This chapter studies a weak variant of the Grothendieck property for Banach spaces \(C(K)\) , called the \(\ell _1\) -Grothendieck property. We characterize the \(\ell _1\) -Grothendieck property in terms of operators onto the space \(c_0\) (this time endowed with the pointwise topology) and in terms of sequences of (finitely supported) Radon measures related to Josefson–Nissenzweig’s theorem. We present a construction of a separable compact space K such that \(C(K)\) has the \(\ell _1\) -Grothendieck property but it does not have the Grothendieck property. Generalizing the classical result of Cembranos and Freniche, stating that the Banach space \(C(K\times L)\) does not have the Grothendieck property for any infinite compact spaces K and L, we show that spaces of the form \(C(K\times L)\) have never even the \(\ell _1\) -Grothendieck property—the proof is constructive and based on tools from probability theory. We also study the \(\ell _1\) -Grothendieck property for spaces \(C(K)\) where K is the limit of an inverse system based on simple extensions of totally disconnected compact spaces.