arXiv subjects
Publications and source records attributed to Maciej Korpalski.
In this working note we study the one-sided positive Banach-Mazur distance between some pairs of $C(K)$ Banach spaces. Building on methods developed in [4], we solve, in particular, one of the problems posed in [2].
The classical Banach spaces $L_\infty[0,1]$ and $\ell_\infty$ are isomorphic. We present here some lower and upper bounds for their Banach-Mazur distance.
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 $\mathcal{K}$ be a class of compact spaces. How many isomorphism types of Banach spaces $C(K)$ are there, for $K\in \mathcal{K}$? We prove that for any uncountable regular cardinal number $κ$, there exist exactly $2^κ$ isomorphism types of spaces $C(K)$ for compact spaces of weight $κ$. We show that, for the class $\mathcal{L}_{ω_1}$ of separable compact linearly ordered spaces of weight $ω_1$, the answer to the above question depends on additional set-theoretic axioms. In particular, assuming the continuum hypothesis, there are $2^{ω_1}$ isomorphism types of $C(L)$, for $L\in \mathcal{L_{ω_1}}$, and assuming a certain axiom proposed by Baumgartner, there is only one type.
We present several results providing lower bounds for the Banach-Mazur distance \[d_{BM}(C(K), C(L))\] between Banach spaces of continuous functions on compact spaces. The main focus is on the case where $C(L)$ represents the classical Banach space $c$ of convergent sequences. In particular, we obtain generalizations and refinements of recent results from \cite{GP24} and \cite{MP25}. Currently, it seems that one of the most interesting questions is when $K = [0, ω]$ is a convergent sequence with a limit and $L = [0,ω]\times 3$ consists of three convergent sequences. In this case, we obtain \[3.53125 \leq d_{BM}(C([0,ω]\times 3),C[0,ω]) \leq 3.87513\]
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 functions on products are isomorphic or embeddable to each other. In particular, for compact lines $K_1, \dots, K_n, L_1, \dots, L_k$ of uncountable character and $k \neq n$, we claim that Banach spaces $C(\prod_{i=1}^n K_i)$ and $C(\prod_{j=1}^k L_j)$ are not isomorphic.
Assume $\text{MA}(κ)$. We show that for every real chain of size $κ$ in the quotient Boolean algebra $P(ω)/fin$ we can find an almost chain of representatives such that every $n\inω$ oscillates at most three times along the almost chain. This is used to show that for every countable discrete extension of a separable compact line $K$ of weight $κ$ there exists an extension operator $E:C(K)\longrightarrow C(L)$ of norm at most three.
We consider a separable compact line $K$ and its extension $L$ consisting of $K$ and a countable number of isolated points. The main object of study is the existence of a bounded extension operator $E: C(K)\to C(L)$. We show that if such an operator exists then there is one for which $\|E\|$ is an odd natural number. We prove that if the topological weight of $K$ is bigger than or equal to the least cardinality of a set $X \subseteq [0,1]$ that cannot be covered by a sequence of closed sets of measure zero then there is an extension $L$ of $K$ admitting no bounded extension operator.