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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

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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, $\ell_\infty(B(H))$.

  • Consequently, we have a complemented subspace isomorphic to $\ell_\infty(H)$.

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.

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  • $\begingroup$ Thank you very much. Since I'm not a mathematician, I will need some time to correctly digest the informations. You wrote that the range of a linear isometric map with left inverse is complemented, is this a general fact? Furthermore, what do you mean by 1-injectivity and 1-complemented? $\endgroup$ – SepulzioNori Feb 7 '17 at 11:11
  • $\begingroup$ @Ilcapitano, yes, it is a general fact, can you prove it? 1-complemented means complemented by a projection of norm 1. 1-injective means 1-complemented in any Banach space (when embedded isometrically). $\endgroup$ – Tomek Kania Feb 7 '17 at 11:15
  • $\begingroup$ I think that you can prove it using isonetricity to say that the left inverse is bounded, and then using the kernel of the left inverse as the complementing subspace. However, the 1-injectivity property is very interesting, even if I think it is very difficult to prove $\endgroup$ – SepulzioNori Feb 7 '17 at 11:25
  • $\begingroup$ Is there any review reference on this? $\endgroup$ – SepulzioNori Feb 7 '17 at 11:43
  • $\begingroup$ @Ilcapitano, it's an application of the Hahn-Banach theorem + compactness. Here's a proof of a more general case: mathoverflow.net/questions/110461/… $\endgroup$ – Tomek Kania Feb 7 '17 at 13:08

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