# What is a name for co-Sobczyk Banach spaces?

Definition. Let us define a Banach space $$X$$ to be co-Sobczyk if every linear bounded operator $$T:Z\to c_0$$ defined on a separable subspace $$Z$$ of $$X$$ extends to a bounded operator $$\bar T:X\to c_0$$.

By the classical Sobczyk Theorem, each separable Banach space is co-Sobczyk. But the class of co-Sobczyk spaces includes many non-separable Banach spaces. In particular, a Banach space $$X$$ is co-Sobczyk if each separable subspace of $$X$$ is contained in a complemented separable subspace. So, all classical Banach spaces $$c_0(\Gamma)$$ and $$\ell_p(\Gamma)$$ for $$1\le p<\infty$$, are co-Sobczyk for any set $$\Gamma$$.

I have a strong feeling that co-Sobczyk spaces have been studied in the theory of non-separable Banach spaces, so asking the MO commubnity for a proper reference and an existing terminology (I suspect that co-Sobczyk spaces are called differently).

Nigel Kalton studied a similar but stronger notion: for $$\lambda \geqslant 1$$, he termed a Banach space $$X$$ to have the $$(\lambda, \mathcal{C})$$-extension property, when for any compact space $$K$$ you may find extensions of operators $$T$$ from subspaces of $$X$$ into $$C(K)$$ to operators from $$X$$ to $$C(K)$$ with norm at most $$\lambda \|T\|$$.
It is thus natural to term your spaces as having the $$(\lambda, c_0)$$-separable extension property if you care about the extension constant.
Update: Correa and Tausk call this separable $$c_0$$-extension property.
• Thank you for the answer. Has this term $(\lambda,c_0)$-separable extension property'' been used in any written paper? Because I thoughtalso about the name "$c_0$-coinjective". Jun 6, 2019 at 10:26
• @TarasBanakh, yes, up to a permutation. Here this is called separable $c_0$-extension property. sciencedirect.com/science/article/pii/S0022247X13002540 Jun 6, 2019 at 10:52