We show for $2\le p<\infty$ and subspaces $X$ of quotients of $L_p$ with a $1$-unconditional finite-dimensional Schauder decomposition that $K(X,{\ell }_p)$ is an $M$-ideal in $L(X,{\ell }_p)$.

We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces $X$ and $Y$ the subspace of all compact operators $𝒦(X,Y)$ is an $M(r_1{r}_2,s_1{s}_2)$-ideal in the space of all continuous linear operators $ℒ(X,Y)$ whenever $𝒦(X,X)$ and $𝒦(Y,Y)$ are $M(r_1,s_1)$- and $M(r_2,s_2)$-ideals in $ℒ(X,X)$ and $ℒ(Y,Y)$, respectively, with $r_1+s_1/2>1$ and $r_2+s_2/2>1$. We also prove that the $M(r,s)$-ideal $𝒦(X,Y)$ in $ℒ(X,Y)$ is separably determined. Among others, our results complete and improve some well-known results...

This work introduces the concept of an M-complete approximate identity (M-cai) for a given operator subspace X of an operator space Y. M-cai’s generalize central approximate identities in ideals in C*-algebras, for it is proved that if X admits an M-cai in Y, then X is a complete M-ideal in Y. It is proved, using ’special’ M-cai’s, that if J is a nuclear ideal in a C*-algebra A, then J is completely complemented in Y for any (isomorphically) locally reflexive operator space Y with J ⊂ Y ⊂ A and...

We investigate several aspects of almost 1-unconditionality. We characterize the metric unconditional approximation property (umap) in terms of “block unconditionality”. Then we focus on translation invariant subspaces $L_E^p\left(\right)$ and $C_E\left(\right)$ of functions on the circle and express block unconditionality as arithmetical conditions on E. Our work shows that the spaces $p_E\left(\right)$, p an even integer, have a singular behaviour from the almost isometric point of view: property (umap) does not interpolate between $L_E^p\left(\right)$ and $L_E^{p+2}\left(\right)$. These...

We characterize Banach lattices under which each b-weakly compact (resp. b-AM-compact, strong type (B)) operator is L-weakly compact (resp. M-weakly compact).