A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if, for every Banach space $X$, every bilinear map $\unicode[STIX]{x1D711}:A\times A\rightarrow X$ satisfying $\unicode[STIX]{x1D711}(a,b)=0$ whenever $a$, $b\in A$ are such that $ab+ba=0$, is of the form $\unicode[STIX]{x1D711}(a,b)=\unicode[STIX]{x1D70E}(ab+ba)$ for some continuous linear map $\unicode[STIX]{x1D70E}$. We show that all $C^{\ast }$-algebras and all group algebras $L^{1}(G)$ of amenable locally compact groups have this property and also discuss some applications.

We consider the fractional critical problem $A_{s}u=K(x)u^{(n+2s)/(n-2s)},u>0$ in $\unicode[STIX]{x1D6FA},u=0$ on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$, where $A_{s},s\in (0,1)$, is the fractional Laplace operator and $K$ is a given function on a bounded domain $\unicode[STIX]{x1D6FA}$ of $\mathbb{R}^{n},n\geq 2$. This is based on A. Bahri’s theory of critical points at infinity in Bahri [Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182 (Longman Scientific & Technical, Harlow, 1989)]. We prove Bahri’s estimates in the fractional setting and we provide existence theorems for the problem when $K$ is close to 1.

We classify tame block algebras of Hecke algebras of classical type over an algebraically closed field of characteristic not equal to two.

We discuss an alternative approach to Fréchet derivatives on Banach spaces inspired by a characterisation of derivatives due to Carathéodory. The approach allows many questions of differentiability to be reduced to questions of continuity. We demonstrate how that simplifies the theory of differentiation, including the rules of differentiation and the Schwarz lemma on the symmetry of second-order derivatives. We also provide a short proof of the differentiable dependence of fixed points in the Banach fixed point theorem.

We carry Sprindžuk’s classification of the complex numbers to the field $\mathbb{Q}_{p}$ of $p$-adic numbers. We establish several estimates for the $p$-adic distance between $p$-adic roots of integer polynomials, which we apply to show that almost all $p$-adic numbers, with respect to the Haar measure, are $p$-adic $\tilde{S}$-numbers of order 1.

First, we generalize the definition of a locally compact topology given by Paterson and Welch for a sequence of locally compact spaces to the case where the underlying spaces are $T_{1}$ and sober. We then consider a certain semilattice of basic open sets for this topology on the space of all paths on a graph and impose relations motivated by the definitions of graph C*-algebra in order to recover the boundary path space of a graph. This is done using techniques of pointless topology. Finally, we generalize the results to the case of topological graphs.

For positive integers $n$ and $k$, let $r_{k}(n)$ denote the number of representations of $n$ as a sum of $k$ squares, where representations with different orders and different signs are counted as distinct. For a given positive integer $m$, by means of some properties of binomial coefficients, we derive some infinite families of congruences for $r_{k}(n)$ modulo $2^{m}$. Furthermore, in view of these arithmetic properties of $r_{k}(n)$, we establish many infinite families of congruences for the overpartition function and the overpartition pair function.

This paper continues the investigation of the structure of uncountable groups whose large subgroups are normal. Moreover, we describe the behaviour of uncountable groups in which every large subgroup is close to normal with the only obstruction of a finite section.

Let $G$ be a second countable locally compact Hausdorff topological group and $P$ be a closed subsemigroup of $G$ containing the identity element $e\in G$. Assume that the interior of $P$ is dense in $P$. Let $\unicode[STIX]{x1D6FC}=\{{\unicode[STIX]{x1D6FC}_{x}\}}_{x\in P}$ be a semigroup of unital normal $\ast$-endomorphisms of a von Neumann algebra $M$ with separable predual satisfying a natural measurability hypothesis. We show that $\unicode[STIX]{x1D6FC}$ is an $E_{0}$-semigroup over $P$ on $M$.