An Elementary Proof Of The Josefson-Nissenzweig Theorem For Banach Spaces C(KxL)
Jerzy Kakol, Wiesław Śliwa
TL;DR
This paper provides an explicit, elementary combinatorial construction of Josefson-Nissenzweig sequences $\mu_n$ on products $X\times Y$ of Tychonoff spaces with infinite compact subspaces, yielding sharp asymptotic bounds for $\sup_{A\times B}|\mu_n(A\times B)|$ and guaranteeing $\mu_n(f) \to 0$ for all $f\in C(X\times Y)$. The authors exploit these sequences to describe complemented copies of $c_0$ in $C(X\times Y)$, extending classical results for Banach spaces of continuous functions and linking $C_p$ and $C_k$ contexts to complemented $c_0$-sums. The work combines an explicit probabilistic-flavored construction with purely combinatorial arguments, providing a constructive alternative to non-constructive proofs and enabling direct descriptions of complemented $c_0$-type subspaces. It also develops a general theory of complemented JN-sequences, giving criteria and constructions that ensure the existence of complemented copies of $(c_0)_p$ in various locally convex spaces and their products.
Abstract
In [8] probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, have been used to show that for every infinite compact spaces K and L there exists a sequence $(μ_n)$ of normalized signed measures on $K\times L$ with finite supports which converges to $0$ with respect to the weak topology of the dual Banach space $C(K\times L).$ In this paper, we return to this construction, limiting ourselves only to elementary combinatorial calculus. The main efects of this construction are additional information about the measures $μ_n$, this is particularly clearly seen (among the others) in the resulting inequalities $$\frac{1}{2\sqrtπ}\frac{1}{\sqrt{n}} <\sup_{A\times B\subset X\times Y} |μ_n(A\times B)|<\frac{2}{\sqrtπ}\frac{1}{\sqrt{n}},$$ $n\in\mathbb{N}$, with $μ_n(f) \to_n 0$ for every $f\in C(X \times Y);$ where X and Y are arbitrary Tychonoff spaces containing infinite compact subsets, respectively. As an application we explicitly describe for Banach spaces $C(X\times Y)$ some complemented subspaces isomorphic to $c_0$. This result generalizes the classical theorem of Cembranos and Freniche, which states that for every infinite compact spaces K and L, the Banach space $C(K\times L)$ contains a complemented copy of the Banach space $c_0.$