Let $A$ be a uniformly complete almost $f$-algebra and a natural number $p\in \{3,4,\cdots \}$. Then ${\Pi }_p\left(A\right)=\{a_1\cdots a_p;a_k\in A,k=1,\cdots ,p\}$ is a uniformly complete semiprime $f$-algebra under the ordering and multiplication inherited from $A$ with ${\Sigma }_p\left(A\right)=\{a^p;0\le a\in A\}$ as positive cone.

We show that, given a Banach space X, the Lipschitz-free space over X, denoted by ℱ(X), is isomorphic to ${\left({\sum }_{n=1}^{\infty }ℱ\left(X\right)\right)}_{\ell ₁}$. Some applications are presented, including a nonlinear version of Pełczyński’s decomposition method for Lipschitz-free spaces and the identification up to isomorphism between ℱ(ℝⁿ) and the Lipschitz-free space over any compact metric space which is locally bi-Lipschitz embeddable into ℝⁿ and which contains a subset that is Lipschitz equivalent to the unit ball of ℝⁿ. We also show that ℱ(M)...

Let O₁,...,Oₙ be open sets in C[0,1], the space of real-valued continuous functions on [0,1]. The product O₁ ⋯ Oₙ will in general not be open, and in order to understand when this can happen we study the following problem: given f₁,..., fₙ ∈ C[0,1], when is it true that f₁ ⋯ fₙ lies in the interior of $B_{\epsilon }\left(f₁\right)\cdots B_{\epsilon }\left(fₙ\right)$ for all ε > 0 ? ($B_{\epsilon }$ denotes the closed ball with radius ε and centre f.) The main result of this paper is a characterization in terms of the walk t ↦ γ(t): = (f₁(t),..., fₙ(t)) in ℝⁿ. It has to...

In [AnsMonic, AnsBoolean], we investigated monic multivariate non-commutative orthogonal polynomials, their recursions, states of orthogonality, and corresponding continued fraction expansions. In this note, we collect a number of examples, demonstrating what these general results look like for the most important states on non-commutative polynomials, namely for various product states. In particular, we introduce a notion of a product-type state on polynomials, which covers all the non-commutative...

Soient $A$ une $C^{*}$-algèbre approximativement finie simple avec unité, $G{L}_1\left(A\right)$ le groupe des inversibles et $U_1\left(A\right)$ le groupe des unitaires de $A$. Nous avons défini dans un précédent travail un homomorphisme ${\Delta }_T$, appelé déterminant universel de $A$, de $G{L}_1\left(A\right)$ sur un groupe abélien associé à $A$. Nous montrons ici que, pour qu’un élément $x$ dans $G{L}_1\left(A\right)$ ou dans $U_1\left(A\right)$ soit produit d’un nombre fini de commutateurs, il (faut et il) suffit que $x\in \mathrm{Ker}\left({\Delta }_T\right).$ Ceci permet en particulier d’identifier le noyau de la projection canonique $K_1\left(A\right)\to K_1^{\mathrm{top}}\left(A\right).$ On établit aussi...