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 µ n : n ∈ ω on K which is weakly* convergent to the zero measure on K and such that µ n = 1 for every n ∈ ω. We call such a sequence of measures 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 ℓ 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 × L always admits a finitely supported Josefson-Nissenzweig sequence. As a corollary we obtain a constructive proof that the space C p (K × 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).

On complemented copies of the space $c_0$ in spaces $C_p(X,E)$

“…Assume that C p (X, C p (X)) contains a complemented copy of C p (X × X). Since X is compact, by [21,Corollary 11.5], C p (X × X) (and hence C p (X, C p (X))) contains a complemented copy of (c 0 ) p . By Corollary 5.3, C p (X) also contains a complemented copy of (c 0 ) p .…”

“…If a sequence (ϕ n ) n∈ω of functionals on a given space C p (X) has the properties as in the above theorem, then it will be called a JN-sequence on X. The existence of JN-sequences on arbitrary spaces was intensively studied in [4], [21] and [22], where several classes of Tychonoff spaces were recognized to (not) admit such sequences (e.g. it appears that for every infinite compact spaces X and Y their product admits a JN-sequence, see [21,Theorem 11.3]).…”

“…The existence of JN-sequences on arbitrary spaces was intensively studied in [4], [21] and [22], where several classes of Tychonoff spaces were recognized to (not) admit such sequences (e.g. it appears that for every infinite compact spaces X and Y their product admits a JN-sequence, see [21,Theorem 11.3]). Note that if X is pseudocompact and C p (X) has the JNP, then every JN-sequence on X satisfies already the conclusion of the Josefson-Nissenzweig theorem for the Banach space C(X) (by the virtue of the Riesz representation of continuous functionals on C(X)).…”

“…By [21,Corollary 11.5], for any infinite compact space X the space C p (X × X) contains a complemented copy of (c 0 ) p . Since C p (βω) does not have any complemented copies of (c 0 ) p , Theorem 5.2 yields the following example.…”

“…(3) If C p (Y ) has the JNP for an infinite Tychonoff space Y , then C p (Y ) is not barrelled (by the Buchwalter-Schmets theorem [9] and [21,Lemma 4.11]), so it does not satisfy the assumptions of Theorem 6.2, even though the space C p (X, C p (Y )) still has the JNP for every space X (by Theorem 5.2).…”