Witnessing the lack of the Grothendieck property in C(K)-spaces via convergent sequences

Abstract

Let K be a compact Hausdorff space and let C(K) be the space of all scalar-valued, continuous functions on K. We show that C(K) is an $$\ell _1(K)$$ -Grothendieck space but not a Grothendieck space exactly when the spaces $$C_p(K)$$ and $$C_p(K \oplus {\mathbb {N}}^{\#})$$ are not linearly isomorphic, where $${\mathbb {N}}^{\#}$$ is the one-point compactificiation of the discrete space of natural numbers. (That is, if C(K) contains a complemented copy of $$c_0$$ , then C(K) fails to be $$\ell _1(K)$$ -Grothendieck if and only if the topologies of pointwise convergence in $$C_p(K)$$ and $$C_p(K \oplus {\mathbb {N}}^{\#})$$ are linearly isomorphic.) Moreover, for infinite compact spaces K and L, there exists a compact space G that has a non-trivial convergent sequence and such that $$C_{p}(K\times L)$$ and $$C_{p}(G)$$ are linearly isomorphic. This extends a remarkable theorem of Cembranos and Freniche. Some examples illustrating the above results are provided.