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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 $\mathbb{N}$?

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    Please do not delete questions after they received extensive answers.2012-04-12
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    @t.b. To be fair, that answer was extensive largely because it answered a different question to the one asked. The original question had a two-dimensional counterexample2012-04-12
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    @Yemon: the present question was deleted, too. [See here](http://math.stackexchange.com/posts/130599/revisions).2012-04-12

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