Related papers: On isomorphisms of Banach spaces of continuous fun…

We prove that there is 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 investigate the following general problem, closely related to the problem of isomorphic classification of Banach spaces $C(K)$ of continuous real-valued functions on a compact space $K$, equipped with the supremum norm: Let $\mathcal{K}$…

We present a necessary condition for a pair of $\mathcal{C}(K)$ spaces to be isomorphic in terms of topological properties of Cantor-Bendixon derivatives of $K$. This in particular gives a completely new information about the perfect…

We present a connected version of the compact $L$-space constructed by Kenneth Kunen under CH. We show that this provides a Corson compact space $K$ such that the Banach space $C(K)$ is isomorphic to no space of continuous function on a…

We investigate isomorphic embeddings $T: C(K)\to C(L)$ between Banach spaces of continuous functions. We show that if such an embedding $T$ is a positive operator then $K$ is an image of $L$ under a upper semicontinuous set-function having…

It is proved that for every compact metric space $K$ there exists a Banach space $X$ whose Calkin algebra $\mathcal{L}(X)/\mathcal{K}(X)$ is homomorphically isometric to $C(K)$. This is achieved by appropriately modifying the…

We show the existence of a compact metric space $K$ such that whenever $K$ embeds isometrically into a Banach space $Y$, then any separable Banach space is linearly isometric to a subspace of $Y$. We also address the following related…

We study the class of compact spaces that appear as structure spaces of separable Banach lattices. In other words, we analyze what $C(K)$ spaces appear as principal ideals of separable Banach lattices. Among other things, it is shown that…

For a metric compact space $L$ and a Banach space $E$, we provide a characterization of the complementability of the Banach space $\mathcal{C}(L)$ of continuous functions on $L$ inside $E$ in terms of the existence of a certain tree in the…

We study Banach spaces $C(K)$ of real-valued continuous functions from the finite product of compact lines. It turns out that the topological character of these compact lines can be used to distinguish whether two spaces of continuous…

Supplementing and expanding classical results, for compact spaces $K$ and $L$, $L$ metric, and their Banach spaces $\mathcal{C}(L)$ and $\mathcal{C}(K)$ of continuous real-valued functions, we provide several characterizations of the…

We give a complete description of homomorphisms between two Banach algebras of Lipschitz functions with values in continuous functions. We also characterize the compactness of those homomorphisms.

We investigate the problem of classifying the Banach spaces $\mathrm{Lip}_0(C(K))$ for Hausdorff compacta $K$. In particular, sufficient conditions are established for a space $\mathrm{Lip}_0(C(K))$ to be isomorphic to…

We construct a totally disconnected compact Hausdorff space N which has clopen subsets M included in L included in N such that N is homeomorphic to M and hence C(N) is isometric as a Banach space to C(M) but C(N) is not isomorphic to C(L).…

A classical result of Cembranos and Freniche states that the C(K, X) spaces contains a complemented copy of c_0 whenever K is an infinite compact Hausdorff space and X is an infinite dimensional Banach space. This paper takes this result as…

We study the classification of spaces of continuous functions $C(K)$ under positive linear maps. For infinite countable compacta, we show that whenever $C(K)$ and $C(L)$ are isomorphic, there exists an isomorphism $T:C(K)\to C(L)$…

The isometric universality of the spaces $C(K)$ for $K$ a non scattered Hausdorff compact does not take into account the ``quality'' of the representation. Indeed, the existence of an isometric copy of a separable Banach space $X$ into…

We prove that if X is an infinite-dimensional Banach space with C^p smooth partitions of unity, then X and X\K are C^p diffeomorphic, for every weakly compact subset K of X.

We prove that for infinite, countable, compact, Hausdorff spaces $K,L$, $C(K)\widehat{\otimes}_\pi C(L)$ is isomorphic to exactly one of the spaces $C(\omega^{\omega^\xi})\widehat{\otimes}_\pi C(\omega^{\omega^\zeta})$, $0\leqslant…

Let $K_1$, $K_2$ be compact Hausdorff spaces and $E_1, E_2$ be Banach spaces not containing a copy of $c_0$. We establish lower estimates of the Banach-Mazur distance between the spaces of continuous functions $\mathcal{C}(K_1, E_1)$ and…