Is $\mathcal{B}(H)$ complemented in $\ell_\infty(I, H)$

Question

Let $H$ be an infinite diensional Hilbert space. Consider unit ball of $H$ as index set, denote it by $I$, then we have an isometric embedding $$ j:\mathcal{B}(H)\to\ell_\infty(I,H):T\mapsto(T(i))_{i\in I} $$ So we have a copy of $\mathcal{B}(H)$ in $\ell_\infty(I,H)$. Is this copy of $\mathcal{B}(H)$ complemented in $\ell_\infty(I,H)$?

I believe it is not, but I can't prove.

Answer 1

The answer to your question is no. No copy of $B(H)$ is complemented in $\ell_\infty(I, H)$. This is because the latter space is a Banach lattice and $B(H)$ lacks the local unconditional structure.

Y. Gordon and D. R. Lewis, Absolutely summing operators and local unconditional structures, Acta Mathematica, 133 (1974), 27–48.

See also

Y. Gordon and D. R. Lewis, Banach ideals on Hilbert spaces, Studia Mathematica, 54 (1975), 161–172.

A Banach space $X$ has the local unconditional structure if and only if $X^{**}$ is complemented in a Banach lattice. (Consult Theorem 17.5 in

J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge University Press, 1995.

for more details.)