Operators on $\ell_\infty(\Gamma)$ and almost disjoint families of subsets

Question

It is well known that given an operator $T:\ell_\infty\to\ell_\infty$ such that $Tx=0$ for each $x\in c_0$ there exists an infinite subset $M$ of the positive integers so that $Tx=0$ for each $x\in \ell_\infty(M)$. See Theorem 2.5.4 in the book on Banach spaces by Albiac and Kalton.

I am interested in a version of this result for $T:\ell_\infty(\Gamma)\to \ell_\infty(\Gamma)$ when $\Gamma$ is uncountable.

From $Tx=0$ for each $x\in c_0(\Gamma)$ I would like to derive the existence of a subset $\Gamma'$ of $\Gamma$ so that $Tx=0$ for each $x\in \ell_\infty(\Gamma')$ with card($\Gamma'$)=card($\Gamma$), or at least with $\Gamma'$ uncountable.

($Q_1$) Is this result true?

To extend the argument in the proof of the countable case, I need the existence of a family $(\Gamma_i)_{i\in I}$ with card$(I)>$ card$(\Gamma)$ of almost disjoint subsets ($\Gamma_i\cap\Gamma_j$ finite for $i\neq j$) of $\Gamma$ with card$(\Gamma_i)=$ card$(\Gamma)$ for each $i$, or at least with all the $\Gamma_i$ uncountable.

($Q_2$) Is it possible to find such a family, at least with some restrictions on card($\Gamma$)?

Answer 1

You cannot prove the statement about a.d. families from Q1 in ZFC only. For instance, it does not hold if $\mathfrak{c}=\omega_1$ and $2^{\omega_1} = \omega_2$. For details see

J.E. Baumgartner, Almost-disjoint sets, the dense set problem and the partition calculus. Ann. Math. Logic 9 (1976), 401–439.

(especially Section 6).

Regarding (Q2) sometimes this is possible. For instance, for cardinals with countable cofinality (see Baumgartner's article I mentioned above).

Btw. I am sure you are familiar with Proposition 1.2 in

H. P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory. Studia Mathematica 37 (1970) 13-36.

Answer 2

Here's a simple reason why you need some assumptions beyond ZFC (failure of GCH) to get such a family on an uncountable $\Gamma$.

Claim: If $2^{< \kappa} = \kappa \geq \omega_1$, then there is no family $\mathcal{A} = \{ A_i : i < \kappa^{+} \}$ of $\kappa$-sized subsets of $\kappa$ whose pairwise intersections are finite (Such families are called strongly almost disjoint).

Proof: By thinning down $\mathcal{A}$, we can assume that for some $\lambda < \kappa$, $A_i \cap \lambda$ is infinite for each $i < \kappa^{+}$. But this means $2^{\lambda} \geq \kappa^{+}$.