# An approximation question on projections

Suppose $\{p_i\}_{i=1}^{m}$ are projections in the d by d matrix algebra $A$ over the complex numbers and satisfy the following condition:

$||Id-\sum_{i=1}^m{p_i}||_2<c$, $||p_ip_j||_2<c, \forall i\neq j$

My first question is

Can we find $q_i, i=1,\cdots, m$ projections in $A$ and a function $f=f(c)$ such that

$$Id=\sum_{i=1}^m{q_i},$$ $$||q_i-p_i||_2<f(c),$$ $$q_iq_j=q_jq_i, \forall i\neq j$$ $$f(c)\to 0 ~~as~~ c\to 0$$?

A more general question is:

If we are given positive operators $p_i,i=1,.., m$ in A, such that they almost commute with each other in the sense that $||p_ip_j-p_jp_i||_2<c$, can we find a unitary u(May depend on c)in A, operators $q_j,j=1,.., m$ which commutes with each other and a function $f(c)$ such that $$||p_j-q_j||_2<f(c)$$ for all j. And $f(c)\to 0 ~~as~~ c\to 0$.

Remarks:

1, Clearly, this is true for $m=2$.

2, if it holds in general in question 1, then we could find a unitary $u\in A$ Such that $up_ju^*$ is close to some diagonal matrix for all j.

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For your question about projections, I think it is true even in an arbitrary C$^*$-algebra. Such results are often called "perturbations". You can find a good account on Section III.3 in Davidson's "C$^*$-algebras by example".

I cannot make sense of your second question: $u=I$ and $f(c)=c$ will always work.

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Thanks, I would check that, and I fixed the typoes in second question. – ougao Mar 26 '14 at 14:57
I am afraid the perturbation you mentioned is a different issue from the question I asked, my question asks whether we can find some abelian algebra close to some given almost commuting system, while perturbation says if we already know this, then blabla... – ougao Mar 26 '14 at 15:07

we can take $q_2=p_1\vee p_2-p_1$, and so on, then you check this works.

Set $q_1=p_1, q_i=p_1\vee \cdots\vee p_{i}-p_1\vee \cdots\vee p_{i-1}, 1<i<m$, $p_m=1-p_1\vee \cdots\vee p_{m-1}$.
Then use the fact $p\vee q-p\sim q-p\wedge q$ so they have the same trace to get an upper bound for $||p_i-q_i||_2$.