compact groups. The complete direct product of a countable collection of nontrivial compact groups furnishes an important example for some of the constructions.

Let G be a locally compact group and let A(G) be its Fourier algebra. We find sufficient conditions for A(G) to be approximately/pseudo-amenable without being amenable. We also study A(G) for its operator approximate/pseudo-amenability.

Tiling sets and spectral sets over finite fields

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

A dual Banach algebra is a Banach algebra that is also a dual Banach space such that multiplication is separately weak\(^*\) continuous. Examples of dual Banach algebras are, among others, von Neumann algebras, the measure algebra M(G). and the Fourier–Stieltjes algebra B(G) of a locally compact group G, or the algebras \(\mathcal {B}(E)\) of all bounded linear operators on a reflexive Banach space E.

Fourier–Stieltjes algebra of a topological group

Let A be an F-algebra and S(A*) be the metric semi-group of normal states of the W*-algebra A*. We show that S(A*) determines A, when A* is of type I. In particular, when A is either the Fourier algebra A(G) or the Fourier–Stieltjes algebra B(G) of a type I locally compact group G, the metric semi-group S(A*) determines G up to opposition. A remark for the general case will also be given.

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.

If G is a locally compact group, denote its Fourier-Stieltjes algebra by B(G) and its Fourier algebra by Λ(G). If G is compact, then B(G) = Λ(G) and σ(B(G)), the spectrum o! B(G), is G. If G is not compact then σ(B(G)) contains partial isometries and projections different from e, the identity of G. More generally, σ(B(G)) is closed under operations that commute with representing and the taking of tensor products. It is shown that σ(B(G)) contains a smallest positive element, zF; and that g G G Cσ(B(G))h+z Fg G σ(B(G))zF is an epimorphism of G into G, the almost periodic compactification of G. A structure theorem is given for the closed, bi-translation, invariant subspaces of B(G). In so doing we introduce the concepts of inverse Fourier transform localized at TΓ, and the standardization of TΓ, where TΓ is a continuous, unitary representation of G.

We define the spine A *(G) of the Fourier–Stieltjes algebra B (G) of a locally compact group G. This algebra encodes information about much of the fine structure of B (G), particularly information about certain homomorphisms and idempotents. We show that A *(G) is graded over a certain semi-lattice, that of non-quotient locally precompact topologies on G. We compute the spine's spectrum G*, which admits a semi-group structure. We discuss homomorphisms from A *(G) to B (H) where H is another locally compact group; and we show that A *(H) contains the image of every completely bounded homomorphism from the Fourier algebra A (H) of any amenable group G. We also show that A *(G) contains all of the idempotents in B (G). Finally, we compute examples for vector groups, abelian lattices, minimally almost periodic groups and the (ax + b)-group; and we explore the complexity of A *(G) for the discrete rational numbers and free groups.

Let $A(G)$ denote the Fourier algebra of the locally compact group $G$. We show that for a large class of groups an ideal $I$ in $A(G)$ has a $\Delta$-weak bounded approximate identity if and only if it has a bounded approximate identity.