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Suppose $I=\bigoplus_nM_{k(n)}(\Bbb C)$,there are many tracial states on $M(I)/I$,where $M(I)$ is a multiplier algebra of $I$. For any free ultrafilter $\omega$ on $\Bbb N$ ,we can construct a tracial state $\tau_{\omega}$ on $M(I)/I$ as follows:$\tau_{\omega}((x_n)+I)=lim_{n \to \omega}tr_{k(n)}(x_n)$,where $tr_{k(n)}$ is a tracial state on $M_{k(n)}(\Bbb C)$.

My question :do there exist other methods to construct tracial states on $M(I)/I$?

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    $\begingroup$ Isn't the multiplier algebra of $I$ just the C*-algebra product (consisting of bounded sequences)? In which case, every pure trace arises from an ultrafilter. (Obviously need pure, as sums of traces are almost never going to arise from ultrafilters.) $\endgroup$ – David Handelman Dec 24 '18 at 19:11
  • $\begingroup$ @David Handelman,do there exist other constructions? $\endgroup$ – mathrookie Dec 25 '18 at 3:02

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