Given a Polish group G, let E(G)$E(G)$ be the right coset equivalence relation G^\omega /c(G)$G^\omega /c(G)$, where c(G)$c(G)$ is the group of all convergent sequences in G. We first established two results:

1. (1) Let G,H$G,H$ be two Polish groups. If H is TSI but G is not, then E(G)\not \le _BE(H)$E(G)\not \le _BE(H)$.

2. (2) Let G be a Polish group. Then the following are equivalent: (a) G is TSI non-archimedean; (b)E(G)\leq _B E_0^\omega $E(G)\leq _B E_0^\omega$; and (c) E(G)\leq _B {\mathbb {R}}^\omega /c_0$E(G)\leq _B {\mathbb {R}}^\omega /c_0$. In particular, E(G)\sim _B E_0^\omega $E(G)\sim _B E_0^\omega$ iff G is TSI uncountable non-archimedean.

A critical theorem presented in this article is as follows: Let G be a TSI Polish group, and let H be a closed subgroup of the product of a sequence of TSI strongly NSS Polish groups. If E(G)\le _BE(H)$E(G)\le _BE(H)$, then there exists a continuous homomorphism S:G_0\rightarrow H$S:G_0\rightarrow H$ such that \ker (S)$\ker (S)$ is non-archimedean, where G_0$G_0$ is the connected component of the identity of G. The converse holds if G is connected, S(G)$S(G)$ is closed in H, and the interval [0,1]$[0,1]$ can be embedded into H.

As its applications, we prove several Rigid theorems for TSI Lie groups, locally compact Polish groups, separable Banach spaces, and separable Fréchet spaces, respectively.

We consider two questions on the geometry of Lipschitz-free p$p$-spaces \mathcal {F}_p$\mathcal {F}_p$, where 0< p\leq 1$0< p\leq 1$, over subsets of finite-dimensional vector spaces. We solve an open problem and show that if (\mathcal {M}, \rho )$(\mathcal {M}, \rho )$ is an infinite doubling metric space (e.g. an infinite subset of an Euclidean space), then \mathcal {F}_p (\mathcal {M}, \rho ^\alpha )\simeq \ell _p$\mathcal {F}_p (\mathcal {M}, \rho ^\alpha )\simeq \ell _p$ for every \alpha \in (0,\,1)$\alpha \in (0,\,1)$ and 0< p\leq 1$0< p\leq 1$. An upper bound on the Banach–Mazur distance between the spaces \mathcal {F}_p ([0, 1]^d, |\cdot |^\alpha )$\mathcal {F}_p ([0, 1]^d, |\cdot |^\alpha )$ and \ell _p$\ell _p$ is given. Moreover, we tackle a question due to Albiac et al. [4] and expound the role of p$p$, d$d$ for the Lipschitz constant of a canonical, locally coordinatewise affine retraction from (K, |\cdot |_1)$(K, |\cdot |_1)$, where K=\cup _{Q\in \mathcal {R}} Q$K=\cup _{Q\in \mathcal {R}} Q$ is a union of a collection \emptyset \neq \mathcal {R} \subseteq \{ Rw + R[0,\,1]^d: w\in \mathbb {Z}^d\}$\emptyset \neq \mathcal {R} \subseteq \{ Rw + R[0,\,1]^d: w\in \mathbb {Z}^d\}$ of cubes in \mathbb {R}^d$\mathbb {R}^d$ with side length R>0$R>0$, into the Lipschitz-free p$p$-space \mathcal {F}_p (V, |\cdot |_1)$\mathcal {F}_p (V, |\cdot |_1)$ over their vertices.

We use new methods, specific for non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a p$p$-Banach space for 0< p<1$0< p<1$ are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy space H_p({\mathbb {D}})$H_p({\mathbb {D}})$ for 0< p<1$0< p<1$ are democratic while, in contrast, no quasi-greedy basis of H_p({\mathbb {D}}^d)$H_p({\mathbb {D}}^d)$ for d\ge 2$d\ge 2$ is, solving thus a problem that was raised in [7]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided.

