We obtain two partial answers to the 3-space problem for isomorphic polyhedrality: (1) every twisted sum of C(alpha), alpha < omega(1), with a separable isomorphically polyhedral space with the BAP, is isomorphically polyhedral. (2) Every twisted sum of c(0)(aleph) and a Banach space having a boundary with property (*) has a boundary with property (*), hence it is isomorphically polyhedral.

The paper studies short exact sequences of Banach modules over the convolution algebra $L_1=L_1(G)$, where $G$ is a compact abelian group. The main tool is the notion of a nonlinear $L_1$-centralizer, which in combination with the Fourier transform, is used to produce sequences of $L_1$-modules $0\rightarrow L_q \rightarrow Z \rightarrow L_p \rightarrow 0$ that are nontrivial as long as the general theory allows it, namely for $p\in (1,\infty], q\in[1,\infty)$. Concrete examples are worked in detail for the circle group, with applications to the Hardy classes, and the Cantor group.

We construct a compact space L and a 1-complemented subspace of the Banach space C(L) which is not isomorphic to a space of continuous functions.

We study in this paper a few remarkable properties of twisted sums Z(kappa, X) of c(0)(kappa) and a Banach space X. We first prove a representation theorem for such twisted sums from which we will obtain, among others, the following: (a) twisted sums of c(0)(kappa) and c(0)(I) are either subspaces of l(infinity)(kappa) or contain a complemented copy of c(0)(kappa(+)); (b) under the hypothesis [p = c], when K is either a suitable Corson compact, a separable Rosenthal compact or a scattered compact of finite height, there is a twisted sum of c(0) and C(K) that is not isomorphic to a space of continuous functions; (c) all twisted sums Z(kappa, X) are isomorphically Lindenstrauss spaces when X is a Lindenstrauss space; (d) all twisted sums Z(kappa, X) are isomorphically polyhedral when X is a polyhedral space with a sigma-discrete boundary, which solves a problem of Castillo and Papini.

The paper studies properties of twisted sums of a Banach space $X$ with $c_0(\kappa)$. We first prove a representation theorem for such twisted sums from which we will obtain, among others, the following: (a) twisted sums of $c_0(I)$ and $c_0(\kappa)$ are either subspaces of $\ell_\infty(\kappa)$ or trivial on a copy of $c_0(\kappa^+)$; (b) under the hypothesis $[\mathfrak p = \mathfrak c]$, when $K$ is either a suitable Corson compact, a separable Rosenthal compact or a scattered compact of finite height, there is a twisted sum of $C(K)$ with $c_0(\kappa)$ that is not isomorphic to a space of continuous functions; (c) all such twisted sums are Lindenstrauss spaces when $X$ is a Lindenstrauss space and $G$-spaces when $X=C(K)$ with $K$ convex, which shows tat a result of Benyamini is optimal; (d) they are isomorphically polyhedral when $X$ is a polyhedral space with property ($\star$), which solves a problem of Castillo and Papini.

Assuming p=c, we show that for every Eberlein compact space L of weight c there exists a short exact sequence 0→c0→X→C(L)→0, where the Banach space X is not isomorphic to a C(K)-space.

We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact $K_{\mathcal A}$ generated by an almost disjoint family $\mathcal A$ of infinite subsets of $\omega$ -- we present a solution to two problems and develop a previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin's axiom the space $C(K_{\mathcal A})$ is uniquely determined up to isomorphism by the cardinality of $\mathcal A$ whenever $|{\mathcal A}|<{\mathfrak c}$, while there are $2^{\mathfrak c}$ nonisomorphic spaces $C(K_{\mathcal A})$ with $|{\mathcal A}|= {\mathfrak c}$. We also investigate Koszmider's problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of $c_0$ and some $C(K)$ spaces.

We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact KA generated by an almost disjoint family A of infinite subsets of ω — we present a solution to two problems and develop a previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin’s axiom the space C(KA) is uniquely determined up to isomorphism by the cardinality of A whenever |A| < c, while there are 2c nonisomorphic spaces C(KA) with |A| = c. We also investigate Koszmider’s problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of c0 and some C(K) spaces.

In this paper we combine topological and functional analysis methods to prove that a non-locally trivial quasi-linear map defined on a C(K) must be nontrivial on a subspace isomorphic to c0. We conclude the paper with a few examples showing that the result is optimal, and providing an application to the existence of nontrivial twisted sums of ℓ1 and c0.