Automorphic distributions are distributions on ${ℝ}^d$, invariant under the linear action of the group $SL(d,\mathbb{Z})$. Combs are characterized by the additional requirement of being measures supported in ${\mathbb{Z}}^d$: their decomposition into homogeneous components involves the family ${\left({𝔈}_{i\lambda }^d\right)}_{\lambda \in ℝ}$, of Eisenstein distributions, and the coefficients of the decomposition are given as Dirichlet series $𝒟\left(s\right)$. Functional equations of the usual (Hecke) kind relative to $𝒟\left(s\right)$ turn out to be equivalent to the invariance of the comb under some modification...

Let G be a locally compact abelian group, M(G) the convolution measure algebra, and X a Banach M(G)-module under the module multiplication μ ∘ x, μ ∈ M(G), x ∈ X. We show that if X is an essential L¹(G)-module, then $\sigma \left(T_{\mu }\right)=\overline{\mu ̂\left(sp\right(X\left)\right)}$ for each measure μ in reg(M(G)), where $T_{\mu }$ denotes the operator in B(X) defined by $T_{\mu }x=\mu \circ x$, σ(·) the usual spectrum in B(X), sp(X) the hull in L¹(G) of the ideal $I_X=f\in L¹\left(G\right)|T_f=0$, μ̂ the Fourier-Stieltjes transform of μ, and reg(M(G)) the largest closed regular subalgebra of M(G); reg(M(G)) contains all...

Let A be a commutative Banach algebra with Gelfand space ∆ (A). Denote by Aut (A) the group of all continuous automorphisms of A. Consider a σ(A,∆(A))-continuous group representation α:G → Aut(A) of a locally compact abelian group G by automorphisms of A. For each a ∈ A and φ ∈ ∆(A), the function ${\phi }_a\left(t\right):=\phi \left({\alpha }_{t}a\right)$ t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum ${\sigma }_w*\left({\phi }_a\right)$ is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define ${Ʌ}_{\phi }^a$ to be the union of all...

The splitting problem is studied for short exact sequences consisting of countable projective limits of DFN-spaces (*) 0 → F → X → G → 0, where F or G are isomorphic to the space of distributions D'. It is proved that every sequence (*) splits for F ≃ D' iff G is a subspace of D' and that, for ultrabornological F, every sequence (*) splits for G ≃ D' iff F is a quotient of D'

It is proved that the class of separable Rosenthal compacta on the Cantor set having a uniformly bounded dense sequence of continuous functions is strongly bounded.

We discuss a strong version of the Dunford-Pettis property, earlier named (DP*) property, which is shared by both ℓ₁ and ${\ell }_{\infty }$. It is equivalent to the Dunford-Pettis property plus the fact that every quotient map onto c₀ is completely continuous. Other weak sequential continuity results on polynomials and analytic mappings related to the (DP*) property are shown.

There are necessary conditions for a point x from the unit sphere to be a denting point of the unit ball of Orlicz spaces equipped with the Orlicz norm generated by arbitrary Orlicz functions. In contrast to results in [12, 17, 16], we present also examples of Orlicz spaces in which strongly extreme points of the unit ball are not denting points.

Jordan $H^{*}$-pairs appear, in a natural way, in the study of Lie $H^{*}$-triple systems ([3]). Indeed, it is shown in [4, Th. 3.1] that the problem of the classification of Lie $H^{*}$-triple systems is reduced to prove the existence of certain $L^{*}$-algebra envelopes, and it is also shown in [3] that we can associate topologically simple nonquadratic Jordan $H^{*}$-pairs to a wide class of Lie $H^{*}$-triple systems and then the above envelopes can be obtained from a suitable classification, in terms of associative $H^{*}$-pairs, of...