Let \mathcal {M}$\mathcal {M}$ be a semifinite von Nemann algebra equipped with an increasing filtration (\mathcal {M}_n)_{n\geq 1}$(\mathcal {M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of \mathcal {M}$\mathcal {M}$ . For 0<p <\infty $0<p <\infty$ , let \mathsf {h}_p^c(\mathcal {M})$\mathsf {h}_p^c(\mathcal {M})$ denote the noncommutative column conditioned martingale Hardy space and \mathsf {bmo}^c(\mathcal {M})$\mathsf {bmo}^c(\mathcal {M})$ denote the column “little” martingale BMO space associated with the filtration (\mathcal {M}_n)_{n\geq 1}$(\mathcal {M}_n)_{n\geq 1}$ .

We prove the following real interpolation identity: if 0<p <\infty $0<p <\infty$ and 0<\theta <1$0<\theta <1$ , then for 1/r=(1-\theta )/p$1/r=(1-\theta )/p$ ,

\begin{align*} \big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{r}^c(\mathcal{M}), \end{align*} $$\begin{align*} \big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{r}^c(\mathcal{M}), \end{align*}$$

For the case of complex interpolation, we obtain that if 0<p<q<\infty $0<p<q<\infty$ and 0<\theta <1$0<\theta <1$ , then for 1/r =(1-\theta )/p +\theta /q$1/r =(1-\theta )/p +\theta /q$ ,

\begin{align*} \big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta}=\mathsf{h}_{r}^c(\mathcal{M}) \end{align*} $$\begin{align*} \big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta}=\mathsf{h}_{r}^c(\mathcal{M}) \end{align*}$$

These extend previously known results from p\geq 1$p\geq 1$ to the full range 0<p<\infty $0<p<\infty$ . Other related spaces such as spaces of adapted sequences and Junge’s noncommutative conditioned L_p$L_p$ -spaces are also shown to form interpolation scale for the full range 0<p<\infty $0<p<\infty$ when either the real method or the complex method is used. Our method of proof is based on a new algebraic atomic decomposition for Orlicz space version of Junge’s noncommutative conditioned L_p$L_p$ -spaces.

We apply these results to derive various inequalities for martingales in noncommutative symmetric quasi-Banach spaces.

The paper alluded to in the title contains the following striking result: Let I$I$ be the unit interval and \Delta$\Delta$ the Cantor set. If X$X$ is a quasi Banach space containing no copy of c_{0}$c_{0}$ which is isomorphic to a closed subspace of a space with a basis and C(I,\,X)$C(I,\,X)$ is linearly homeomorphic to C(\Delta ,\, X)$C(\Delta ,\, X)$, then X$X$ is locally convex, i.e., a Banach space. We will show that Kalton result is sharp by exhibiting non-locally convex quasi Banach spaces X$X$ with a basis for which C(I,\,X)$C(I,\,X)$ and C(\Delta ,\, X)$C(\Delta ,\, X)$ are isomorphic. Our examples are rather specific and actually, in all cases, X$X$ is isomorphic to C(K,\,X)$C(K,\,X)$ if K$K$ is a metric compactum of finite covering dimension.

We prove that if M$M$ is a \text{JBW}^{\ast }$\text{JBW}^{\ast }$-triple and not a Cartan factor of rank two, then M$M$ satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of M$M$ onto the unit sphere of another real Banach space Y$Y$ extends to a surjective real linear isometry from M$M$ onto Y$Y$.

The General Curve Lemma is a tool of Infinite-Dimensional Analysis that enables refined studies of differentiability properties of maps between real locally convex spaces to be made. In this article, we generalize the General Curve Lemma in two ways. First, we remove the condition of local convexity in the real case. Second, we adapt the lemma to the case of curves in topological vector spaces over ultrametric fields.

We consider order and type properties of Marcinkiewicz and Lorentz function spaces. We show that if $0<p<1$, a $p$-normable quasi-Banach space is natural (i.e. embeds into a $q$-convex quasi-Banach lattice for some $q>0$) if and only if it is finitely representable in the space $L_{p,\infty}.$ We also show in particular that the weak Lorentz space $L_{1,\infty}$ do not have type $1$, while a non-normable Lorentz space $L_{1,p}$ has type $1$. We present also criteria for upper $r$-estimate and $r$-convexity of Marcinkiewicz spaces.

We exhibit examples of F$F$-spaces with trivial dual which are isomorphic to its quotient by a line, thus solving a problem in Rolewicz's Metric Linear Spaces.