Bounded approximation properties in terms of $C[0,1]$ - Lima, Åsvald; Lima, Vegard; Oja, Eve (MATHEMATICA SCANDINAVICA, 2012)
Let $X$ be a Banach space and let $\mathcal I$ be the Banach operator ideal of integral operators. We prove that $X$ has the $λ$-bounded approximation property ($λ$-BAP) if and only if for every operator $T\in \mathcal I(X,C[0,1]^*)$ there exists a net $(S_α)$ of finite-rank operators on $X$ such that $S_α→ I_X$ pointwise and 26767 \limsup_α\|TS_α\|_{\mathcal I}≤λ\|T\|_{\mathcal I}. 26767 We also prove that replacing $\mathcal I$ by the ideal $\mathcal N$ of nuclear operators yields a condition which is equivalent to the weak $λ$-BAP.