We establish an extension of the Banach–Stone theorem to a class of isomorphisms more general than isometries in a noncompact framework. Some applications are given. In particular, we give a canonical representation of some (not necessarily linear) operators between products of function spaces. Our results are established for an abstract class of function spaces included in the space of all continuous and bounded functions defined on a complete metric space.

If $k$ is a positive integer, a group $G$ is said to have the $FE_{k}$ -property if for each element $g$ of $G$ there exists a normal subgroup of finite index $X(g)$ such that the subgroup $\langle g,x\rangle$ is nilpotent of class at most $k$ for all $x\in X(g)$ . Thus, $FE_{1}$ -groups are precisely those groups with finite conjugacy classes ($FC$ -groups) and the aim of this paper is to extend properties of $FC$ -groups to the case of groups with the $FE_{k}$ -property for $k>1$ . The class of $FE_{k}$ -groups contains the relevant subclass $FE_{k}^{\ast }$ , consisting of all groups $G$ for which to every element $g$ there corresponds a normal subgroup of finite index $Y(g)$ such that $\langle g,U\rangle$ is nilpotent of class at most $k$ , whenever $U$ is a nilpotent subgroup of class at most $k$ of $Y(g)$ .

We show that the family of all holomorphic functions $f$ in a domain $D$ satisfying

$$\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|}(z)\leq C\quad \text{for all }z\in D\end{eqnarray}$$

$k$

$C>0$

$\unicode[STIX]{x1D6FC}>1$

$k\geq 2$

$$\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|^{\unicode[STIX]{x1D6FC}}}(z)\leq C\quad \text{for all }z\in D\end{eqnarray}$$

We show that closed $\mathbb{S}\text{ol}^{3}\times \mathbb{E}^{1}$ -manifolds are Seifert fibred, with general fibre the torus, and base one of the flat 2-orbifolds $T,Kb,\mathbb{A},\mathbb{M}b,S(2,2,2,2),P(2,2)$ or $\mathbb{D}(2,2)$ , and outline how such manifolds may be classified.

We show that any regular pseudocomplemented Kleene algebra defined on an algebraic lattice is isomorphic to a rough set Kleene algebra determined by a tolerance induced by an irredundant covering.

We prove a general large-sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral gap property. In such a context the point is to exhibit a strong uniform expansion property for a suitable family of Cayley graphs on quotients. In our combinatorial approach, this is replaced by a result of Alon–Roichman about expanding properties of random Cayley graphs. Applying the general setting we show, for instance, that with high probability (in a strong explicit sense) random coloured subsets of integers contain monochromatic (nonempty) subsets summing to $0$ , and that a random colouring of the edges of a complete graph contains a monochromatic triangle.

We shall introduce the notions of strong Morita equivalence for unital inclusions of unital $C^{\ast }$ -algebras and conditional expectations from an equivalence bimodule onto its closed subspace with respect to conditional expectations from unital $C^{\ast }$ -algebras onto their unital $C^{\ast }$ -subalgebras. Also, we shall study their basic properties.