Suppose $X$ is a Banach space with an unconditional basis $(e_n)$. Then, one may easily define a Boolean algebra of projections in $\mathcal{L}(E)$ which is isomorphic to the power-set of $\mathbb{N}$ (by a Boolean algebra of projections I understand a family of bounded idempotents on a Banach space which is a Boolean algebra under operations $P\wedge Q = PQ$ and $P\vee Q = P+Q-PQ$, zero-element equal to zero operator and unit equal to the identity on $X$).
What are sufficient conditions for a Banach space to have a complete Boolean algebra of projections? Is there a separable Banach space $X$ without unconditional basis with some Boolean algebra of projections isomorphic to the power-set of $\mathbb{N}$?