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Given a compact Hausdorff space \(K\) the geometry of the Banach space \(C(K\times K)\), besides
being interesting by itself, is additionally important
because \(C(K\times K)\) is isomorphic to the injective tensor product \(C(K)\hat\otimes_\varepsilon C(K)\)
and hence relevant to the investigations of the properties of these tensor products
\(X\hat\otimes_\varepsilon Y\) in terms of the properties of \(X\) and \(Y\).
It is well known, by a surprising and celebrated result of P. Cembranos and F. Freniche from 1984, that if \(C(K)\) contains a copy of \(c_0\) (i.e., C(K)
is infinite dimensional), then \(C(K\times K)\) always contains a complemented copy of \(c_0\).
A nonseparable version of this result has been recently obtained
by E. M. Galego and J. Hagler who proved that it is relatively consistent with ZFC that if \(C(K)\) has density \(\omega_1\) and \(C(K)\) has a copy of \(c_0(\omega_1)\), then \(C(K\times K)\) has a complemented copy \(c_0(\omega_1)\).
Their proof relies on Todorcevic's analysis of nonseparable Banach spaces under the assumption of
an additional set-theoretic axiom known as Martin's Maximum.
In this paper, we show that this nonseparable version of Cembranos' and Freniche's theorem indeed is sensitive to
additional set-theoretic assumptions. We prove that
under the presence of the Ostaszewski's combinatorial principle \(\clubsuit\), there is a scattered compact space \(K\) such that \(C(K)\) has density \(\omega_1\), \(C(K)\) contains a copy of \(c_0(\omega_1)\), however, \(C(K\times K)\) contains no complemented copy of \(c_0(\omega_1)\).
We also show show that for most known until now constructions of scattered compacta \(K\) similar to the one
we consider, that is, with a copy of \(c_0(\omega_1)\) but without any
complemented copy, the space \(C(K\times K)\) contains
a complemented copy of \(c_0(\omega_1)\).
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