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Let $\mathcal{A}$ be a sub-algebra of $B(H)$ such that $\mathcal{A}$ generated by all its idempotents and $\mathcal{A}$ is closed under weak operator topology. Suppose that there exist idempotents $P_1$ and $P_2$ in $\mathcal{A}$ such that $P_1P_2\neq P_2P_1$. Can we say there exists idempotent $P_3 \in \mathcal{A}$ such that $P_1P_3\neq P_3P_1$ and $P_3P_2\neq P_2P_3$?

If yes, how can prove it? If no, is there any example of a sub-algebra to reject it?

Please help me to solve this question.

Your prompt reply will be greatly appreciated.

Thanks in advance.

joker
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    When you say $\mathcal{A}$ is "generated" by it idempotents, what do you mean? Generated with respect to algebraic operations and weak operator limits? Or just generated as an algebra? Or something else? – Eric Wofsey Nov 19 '15 at 03:18
  • respect to weak operator limit. – joker Nov 19 '15 at 14:39

1 Answers1

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It is proved in [3] (see below) that :

Theorem. Let $H$ be a Hilbert space. There exist three projections which generate $B(H)$. The number three cannot be reduced if $H$ has dimensionality 3 or greater.

Actually, the discussion of generators of certain von Neumann algebras has a long history. As for some references that are related to your question, you may take a look at these:

References:

1- On strong generation of $B(H)$ by two commutative C*-algebras(1997)

2- Generators of certain von Neumann algebras(1967)

3- Generators of the ring of bounded operators (by C. Davis)

ABB
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