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We know that a $C^*$-algebra $A$ has real rank zero iff every self-adjoint element of $A$ can be approximated in norm by self-adjoint element with finite spectra. My question is:

If we have two self-adjoint elements $T, S$ in $A$ with the same spectrum(may be infinite), Now can we also find two self-adjoint elements in $A$ with the same finte spectrum approximated $T, S$ in norm( within the same $\epsilon>0$)?

Hope some help or suggestion, thanks!

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Do you want any relation between $S$ and $T$ and between the two approximating elements? Are you referring to the joint spectrum of commuting elements? If not, then there is no difficulty approximating two elements instead of one, so I suspect I am not following you. –  Terry Loring May 8 '12 at 15:45
    
Thanks, I can prove this. –  Aviv May 9 '12 at 9:36
    
Do not edit your questions so that they remove meaningful information. I am reverting this to your previous version. –  Yemon Choi May 20 '12 at 12:08
    
If you have since solved the problem, add this extra information as an update. Please do not delete the record of what you originally asked. –  Yemon Choi May 20 '12 at 12:11
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1 Answer

Yes, you can jiggle the two approximating operators so that they have the same finite spectrum. Use functional calculus.

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Thanks, Can you say any more, or some reference? –  Aviv May 8 '12 at 16:14
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You need to learn about functional calculus. If you didn't really care that the two approximating operators have the same spectrum (as I gather from your response to Terry's answer) then you don't need functional calculus, you need to brush up on epsilon-delta arguments. –  Nik Weaver May 8 '12 at 18:09
    
Thanks, I can prove this. –  Aviv May 9 '12 at 9:36
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