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Let $X$ be a locally compact Hausdorff space, such that $C_0(X)$ is an injective Banach space, i.e. a $\mathfrak{P}_\lambda$ space for some $\lambda\geq 1$. Is it true that $X$ is compact? If additi …

Does every contractively complemented subspace of $c_0(I)$ is isometric to $c_0(J)$ for some $J\subseteq I$? May be someone has a counterexample?

Can someone give me an example of a Banach space $X$ and contractive projection $P\in\mathcal{B}(X)$ such that $\ker P$ is not a range of any contractive projection $Q\in\mathcal{B}(X)$?

I was looking for an article mimicing result of Pelczynski for $\ell_p$. I have found this one Rodriguez-Salinas, B. (1994). On the Complemented Subspaces of $c_0(I)$ and $\ell_p(I)$ for $1 < p < \i …

Let $n\in\mathbb{N}$ and $p,q\in(1,+\infty)$ with $p^{-1}+q^{-1}=1$. Consider isometric embedding between $\mathbb{C}$-Banach spaces $$ \rho:\ell_p^n\to\ell_\infty(S, \ell_1^n),x\mapsto(f\cdot x)_{f\i …

Let $\ell_{1,0}(S)=(c_{0,0}(S),\Vert\cdot\Vert_1)$ be a space of functions on a set $S$ with finite support, endowed with $\ell_1$ norm. Could you answer the at least one of the following questions …

A short variant of Pitt's theorem is the followig: for $1\leq p < r <\infty$ holds $$ \mathcal{B}(\ell_r(\mathbb{N}),\ell_p(\mathbb{N}))=\mathcal{K}(\ell_r(\mathbb{N}),\ell_p(\mathbb{N})) $$ Now let $ …

I would like to know a simple proof of isometric injectivity of $L_\infty$. The proof I've found in Topics in Banach space theory. F. Albiac, N. Kalton uses two deep result. $L_\infty$ as commutati …

Could someone give a reference or construct an example of closed subspace of $Y\subset L_1[0,1]$ such that $\operatorname{dist}(x,Y)$ is not attained of for any $x\notin Y$. I read somewhere that $Y$ …

I would like to know if there exist an explicit decription of ultrapowers of $c_0(\ell_1)$ and $\ell_1(c_0)$. The best option would be -- "they are complemented subspaces of $C(K, L_1(\mu))$ and $L_1( …

I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case …

I'm looking for articles describing or proving nonexistence of isometric embeddings of $m$-dimensional space $\ell_q^m$ into $L_p$ and $\ell_p$ for $q,p\in[1,+\infty]$. Since $\ell_q^m$ is finite di …

This fact is an easy consequence of results of the paper Classes of Banach spaces with unique isometric preduals. by Leon Brown and Takashi Ito, but it looks like an overkill. Does anyone know a simpl …

The scale of Schatten-von Neumann classes is noncommutatitve analog of classical $\ell_p$-spaces. A lot of researchers devoted their lives to study Banach geometric structure of these spaces. Differen …

It is an old question if every injective Banach space is isomorphic as Banach space to $C(X)$-space. I would like to know if the weakened module version of this question is answered. More precisely: …