Extension of operators from subspaces of $c_ 0(\Gamma )$ into $C(K)$ spaces

Abstract:

It is shown that for every $\varepsilon > 0$, every bounded linear operator $T$ from a subspace $X$ of ${c_0}\left ( \Gamma \right )$ into a $C\left ( K \right )$ space has an extension ${\mathbf {T}}$ from ${c_0}\left ( \Gamma \right )$ into the $C\left ( K \right )$ space such that $\left \| {\mathbf {T}} \right \| \leq \left ( {1 + \varepsilon } \right )\left \| T \right \|$. Even when $\Gamma$ is countable, $T$ is compact, and $X$ has codimension 1 in ${c_0}$, the "$\varepsilon$" cannot be replaced by 0. These results answer questions raised by J. Lindenstrauss and A. Pełczynski in 1971.