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.