Let G be a locally compact group, and consider the weakly almost periodic functionals on M(G), the measure algebra of G, denoted by WAP(M(G)). This is a C*-subalgebra of the commutative C*-algebra M(G)*, and so has character space, say $K_{WAP}$. In this paper, we investigate properties of $K_{WAP}$. We present a short proof that $K_{WAP}$ can naturally be turned into a semigroup whose product is separately continuous; at the Banach algebra level, this product is simply the natural one induced by the Arens products. This...

We introduce and study the notions of w*-approximate Connes amenability and pseudo-Connes amenability for dual Banach algebras. We prove that the dual Banach sequence algebra ℓ¹ is not w*-approximately Connes amenable. We show that in general the concepts of pseudo-Connes amenability and Connes amenability are distinct. Moreover the relations between these new notions are also discussed.

We show that the dual version of our factorization [J. Funct. Anal. 261 (2011)] of contractive homomorphisms φ: L¹(F) → M(G) between group/measure algebras fails to hold in the dual, Fourier/Fourier-Stieltjes algebra, setting. We characterize the contractive w*-w* continuous homomorphisms between measure algebras and (reduced) Fourier-Stieltjes algebras. We consider the problem of describing all contractive homomorphisms φ: L¹(F) → L¹(G).

Let $X$ be a hypergroup. In this paper, we define a locally convex topology $\beta$ on $L\left(X\right)$ such that ${(L\left(X\right),\beta )}^{*}$ with the strong topology can be identified with a Banach subspace of $L{\left(X\right)}^{*}$. We prove that if $X$ has a Haar measure, then the dual to this subspace is $L_C{\left(X\right)}^{**}{=cl\{F\in L\left(X\right)}^{**};F$ has compact carrier}. Moreover, we study the operators on $L{\left(X\right)}^{*}$ and $L_0^{\infty }\left(X\right)$ which commute with translations and convolutions. We prove, among other things, that if $wap\left(L\right(X\left)\right)$ is left stationary, then there is a weakly compact operator $T$ on $L{\left(X\right)}^{*}$ which commutes with convolutions if and...