I am currently interested in the problem of complemented subspaces in a Banach space. Specifically, I am interested in understanding the problem when the Banach space in question is the space $\mathcal{B}(\mathcal{H})$ of bounded linear operators on an infinite-dimensional, separable Hilbert space.
Since I am not an expert, I would like to know if there are some "well-known" results and/or some references on the problem.
Thank You
Here are some known results:
Trivially, $B(H)$ itself is an example.
Moreover, $H$ is complemented in $B(H)$. Indeed, fix a norm-one vector $y\in H$. Then the map $x\mapsto x\otimes y$ is linear isometric and has left-inverse. Therefore, its range is complemented.
Now, fix an orthonormal basis of $H$ and consider all multiplication operators associated to it. This family is isometric to $\ell_\infty$, hence by 1-injectivity of $\ell_\infty$, it is 1-complemented.
By combining the above, we also get $\ell_\infty \oplus H$ as a complemented subspace.
Using Pełczyński's decomposition method, one can prove that $B(H)$ is isomorphic as a Banach space to its $\ell_\infty$-sum, $\ell_\infty(B(H))$.
Lindenstrauss and Haagerup observed that $B(H)$ is also Banach-space isomorphic to $(\bigoplus_n M_n)_{\ell_\infty}$.
Christensen and Sinclair proved that if $M$ is an injective (infinite-dimensional) factor in $B(H)$, then it is isomorphic to $B(H)$ as a Banach space, so we won't get anything new looking at obvious complemented von Neumann subalgebras.
E. Christensen and A. M. Sinclair, Completely bounded isomorphisms of injective von Neumann algebras, Proc. Edinburgh Math. Soc. (2) 32 (1989), 317-327.
Blower proved that $B(H)$ is a primary Banach space.
G. Blower, The Banach space $B(\ell_2)$ is primary, Bull. London Math. Soc., 22 (1990), 76–182.
It seems to me that this everything that is known on this matter.