We provide a deep investigation of the notions of - and ${}^{\infty }$-hypoellipticity for partial differential operators with constant Colombeau coefficients. This involves generalized polynomials and fundamental solutions in the dual of a Colombeau algebra. Sufficient conditions and necessary conditions for - and ${}^{\infty }$-hypoellipticity are given

Let $ℛ$ be a semiprime ring with unity $e$ and $\phi$, $\varphi$ be automorphisms of $ℛ$. In this paper it is shown that if $ℛ$ satisfies $$2𝒟\left(x^n\right)=𝒟\left(x^{n-1}\right)\phi \left(x\right)+\varphi \left(x^{n-1}\right)𝒟\left(x\right)+𝒟\left(x\right)\phi \left(x^{n-1}\right)+\varphi \left(x\right)𝒟\left(x^{n-1}\right)$$ for all $x\in ℛ$ and some fixed integer $n\ge 2$, then $𝒟$ is an ($\phi$, $\varphi$)-derivation. Moreover, this result makes it possible to prove that if $ℛ$ admits an additive mappings $𝒟,𝒢:ℛ\to ℛ$ satisfying the relations $$\begin{array}{c}2𝒟\left(x^n\right)=𝒟\left(x^{n-1}\right)\phi \left(x\right)+\varphi \left(x^{n-1}\right)𝒢\left(x\right)+𝒢\left(x\right)\phi \left(x^{n-1}\right)+\varphi \left(x\right)𝒢\left(x^{n-1}\right),\\ 2𝒢\left(x^n\right)=𝒢\left(x^{n-1}\right)\phi \left(x\right)+\varphi \left(x^{n-1}\right)𝒟\left(x\right)+𝒟\left(x\right)\phi \left(x^{n-1}\right)+\varphi \left(x\right)𝒟\left(x^{n-1}\right),\end{array}$$ for all $x\in ℛ$ and some fixed integer $n\ge 2$, then $𝒟$ and $𝒢$ are ($\phi$, $\varphi$)derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.

The main result of the note is a characterization of 1-amenability of Banach algebras of approximable operators for a class of Banach spaces with 1-unconditional bases in terms of a new basis property. It is also shown that amenability and symmetric amenability are equivalent concepts for Banach algebras of approximable operators, and that a type of Banach space that was long suspected to lack property 𝔸 has in fact the property. Some further ideas on the problem of whether or not amenability (in...

Let 1 ≤ p < ∞, $={\left(Xₙ\right)}_{n\in \mathbb{N}}$ be a sequence of Banach spaces and $l_p\left(\right)$ the coresponding vector valued sequence space. Let $={\left(Xₙ\right)}_{n\in \mathbb{N}}$, $={\left(Yₙ\right)}_{n\in \mathbb{N}}$ be two sequences of Banach spaces, $={\left(Vₙ\right)}_{n\in \mathbb{N}}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator $M_{}:l_p\left(\right)\to l_q\left(\right)$ by $M_{}\left({\left(xₙ\right)}_{n\in \mathbb{N}}\right):={\left(Vₙ\left(xₙ\right)\right)}_{n\in \mathbb{N}}$. We give necessary and sufficient conditions for $M_{}$ to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞.

We show that under some conditions, 3-weak amenability of the (2n)th dual of a Banach algebra A for some n ≥ 1 implies 3-weak amenability of A.