Let $B(H)$ is the C*algebra of all bounded linear operators on Hilbert space $H$. Are there a closed twosided ideal $I$ and a subalgebra $A$ of $B(H)$ such that $B(H)=I \oplus A$ (direct sum I and A)?

$K(H)$, the compact operators on $H$, is the only proper closed ideal in $B(H)$ when $H$ is a separable infinite dimensional Hilbert space, and $K(H)$ is not complemented in $B(H)$ (because if it were the diagonal compact operators would be complemented in the diagonal bounded operators, which is to say that $c_0$ would be complemented in $\ell_\infty$). 


Let me complement Bill's answer. It is basically the same idea. Gramsch and Luft described the lattice of closed ideals of $B(H)$ where is $H$ a nonseparable Hilbert space.
They proved that for a nonseparable Hilbert space $H$ with density character ${\rm dens}\; H$ (this is the minimal cardinality of a dense subset) all the closed ideals of $B(H)$ are of the form $\{0\}$, $K(H)$ (the compact operators) and $$K_\lambda(H) = \{T\in B(H)\colon \mbox{dens }T[H]< \lambda\}$$ where $\lambda\leqslant\kappa^+$. Certainly, $$B(H) = K_{({\rm dens}\; H)^+}(H).$$ Hence, we are interested in the case only where $\lambda\leqslant {\rm dens}\; H$. Fix an orthonormal basis for $H$ and identify operators on $H$ with matrices with respect to this basis. Consider the Banach space $\ell_\infty^\lambda({\rm dens}\; H)$ of all bounded complexvalued functions on the cardinal ${\rm dens}\; H$ with at most $<\lambda$ nonzero entries, endowed with the supnorm. If you intersect the ideal $K_\lambda(H)$ with the diagonal masa (that is, $\ell_\infty({\rm dens}\; H)$) then you'll get precisely $\ell_\infty^\lambda({\rm dens}\; H)$. The diagonal masa is complemented because $\ell_\infty({\rm dens}\; H)$ is an injective Banach space. Consequently, if $K_\lambda(H)$ was complemented in $B(H)$ then $\ell_\infty^\lambda({\rm dens}\; H)$ would be complemented in $\ell_\infty({\rm dens}\; H)$. This is however a contradiction because $\ell_\infty^\lambda({\rm dens}\; H)$ is not injective (see this paper for a discussion of this space and its injectivitylike properties). Proof of noninjectivity of $\ell_\infty^\lambda(\kappa)$: Assume $\ell_\infty^\lambda(\kappa)$ is injective. Manifestly, it contains $c_0(\kappa)$. Then by Rosenthal's theorem
it would contain a copy of $\ell_\infty(\kappa)$ and by Pełczyński's decomposition method, it would be isomorphic to $\ell_\infty(\kappa)$; a contradiction. EDIT: As Bill pointed out we have to be careful why this is indeed impossible. Under GCH this is evident but life would be too easy with GCH. Actually, to see this is a result in ZFC, one has to tweak the argument in Paragraph d) on page 12 of the hyperlinked paper by replacing $\mathbb{N}$ with $\lambda$ and $\aleph_1$ with $\lambda$. Then the whole proof carries over. (Frankly, I learnt it from one of the authors of this paper some time ago and presumed that it is wellknown. My apologies for that.) 

