The Josefson--Nissenzweig theorem, Grothendieck property, and finitely supported measures on compact spaces

Abstract

The celebrated Josefson-Nissenzweig theorem implies that for a Banach space $C(K)$ of continuous real-valued functions on an infinite compact space $K$ there exists a sequence of Radon measures $\langleμ_n\colon\ n\inω\rangle$ on $K$ which is weakly* convergent to the zero measure on $K$ and such that $\big\|μ_n\big\|=1$ for every $n\inω$. We call such a sequence of measures \textit{a Josefson-Nissenzweig sequence}. In this paper we study the situation when the space $K$ admits a Josefson-Nissenzweig sequence of measures such that its every element has finite support. We prove among the others that $K$ admits such a Josefson-Nissenzweig sequence if and only if $C(K)$ does not have the Grothendieck property restricted to functionals from the space $\ell_1(K)$. We also investigate miscellaneous analytic and topological properties of finitely supported Josefson-Nissenzweig sequences on general Tychonoff spaces. We prove that various properties of compact spaces guarantee the existence of finitely supported Josefson-Nissenzweig sequences. One such property is, e.g., that a compact space can be represented as the limit of an inverse system of compact spaces based on simple extensions. An immediate consequence of this result is that many classical consistent examples of Efimov spaces, i.e. spaces being counterexamples to the famous Efimov problem, admit such sequences of measures. Similarly, we show that if $K$ and $L$ are infinite compact spaces, then their product $K\times L$ always admits a finitely supported Josefson--Nissenzweig sequence. As a corollary we obtain a constructive proof that the space $C_p(K\times L)$ contains a complemented copy of the space $c_0$ endowed with the pointwise topology--this generalizes results of Cembranos and Freniche. Finally, we provide a direct proof of the Josefson-Nissenzweig theorem for the case of Banach spaces $C(K)$.