5 Rollback to Revision 2 - reverted deletions by OP that rendered qn meaningless for new readers

Elements

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!

4 deleted 212 characters in body; edited tags; edited title

# simultaneously Approximated problems.problems.(closed)

If $RR(A)=0$. If we have two self-adjoint elements $T, S$ in $A$ with the same spectrum(may be infinite), Now I know that we can also find two self-adjoint elements in $A$ with the same finte spectrum

Elements approximated $T, S$ in normrespectively.

3 deleted 214 characters in body; edited title

We know that a $C^*$-algebra $A$ has real rank zero iff every self-adjoint element of
If $A$ can be approximated in norm by self-adjoint element with finite spectra. My question is:RR(A)=0$. If we have two self-adjoint elements$T, S$in$A$with the same spectrum(may be infinite), Now can I know that we can also find two self-adjoint elements in$A$with the same finte spectrum approximated$T, S$in norm ( within the same$\epsilon>0\$)?